No­ta­tions

The be­low are the sim­plest pos­si­ble de­scrip­tions of var­i­ous sym­bols, just to help you keep read­ing if you do not re­mem­ber/know what they stand for. Don't cite them on a math test and then blame this book for your grade.

Watch it. There are so many ad hoc us­ages of sym­bols, some will have been over­looked here. Al­ways use com­mon sense first in guess­ing what a sym­bol means in a given con­text.

The quoted val­ues of phys­i­cal con­stants are usu­ally taken from NIST CO­DATA in 2012 or later. The fi­nal digit of the listed value is nor­mally doubt­ful. (It cor­re­sponds to the first nonzero digit of the stan­dard de­vi­a­tion). Num­bers end­ing in triple dots are ex­act and could be writ­ten down to more dig­its than listed if needed.

$\setbox 0=\hbox{$\cdot$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

A dot might in­di­cate

• A dot prod­uct be­tween vec­tors, if in be­tween them.
• A time de­riv­a­tive of a quan­tity, if on top of it.

And also many more pro­saic things (punc­tu­a­tion signs, dec­i­mal points, ...).

$\setbox 0=\hbox{$\times$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Mul­ti­pli­ca­tion sym­bol. May in­di­cate:

• An em­phatic mul­ti­pli­ca­tion.
• Mul­ti­pli­ca­tion con­tin­ued on the next line or from the pre­vi­ous line.
• A vec­to­r­ial prod­uct be­tween vec­tors. In in­dex no­ta­tion, the $i$-th com­po­nent of $\vec{v}\times\vec{w}$ equals
  
  $$(\vec{v}\times\vec{w})_i= v_{\overline{\imath}}w_{\overlin... ...th}}}- v_{\overline{\overline{\imath}}}w_{\overline{\imath}}$$
  
  
  where ${\overline{\imath}}$ is the in­dex fol­low­ing $i$ in the se­quence 123123..., and ${\overline{\overline{\imath}}}$ the one pre­ced­ing it (or sec­ond fol­low­ing). Al­ter­na­tively, eval­u­ate the de­ter­mi­nant
  
  $$\vec{v}\times\vec{w} = \left\vert \begin{array}{ccc} ... ...t k}\ v_x&v_y&v_z\ w_x&w_y&w_z \end{array} \right\vert$$
  
  

$\setbox 0=\hbox{$!$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Might be used to in­di­cate a fac­to­r­ial. Ex­am­ple: 5! $\vphantom0\raisebox{1.5pt}{$=$}$ 1 $\times$ 2 $\times$ 3 $\times$ 4 $\times$ 5 $\vphantom0\raisebox{1.5pt}{$=$}$ 120.

The func­tion that gen­er­al­izes $n!$ to non­in­te­ger val­ues of $n$ is called the gamma func­tion; $n!$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\Gamma(n+1)$. The gamma func­tion gen­er­al­iza­tion is due to, who else, Euler. (How­ever, the fact that $n!$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\Gamma(n+1)$ in­stead of $n!$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\Gamma(n)$ is due to the id­iocy of Le­gendre.) In Le­gendre-re­sis­tant no­ta­tion,

$$n!=\int_0^{\infty}t^ne^{-t}{ \rm d}{t}$$

Straight­for­ward in­te­gra­tion shows that 0! is 1 as it should, and in­te­gra­tion by parts shows that $(n+1)!$ $\vphantom0\raisebox{1.5pt}{$=$}$ $(n+1)n!$, which en­sures that the in­te­gral also pro­duces the cor­rect value of $n!$ for any higher in­te­ger value of $n$ than 0. The in­te­gral, how­ever, ex­ists for any real value of $n$ above $\vphantom{0}\raisebox{1.5pt}{$-$}$1, not just in­te­gers. The val­ues of the in­te­gral are al­ways pos­i­tive, tend­ing to pos­i­tive in­fin­ity for both $n\downarrow-1$, (be­cause the in­te­gral then blows up at small val­ues of $t$), and for $n\uparrow\infty$, (be­cause the in­te­gral then blows up at medium-large val­ues of $t$). In par­tic­u­lar, Stir­ling’s for­mula says that for large pos­i­tive $n$, $n!$ can be ap­prox­i­mated as

$$n! \sim \sqrt{2\pi n} n^n e^{-n} \left[1 + \ldots\right]$$

where the value in­di­cated by the dots be­comes neg­li­gi­bly small for large $n$. The func­tion $n!$ can be ex­tended fur­ther to any com­plex value of $n$, ex­cept the neg­a­tive in­te­ger val­ues of $n$, where $n!$ is in­fi­nite, but is then no longer pos­i­tive. Euler’s in­te­gral can be done for $n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-\frac12$ by mak­ing the change of vari­ables $\sqrt{t}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $u$, pro­duc­ing the in­te­gral $\int_0^\infty2e^{-u^2}{ \rm d}{u}$, or $\int_{-\infty}^{\infty}e^{-u^2}{ \rm d}{u}$, which equals $\sqrt{\int_{-\infty}^{\infty}e^{-x^2}{ \rm d}{x}\int_{-\infty}^{\infty}e^{-y^2}{ \rm d}{y}}$ and the in­te­gral un­der the square root can be done an­a­lyt­i­cally us­ing po­lar co­or­di­nates. The re­sult is that

$$(-\frac12)! = \int_{-\infty}^{\infty}e^{-u^2}{ \rm d}{u} = \sqrt{\pi}$$

To get $\frac12!$, mul­ti­ply by $\frac12$, since $n!$ $\vphantom0\raisebox{1.5pt}{$=$}$ $n(n-1)!$.

A dou­ble ex­cla­ma­tion mark may mean every sec­ond item is skipped, e.g. 5!! $\vphantom0\raisebox{1.5pt}{$=$}$ 1 $\times$ 3 $\times$ 5. In gen­eral, $(2n+1)!!$ $\vphantom0\raisebox{1.5pt}{$=$}$ $(2n+1)!$/$2^nn!$. Of course, 5!! should log­i­cally mean (5!)!. Logic would in­di­cate that 5 $\times$ 3 $\times$ 1 should be in­di­cated by some­thing like 5!’. But what is logic in physics?

$\setbox 0=\hbox{$\vert$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The mag­ni­tude or ab­solute value of the num­ber or vec­tor, if en­closed be­tween a pair of them.
• The de­ter­mi­nant of a ma­trix, if en­closed be­tween a pair of them.
• The norm of the func­tion, if en­closed be­tween two pairs of them.
• The end of a bra or start of a ket.
• A vi­sual sep­a­ra­tor in in­ner prod­ucts.

$\setbox 0=\hbox{$\vert\ldots\rangle$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

A ket is used to in­di­cate some state. For ex­am­ple, ${\left\vert l\:m\right\rangle}$ in­di­cates an an­gu­lar mo­men­tum state with az­imuthal quan­tum num­ber $l$ and mag­netic quan­tum num­ber $m$. Sim­i­larly, ${\left\vert\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em ... ...n-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em\right\rangle}$ is the spin-down state of a par­ti­cle with spin $\frac12$. Other com­mon ones are ${\left\vert{\underline x}\right\rangle}$ for the po­si­tion eigen­func­tion ${\underline x}$, i.e. $\delta(x-{\underline x})$, ${\left\vert 1{\rm {s}}\right\rangle}$ for the 1s or $\psi_{100}$ hy­dro­gen state, ${\left\vert 2{\rm {p}}_z\right\rangle}$ for the 2p$_z$ or $\psi_{210}$ state, etcetera. In short, what­ever can in­di­cate some state can be pushed into a ket.

$\setbox 0=\hbox{$\langle\ldots\vert$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

A bra is like a ket ${\left\vert\ldots\right\rangle}$, but ap­pears in the left side of in­ner prod­ucts, in­stead of the right one.

$\setbox 0=\hbox{$\uparrow$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

In­di­cates the spin up state. Math­e­mat­i­cally, equals the func­tion ${\uparrow}(S_z)$ which is by de­f­i­n­i­tion equal to 1 at $S_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12\hbar$ and equal to 0 at $S_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-\frac12\hbar$. A spa­tial wave func­tion mul­ti­plied by $\uparrow$ is a par­ti­cle in that spa­tial state with its spin up. For mul­ti­ple par­ti­cles, the spins are listed with par­ti­cle 1 first.

$\setbox 0=\hbox{$\downarrow$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

In­di­cates the spin down state. Math­e­mat­i­cally, equals the func­tion ${\downarrow}(S_z)$ which is by de­f­i­n­i­tion equal to 0 at $S_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12\hbar$ and equal to 1 at $S_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-\frac12\hbar$. A spa­tial wave func­tion mul­ti­plied by $\downarrow$ is a par­ti­cle in that spa­tial state with its spin down. For mul­ti­ple par­ti­cles, the spins are listed with par­ti­cle 1 first.

$\setbox 0=\hbox{$\sum$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Sum­ma­tion sym­bol. Ex­am­ple: if in three di­men­sional space a vec­tor $\vec{f}$ has com­po­nents $f_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2, $f_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, $f_3$ $\vphantom0\raisebox{1.5pt}{$=$}$ 4, then $\sum_{{\rm {all }}i}f_i$ stands for $2+1+4$ $\vphantom0\raisebox{1.5pt}{$=$}$ 7.

One im­por­tant thing to re­mem­ber: the sym­bol used for the sum­ma­tion in­dex does not make a dif­fer­ence: $\sum_{{\rm {all }}j}f_j$ is ex­actly the same as $\sum_{{\rm {all }}i}f_i$. So freely re­name the in­dex, but al­ways make sure that the new name is not al­ready used for some­thing else in the part that it ap­pears in. If you use the same name for two dif­fer­ent things, it be­comes a mess.

Re­lated to that, $\sum_{{\rm {all }}i}f_i$ is not some­thing that de­pends on an in­dex $i$. It is just a com­bined sim­ple num­ber. Like 7 in the ex­am­ple above. It is com­monly said that the sum­ma­tion in­dex sums away.

$\setbox 0=\hbox{$\prod$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(Not to be con­fused with ${\mit\Pi}$ fur­ther down.) Mul­ti­pli­ca­tion sym­bol. Ex­am­ple: if in three di­men­sional space a vec­tor $\vec{f}$ has com­po­nents $f_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2, $f_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, $f_3$ $\vphantom0\raisebox{1.5pt}{$=$}$ 4, then $\prod_{{\rm {all }}i}f_i$ stands for $2\times1\times4$ $\vphantom0\raisebox{1.5pt}{$=$}$ 6.

One im­por­tant thing to re­mem­ber: the sym­bol used for the mul­ti­pli­ca­tions in­dex does not make a dif­fer­ence: $\prod_{{\rm {all }}j}f_j$ is ex­actly the same as $\prod_{{\rm {all }}i}f_i$. So freely re­name the in­dex, but al­ways make sure that the new name is not al­ready used for some­thing else in the part that it ap­pears in. If you use the same name for two dif­fer­ent things, it be­comes a mess.

Re­lated to that, $\prod_{{\rm {all }}i}f_i$ is not some­thing that de­pends on an in­dex $i$. It is just a com­bined sim­ple num­ber. Like 6 in the ex­am­ple above. It is com­monly said that the mul­ti­pli­ca­tion in­dex fac­tors away. (By who?)

$\setbox 0=\hbox{$\int$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

In­te­gra­tion sym­bol, the con­tin­u­ous ver­sion of the sum­ma­tion sym­bol. For ex­am­ple,

$$\int_{\mbox{\scriptsize all }x} f(x){ \rm d}x$$

is the sum­ma­tion of $f(x){ \rm d}{x}$ over all in­fin­i­tes­i­mally small frag­ments ${\rm d}{x}$ that make up the en­tire $x$-range. For ex­am­ple, $\int_{x=0}^2(2+x){ \rm d}{x}$ equals 3 $\times$ 2 $\vphantom0\raisebox{1.5pt}{$=$}$ 6; the av­er­age value of $2+x$ be­tween $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 and $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2 is 3, and the sum of all the in­fin­i­tes­i­mally small seg­ments ${\rm d}{x}$ gives the to­tal length 2 of the range in $x$ from 0 to 2.

One im­por­tant thing to re­mem­ber: the sym­bol used for the in­te­gra­tion vari­able does not make a dif­fer­ence: $\int_{{\rm {all }}y}f(y){ \rm d}{y}$ is ex­actly the same as $\int_{{\rm {all }}x}f(x){ \rm d}{x}$. So freely re­name the in­te­gra­tion vari­able, but al­ways make sure that the new name is not al­ready used for some­thing else in the part it ap­pears in. If you use the same name for two dif­fer­ent things, it be­comes a mess.

Re­lated to that $\int_{{\rm {all }}x}f(x){ \rm d}{x}$ is not some­thing that de­pends on a vari­able $x$. It is just a com­bined num­ber. Like 6 in the ex­am­ple above. It is com­monly said that the in­te­gra­tion vari­able in­te­grates away.

$\setbox 0=\hbox{$\to$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• An ap­proach­ing process. $\lim_{\varepsilon\to0}$ in­di­cates for prac­ti­cal pur­poses the value of the ex­pres­sion fol­low­ing the $\lim$ when $\varepsilon$ is ex­tremely small. Sim­i­larly, $\lim_{r\to\infty}$ in­di­cates the value of the fol­low­ing ex­pres­sion when $r$ is ex­tremely large.
• The fact that the left side leads to, or im­plies, the right-hand side.

$\setbox 0=\hbox{$\vec{\phantom{a}}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Vec­tor sym­bol. An ar­row above a let­ter in­di­cates it is a vec­tor. A vec­tor is a quan­tity that re­quires more than one num­ber to be char­ac­ter­ized. Typ­i­cal vec­tors in physics in­clude po­si­tion ${\skew0\vec r}$, ve­loc­ity $\vec{v}$, lin­ear mo­men­tum ${\skew0\vec p}$, ac­cel­er­a­tion $\vec{a}$, force $\vec{F}$, an­gu­lar mo­men­tum $\vec{L}$, etcetera.

$\setbox 0=\hbox{$\widehat{\phantom{a}}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

A hat over a let­ter in this book in­di­cates that it is the op­er­a­tor, turn­ing func­tions into other func­tions.

$\setbox 0=\hbox{$'$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• A de­riv­a­tive of a func­tion. Ex­am­ples: $1'$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, $x'$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, $\sin'(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\cos(x)$, $\cos'(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-\sin(x)$, $(e^x)'$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e^x$.
• A small or mod­i­fied quan­tity.
• A quan­tity per unit length.

$\setbox 0=\hbox{$\nabla$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The spa­tial dif­fer­en­ti­a­tion op­er­a­tor nabla. In Carte­sian co­or­di­nates:

$$\nabla \equiv \left( \frac{\partial}{\partial x}, \f... ...partial}{\partial y} + {\hat k}\frac{\partial}{\partial z}$$

Nabla can be ap­plied to a scalar func­tion $f$ in which case it gives a vec­tor of par­tial de­riv­a­tives called the gra­di­ent of the func­tion:

$$\mathop{\rm grad}\nolimits f = \nabla f = {\hat\imath}\f... ...al f}{\partial y} + {\hat k}\frac{\partial f}{\partial z}.$$

Nabla can be ap­plied to a vec­tor in a dot prod­uct mul­ti­pli­ca­tion, in which case it gives a scalar func­tion called the di­ver­gence of the vec­tor:

$$\div\vec v = \nabla\cdot\vec v = \frac{\partial v_x}{\pa... ...partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}$$

or in in­dex no­ta­tion

$$\div\vec v = \nabla\cdot\vec v = \sum_{i=1}^3 \frac{\partial v_i}{\partial x_i}$$

Nabla can also be ap­plied to a vec­tor in a vec­to­r­ial prod­uct mul­ti­pli­ca­tion, in which case it gives a vec­tor func­tion called the curl or rot of the vec­tor. In in­dex no­ta­tion, the $i$-th com­po­nent of this vec­tor is

$$\left(\mathop{\rm curl}\nolimits \vec v\right)_i = \left... ...line{\imath}}}}{\partial x_{{\overline{\overline{\imath}}}}}$$

where ${\overline{\imath}}$ is the in­dex fol­low­ing $i$ in the se­quence 123123..., and ${\overline{\overline{\imath}}}$ the one pre­ced­ing it (or the sec­ond fol­low­ing it).

The op­er­a­tor $\nabla^2$ is called the Lapla­cian. In Carte­sian co­or­di­nates:

$$\nabla^2 \equiv \frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}+ \frac{\partial^2}{\partial z^2}$$

Some­times the Lapla­cian is in­di­cated as $\Delta$. In rel­a­tivis­tic in­dex no­ta­tion it is equal to $\partial_i\partial^i$, with maybe a mi­nus sign de­pend­ing on who you talk with.

In non Carte­sian co­or­di­nates, don’t guess; look these op­er­a­tors up in a ta­ble book, [41, pp. 124-126]: . For ex­am­ple, in spher­i­cal co­or­di­nates,

$$\nabla = {\hat\imath}_r \frac{\partial}{\partial r} + {\... ...hi \frac{1}{r \sin\theta} \frac{\partial}{\partial \phi} %$$ (N.2)

That al­lows the gra­di­ent of a scalar func­tion $f$, i.e. $\nabla{f}$, to be found im­me­di­ately. But if you ap­ply $\nabla$ on a vec­tor, you have to be very care­ful be­cause you also need to dif­fer­en­ti­ate ${\hat\imath}_r$, ${\hat\imath}_\theta$, and ${\hat\imath}_\phi$. In par­tic­u­lar, the cor­rect di­ver­gence of a vec­tor $\vec{v}$ is

$$\nabla \cdot \vec v = \frac{1}{r^2} \frac{\partial r^2 v_r... ...rac{1}{r\sin\theta} \frac{\partial v_\phi}{\partial\phi} %$$ (N.3)

The curl $\nabla$ $\times$ $\vec{v}$ of the vec­tor is

$$\frac{{\hat\imath}_r}{r\sin\theta} \left( \frac{\partial... ...rtial r} - \frac{\partial v_r}{\partial\theta} \right) %$$ (N.4)

Fi­nally the Lapla­cian is:

$$\nabla^2 = \frac{1}{r^2} \left\{ \frac{\partial}{\part... ...n^2\theta} \frac{\partial^2}{\partial \phi^2} \right\} %$$ (N.5)

See also spher­i­cal co­or­di­nates.

Cylin­dri­cal co­or­di­nates are usu­ally in­di­cated as $r$, $\theta$ and $z$. Here $z$ is the Carte­sian co­or­di­nate, while $r$ is the dis­tance from the $z$-axis and $\theta$ the an­gle around the $z$ axis. In two di­men­sions, i.e. with­out the $z$ terms, they are usu­ally called po­lar co­or­di­nates. In cylin­dri­cal co­or­di­nates:

$$\nabla = {\hat\imath}_r \frac{\partial}{\partial r} + {\... ...ial \theta} + {\hat\imath}_z \frac{\partial}{\partial z} %$$ (N.6)

$$\nabla \cdot \vec v = \frac{1}{r} \frac{\partial r v_r}{\p... ...theta}{\partial\theta} + \frac{\partial v_z}{\partial z} %$$ (N.7)

$$\nabla\times\vec{v} = {\hat\imath}_r \left( \frac{1}{r... ...rtial r} - \frac{\partial v_r}{\partial\theta} \right) %$$ (N.8)

$$\nabla^2 = \frac{1}{r} \frac{\partial}{\partial r} \le... ...}{\partial \theta^2} + \frac{\partial^2}{\partial z^2} %$$ (N.9)

$\setbox 0=\hbox{$\mathop{\Box}\nolimits $}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The D'Alem­bert­ian is de­fined as

$$\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\pa... ...partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2}$$

where $c$ is a con­stant called the wave speed. In rel­a­tivis­tic in­dex no­ta­tion, $\mathop{\Box}\nolimits$ is equal to $-\partial_\mu\partial^\mu$.

$\setbox 0=\hbox{$^*$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

A su­per­script star nor­mally in­di­cates a com­plex con­ju­gate. In the com­plex con­ju­gate of a num­ber, every ${\rm i}$ is changed into a $-{\rm i}$.

$\raisebox{.3pt}{$<$}$    

Less than.

$\raisebox{-.3pt}{$\leqslant$}$    

Less than or equal.

$\setbox 0=\hbox{$\langle\ldots\rangle$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• An in­ner prod­uct.
• An ex­pec­ta­tion value.

$\raisebox{.3pt}{$>$}$    

Greater than.

$\raisebox{-.5pt}{$\geqslant$}$    

Greater than or equal.

$\setbox 0=\hbox{$[\ldots]$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• A group­ing of terms in a for­mula.
• A com­mu­ta­tor. For ex­am­ple, $[A,B]$ $\vphantom0\raisebox{1.5pt}{$=$}$ $AB-BA$.

$\vphantom0\raisebox{1.5pt}{$=$}$    

Equals sign. The quan­tity to the left is the same as the one to the right.

$\vphantom0\raisebox{1.5pt}{$\equiv$}$    

Em­phatic equals sign. Typ­i­cally means “by de­f­i­n­i­tion equal” or every­where equal.

$\vphantom0\raisebox{1.1pt}{$\approx$}$    

In­di­cates ap­prox­i­mately equal. Read it as “is ap­prox­i­mately equal to.”

$\vphantom0\raisebox{1.5pt}{$\sim$}$    

In­di­cates ap­prox­i­mately equal. Of­ten used when the ap­prox­i­ma­tion ap­plies only when some­thing is small or large. Read it as is ap­prox­i­mately equal to or as “is as­ymp­tot­i­cally equal to.”

$\setbox 0=\hbox{$\propto$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Pro­por­tional to. The two sides are equal ex­cept for some un­known con­stant fac­tor.

$\setbox 0=\hbox{$\alpha$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(al­pha) May in­di­cate:

• The fine struc­ture con­stant, $e^2$/$4\pi\epsilon_0{\hbar}c$, equal to 7.297 352 570 10$\POW9,{-3}$, or about 1/137, in value.
• A Dirac equa­tion ma­trix.
• A nu­clear de­cay mode in which a he­lium-4 nu­cleus is emit­ted.
• In­ter­nal con­ver­sion rate as frac­tion of the gamma de­cay rate.
• Some con­stant.
• Some an­gle.
• An eigen­func­tion of a generic op­er­a­tor $A$.
• A sum­ma­tion in­dex.
• Com­po­nent in­dex of a vec­tor.

$\setbox 0=\hbox{$\beta$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(beta) May in­di­cate:

• A nu­clear de­cay mode in which an elec­tron ($\beta^-$) or positron ($\beta^+$) is emit­ted. Some­times $\beta^+$ is taken to also in­clude elec­tron cap­ture.
• A nu­clear vi­bra­tional mode that main­tains the ax­ial sym­me­try of the nu­cleus.
• Some con­stant.
• Some an­gle.
• An eigen­func­tion of a generic op­er­a­tor $B$.
• A sum­ma­tion in­dex.

$\setbox 0=\hbox{$\Gamma$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(Gamma) May in­di­cate:

• The Gamma func­tion. Look un­der ! for de­tails.
• The width or un­cer­tainty in en­ergy of an ap­prox­i­mate en­ergy eigen­state.
• Ori­gin in wave num­ber space.

$\setbox 0=\hbox{$\gamma$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(gamma) May in­di­cate:

• Gy­ro­mag­netic ra­tio.
• Stan­dard sym­bol for a pho­ton of elec­tro­mag­netic ra­di­a­tion.
• A nu­clear de-ex­ci­ta­tion mode in which a pho­ton is emit­ted.
• A nu­clear vi­bra­tional mode that messes up the ax­ial sym­me­try of the nu­cleus.
• Sum­ma­tion in­dex.
• In­te­gral in the tun­nel­ing WKB ap­prox­i­ma­tion.

$\setbox 0=\hbox{$\Delta$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(cap­i­tal delta) May in­di­cate:

• An in­cre­ment in the quan­tity fol­low­ing it.
• A delta par­ti­cle.
• Of­ten used to in­di­cate the Lapla­cian $\nabla^2$.

$\setbox 0=\hbox{$\delta$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(delta) May in­di­cate:

• With two sub­scripts, the Kro­necker delta, which by de­f­i­n­i­tion is equal to one if its two sub­scripts are equal, and zero in all other cases.

• With­out two sub­scripts, the “Dirac delta func­tion”, which is in­fi­nite when its ar­gu­ment is zero, and zero if it is not. In ad­di­tion the in­fin­ity is such that the in­te­gral of the delta func­tion over its sin­gle nonzero point is unity. The delta func­tion is not a nor­mal func­tion, but a dis­tri­b­u­tion. It is best to think of it as the ap­prox­i­mate func­tion shown in the right hand side of fig­ure 7.10 for a very, very, small pos­i­tive value of $\varepsilon$.

  One of­ten im­por­tant way to cre­ate a three-di­men­sional delta func­tion in spher­i­cal co­or­di­nates is to take the Lapla­cian of the func­tion $\vphantom{0}\raisebox{1.5pt}{$-$}$1/$4{\pi}r$. Chap­ter 13.3 ex­plains why. In two di­men­sions, take the Lapla­cian of $\ln(r)$/$2\pi$ to get a delta func­tion.

• Of­ten used to in­di­cate a small amount of the fol­low­ing quan­tity, or of a small change in the fol­low­ing quan­tity. There are nu­anced dif­fer­ences in the us­age of $\delta$, $\partial$ and ${\rm d}$ that are too much to go in here.

• Of­ten used to in­di­cate a sec­ond small quan­tity in ad­di­tion to $\varepsilon$.

$\setbox 0=\hbox{$\partial$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(par­tial) In­di­cates a van­ish­ingly small change or in­ter­val of the fol­low­ing vari­able. For ex­am­ple, $\partial{f}$/$\partial{x}$ is the ra­tio of a van­ish­ingly small change in func­tion $f$ di­vided by the van­ish­ingly small change in vari­able $x$ that causes this change in $f$. Such ra­tios de­fine de­riv­a­tives, in this case the par­tial de­riv­a­tive of $f$ with re­spect to $x$.

Also used in rel­a­tivis­tic in­dex no­ta­tion, chap­ter 1.2.5.

$\setbox 0=\hbox{$\epsilon$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(ep­silon) May in­di­cate:

• $\epsilon_0$ is the per­mit­tiv­ity of space. Equal to 8.854 187 817... 10$\POW9,{-12}$ C$\POW9,{2}$/J m. The ex­act value is 1$\raisebox{.5pt}{$/$}$​$4{\pi}c^2$ 10$\POW9,{7}$ C$\POW9,{2}$/J m, be­cause of the ex­act SI de­f­i­n­i­tions of am­pere and speed of light.
• Scaled en­ergy.
• Or­bital en­ergy.
• La­grangian mul­ti­plier.
• A small quan­tity, if sym­bol $\varepsilon$ is not avail­able.

$\setbox 0=\hbox{$\varepsilon$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(vari­ant of ep­silon) May in­di­cate:

• A very small quan­tity.
• The slop in en­ergy con­ser­va­tion dur­ing a de­cay process.

$\setbox 0=\hbox{$\eta$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(eta) May be used to in­di­cate a $y$-po­si­tion of a par­ti­cle.

$\setbox 0=\hbox{$\Theta$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(cap­i­tal theta) Used in this book to in­di­cate some func­tion of $\theta$ to be de­ter­mined.

$\setbox 0=\hbox{$\theta$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(theta) May in­di­cate:

• In spher­i­cal co­or­di­nates, the an­gle from the cho­sen $z$ axis, with apex at the ori­gin.
• $z$-po­si­tion of a par­ti­cle.
• A generic an­gle, like the one be­tween the vec­tors in a cross or dot prod­uct.
• In­te­gral act­ing as an an­gle in the clas­si­cal WKB ap­prox­i­ma­tion.
• In­te­gral act­ing as an an­gle in the adi­a­batic ap­prox­i­ma­tion.

$\setbox 0=\hbox{$\vartheta$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(vari­ant of theta) An al­ter­nate sym­bol for $\theta$.

$\setbox 0=\hbox{$\kappa$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(kappa) May in­di­cate:

• A con­stant that phys­i­cally cor­re­sponds to some wave num­ber.
• A sum­ma­tion in­dex.
• Ther­mal con­duc­tiv­ity.

$\setbox 0=\hbox{$\Lambda$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(Lambda) May in­di­cate:

• Lorentz trans­for­ma­tion ma­trix.

$\setbox 0=\hbox{$\lambda$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(lambda) May in­di­cate:

• Wave length.
• De­cay con­stant.
• A generic eigen­value.
• En­try of a Lorentz trans­for­ma­tion.
• Scaled square mo­men­tum.
• Some mul­ti­ple of some­thing.

$\setbox 0=\hbox{$\mu$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(mu) May in­di­cate:

• Mag­netic di­pole mo­ment:
  Al­pha par­ti­cle: 0 (spin is zero).
  Deuteron: 0.433 073 49 10$\POW9,{-26}$ J/T or 0.857 438 231 $\mu_{\rm N}$.
  Elec­tron: $\vphantom{0}\raisebox{1.5pt}{$-$}$9.284 764 3 10$\POW9,{-24}$ J/T or $\vphantom{0}\raisebox{1.5pt}{$-$}$1.001 159 652 180 8 $\mu_{\rm B}$.
  He­lion: $\vphantom{0}\raisebox{1.5pt}{$-$}$1.074 617 49 10$\POW9,{-26}$ J/T or $\vphantom{0}\raisebox{1.5pt}{$-$}$2.127 625 306 $\mu_{\rm N}$.
  Neu­tron: $\vphantom{0}\raisebox{1.5pt}{$-$}$0.966 236 5 10$\POW9,{-26}$ J/T or $\vphantom{0}\raisebox{1.5pt}{$-$}$1.913 042 7 $\mu_{\rm N}$.
  Pro­ton: 1.410 606 74 10$\POW9,{-26}$ J/T or 2.792 847 36 $\mu_{\rm N}$.
  Tri­ton: 1.504 609 45 10$\POW9,{-26}$ J/T or 2.978 962 45 $\mu_{\rm N}$.
• $\mu_{\rm B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e\hbar$/$2m_{\rm e}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 9.274 009 7 10$\POW9,{-24}$ J/T or 5.788 381 807 10$\POW9,{-5}$ eV/T is the Bohr mag­ne­ton.
• $\mu_{\rm N}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e\hbar$/$2m_{\rm p}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 5.050 783 5 10$\POW9,{-27}$ J/T or 3.152 451 261 10$\POW9,{-8}$ eV/T is the nu­clear mag­ne­ton.
• A sum­ma­tion in­dex.
• Chem­i­cal po­ten­tial/mo­lar Gibbs free en­ergy.

$\setbox 0=\hbox{$\nu$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(nu) May in­di­cate:

• Elec­tron neu­trino.
• Scaled en­ergy eigen­func­tion num­ber in solids.
• A sum­ma­tion in­dex.
• Strength of a delta func­tion po­ten­tial.

$\setbox 0=\hbox{$\xi$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(xi) May in­di­cate:

• Scaled ar­gu­ment of the one-di­men­sional har­monic os­cil­la­tor eigen­func­tions.
• $x$-po­si­tion of a par­ti­cle.
• A sum­ma­tion or in­te­gra­tion in­dex.

$\setbox 0=\hbox{${\mit\Pi}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(Oblique Pi) (Not to be con­fused with $\prod$ de­scribed higher up.) Par­ity op­er­a­tor. Re­places ${\skew0\vec r}$ by $-{\skew0\vec r}$. That is equiv­a­lent to a mir­ror­ing in a mir­ror through the ori­gin, fol­lowed by a 180$\POW9,{\circ}$ ro­ta­tion around the axis nor­mal to the mir­ror.

$\setbox 0=\hbox{$\pi$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(pi) May in­di­cate:

• A con­stant with value 3.141 592 653 589 793 238 462....
  The area of a cir­cle of ra­dius $r$ is ${\pi}r^2$ and its perime­ter is $2{\pi}r$.
  The vol­ume of a sphere of ra­dius $r$ is $\frac43{\pi}r^3$ and its sur­face is $4{\pi}r^2$.
  A 180$\POW9,{\circ}$ an­gle ex­pressed in ra­di­ans is $\pi$.
  Note also that $e^{\pm{\rm i}\pi}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$1 and $e^{\pm{\rm i}2\pi}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.
• A chem­i­cal bond that looks from the side like a p state.
• A par­ti­cle in­volved in the forces keep­ing the nu­clei of atoms to­gether ($\pi$-me­son or pion for short).
• Par­ity.

$\setbox 0=\hbox{$\tilde\pi $}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Canon­i­cal mo­men­tum den­sity.

$\setbox 0=\hbox{$\rho$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(rho) May in­di­cate:

• Elec­tric charge per unit vol­ume.
• Scaled ra­dial co­or­di­nate.
• Ra­dial co­or­di­nate.
• Eigen­func­tion of a ro­ta­tion op­er­a­tor ${\cal R}$.
• Mass-base den­sity.
• En­ergy den­sity of elec­tro­mag­netic ra­di­a­tion.

$\setbox 0=\hbox{$\sigma$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(sigma) May in­di­cate:

• A stan­dard de­vi­a­tion of a value.
• A chem­i­cal bond that looks like an s state when seen from the side.
• Pauli spin ma­trix.
• Sur­face ten­sion.
• Elec­tri­cal con­duc­tiv­ity.
• $\sigma_{\rm {B}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 5.670 37 W/m$\POW9,{2}$ K$\POW9,{4}$ is the Ste­fan-Boltz­mann

$\setbox 0=\hbox{$\tau$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(tau) May in­di­cate:

• A time or time in­ter­val.
• Life time or half life.
• Some co­ef­fi­cient.

$\setbox 0=\hbox{$\Phi$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(cap­i­tal phi) May in­di­cate:

• Some func­tion of $\phi$ to be de­ter­mined.
• The mo­men­tum-space wave func­tion.
• Rel­a­tivis­tic elec­tro­mag­netic po­ten­tial.

$\setbox 0=\hbox{$\phi$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(phi) May in­di­cate:

• In spher­i­cal co­or­di­nates, the an­gle around the cho­sen $z$ axis. In­creas­ing $\phi$ by $2\pi$ en­cir­cles the $z$-axis ex­actly once.
• A phase an­gle.
• Some­thing equiv­a­lent to an an­gle.
• Field op­er­a­tor $\phi({\skew0\vec r})$ an­ni­hi­lates a par­ti­cle at po­si­tion ${\skew0\vec r}$ while $\phi^\dagger({\skew0\vec r})$ cre­ates one at that po­si­tion.

$\setbox 0=\hbox{$\varphi$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(vari­ant of phi) May in­di­cate:

• A change in an­gle $\phi$.
• An al­ter­nate sym­bol for $\phi$.
• An elec­tro­sta­tic po­ten­tial.
• An elec­tro­sta­tic quan­tum field.
• A hy­po­thet­i­cal se­lec­to­sta­tic quan­tum field.

$\setbox 0=\hbox{$\chi$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(chi) May in­di­cate

• Spinor com­po­nent.
• Gauge func­tion of elec­tro­mag­netic field.

$\setbox 0=\hbox{$\Psi$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(cap­i­tal psi) Up­per case psi is used for the wave func­tion.

$\setbox 0=\hbox{$\psi$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(psi) Typ­i­cally used to in­di­cate an en­ergy eigen­func­tion. De­pend­ing on the sys­tem, in­dices may be added to dis­tin­guish dif­fer­ent ones. In some cases $\psi$ might be used in­stead of $\Psi$ to in­di­cate a sys­tem in an en­ergy eigen­state. Let me know and I will change it. A sys­tem in an en­ergy eigen­state should be writ­ten as $\Psi$ $\vphantom0\raisebox{1.5pt}{$=$}$ $c\psi$, not $\psi$, with $c$ a con­stant of mag­ni­tude 1.

$\setbox 0=\hbox{$\Omega$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(Omega) May in­di­cate:

• Solid an­gle. See an­gle and “spher­i­cal co­or­di­nates.”

$\setbox 0=\hbox{$\omega$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

(omega) May in­di­cate:

• An­gu­lar fre­quency of the clas­si­cal har­monic os­cil­la­tor. Equal to $\sqrt{c/m}$ where $c$ is the spring con­stant and $m$ the mass.
• An­gu­lar fre­quency of a sys­tem.
• An­gu­lar fre­quency of light waves.
• Per­tur­ba­tion fre­quency,
• Any quan­tity hav­ing units of fre­quency, 1/s.

$\setbox 0=\hbox{$A$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Re­peat­edly used to in­di­cate the op­er­a­tor for a generic phys­i­cal quan­tity $a$, with eigen­func­tions $\alpha$.
• Elec­tro­mag­netic vec­tor po­ten­tial, or four vec­tor po­ten­tial.
• Ein­stein $A$ co­ef­fi­cient.
• Some generic ma­trix.
• Some con­stant.
• Area.

Å    

Ångstrom. Equal to 10$\POW9,{-10}$ m.

$\setbox 0=\hbox{$a$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The value of a generic phys­i­cal quan­tity with op­er­a­tor $A$
• The am­pli­tude of the spin-up state
• The am­pli­tude of the first state in a two-state sys­tem.
• Ac­cel­er­a­tion.
• Start point of an in­te­gra­tion in­ter­val.
• The first of a pair of par­ti­cles.
• Some co­ef­fi­cient.
• Some con­stant.
• Ab­sorp­tiv­ity of elec­tro­mag­netic ra­di­a­tion.
• An­ni­hi­la­tion op­er­a­tor $\widehat a$ or cre­ation op­er­a­tor $\widehat a^\dagger$.
• Bohr ra­dius of he­lium ion.

$\setbox 0=\hbox{$a_0$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Bohr ra­dius, $4\pi\epsilon_0\hbar^2$/${m_{\rm e}}e^2$ or 0.529 177 210 9 Å, with Å$\vphantom0\raisebox{1.5pt}{$=$}$ 10$\POW9,{-10}$ m. Com­pa­ra­ble in size to atoms, and a good size to use to sim­plify var­i­ous for­mu­lae.
• The ini­tial value of a co­ef­fi­cient $a$.

ab­solute    

May in­di­cate:

• The ab­solute value of a real num­ber $a$ is in­di­cated by $\vert a\vert$. It equals $a$ is $a$ is pos­i­tive or zero and $-a$ if $a$ is neg­a­tive.
• The ab­solute value of a com­plex num­ber $a$ is in­di­cated by $\vert a\vert$. It equals the length of the num­ber plot­ted as a vec­tor in the com­plex plane. This sim­pli­fies to above de­f­i­n­i­tion if $a$ is real.
• An ab­solute tem­per­a­ture is a tem­per­a­ture mea­sured from ab­solute zero. At ab­solute zero all sys­tems are in their ground state. Ab­solute zero is $-$273.15 $\POW9,{\circ}$C in de­grees Cen­tri­grade (Cel­sius). The SI ab­solute tem­per­a­ture scale is de­grees Kelvin, K. Ab­solute zero tem­per­a­ture is 0 K, while 0 $\POW9,{\circ}$C is 273.15 K.

adi­a­batic    

An adi­a­batic process is a process in which there is no heat trans­fer with the sur­round­ings. If the process is also re­versible, it is called isen­tropic. Typ­i­cally, these processes are fairly quick, in or­der not to give heat con­duc­tion enough time to do its stuff, but not so ex­ces­sively quick that they be­come ir­re­versible.

Adi­a­batic processes in quan­tum me­chan­ics are de­fined quite dif­fer­ently to keep stu­dents on their toes. See chap­ter 7.1.5. These processes are very slow, to give the sys­tem all pos­si­ble time to ad­just to its sur­round­ings. Of course, quan­tum physi­cist were not aware that the same term had al­ready been used for a hun­dred years or so for rel­a­tively fast processes. They as­sumed they had just in­vented a great new term!

ad­joint    

The ad­joint $A^H$ or $A^\dagger$ of an op­er­a­tor is the one you get if you take it to the other side of an in­ner prod­uct. (While keep­ing the value of the in­ner prod­uct the same re­gard­less of what­ever two vec­tors or func­tions may be in­volved.) Her­mit­ian op­er­a­tors are self-ad­joint;they do not change if you take them to the other side of an in­ner prod­uct. Skew-Her­mit­ianop­er­a­tors just change sign. Uni­tary op­er­a­torschange into their in­verse when taken to the other side of an in­ner prod­uct. Uni­tary op­er­a­tors gen­er­al­ize ro­ta­tions of vec­tors: an in­ner prod­uct of vec­tors is the same whether you ro­tate the first vec­tor one way, or the sec­ond vec­tor the op­po­site way. Uni­tary op­er­a­tors pre­serve in­ner prod­ucts (when ap­plied to both vec­tors or func­tions). Fourier trans­forms are uni­tary op­er­a­tors on ac­count of the Par­se­val equal­ity that says that in­ner prod­ucts are pre­served.

am­pli­tude    

Every­thing in quan­tum me­chan­ics is an am­pli­tude. How­ever, most im­por­tantly, the quan­tum am­pli­tude gives the co­ef­fi­cient of a state in a wave func­tion. For ex­am­ple, the usual quan­tum wave func­tion gives the quan­tum am­pli­tude that the par­ti­cle is at the given po­si­tion.

an­gle    

Con­sider two semi-in­fi­nite lines ex­tend­ing from a com­mon in­ter­sec­tion point. Then the an­gle be­tween these lines is de­fined in the fol­low­ing way: draw a unit cir­cle in the plane of the lines and cen­tered at their in­ter­sec­tion point. The an­gle is then the length of the cir­cu­lar arc that is in be­tween the lines. More pre­cisely, this gives the an­gle in ra­di­ans, rad. Some­times an an­gle is ex­pressed in de­grees, where $2\pi$ rad is taken to be 360$\POW9,{\circ}$. How­ever, us­ing de­grees is usu­ally a very bad idea in sci­ence.

In three di­men­sions, you may be in­ter­ested in the so-called solid an­gle $\Omega$ in­side a con­i­cal sur­face. This an­gle is de­fined in the fol­low­ing way: draw a sphere of unit ra­dius cen­tered at the apex of the con­i­cal sur­face. Then the solid an­gle is the area of the spher­i­cal sur­face that is in­side the cone. Solid an­gles are in stera­di­ans. The cone does not need to be a cir­cu­lar one, (i.e. have a cir­cu­lar cross sec­tion), for this to ap­ply. In fact, the most com­mon case is the solid an­gle cor­re­spond­ing to an in­fin­i­tes­i­mal el­e­ment ${\rm d}\theta$ $\times$ ${\rm d}\phi$ of spher­i­cal co­or­di­nate sys­tem an­gles. In that case the sur­face of the unit sphere in­side the con­i­cal sur­face is is ap­prox­i­mately rec­tan­gu­lar, with sides ${\rm d}\theta$ and $\sin(\theta){\rm d}\phi$. That makes the en­closed solid an­gle equal to ${\rm d}\Omega$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sin(\theta){\rm d}\theta{\rm d}\phi$.

$\setbox 0=\hbox{$B$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Re­peat­edly used to in­di­cate a generic sec­ond op­er­a­tor or ma­trix.
• Ein­stein $B$ co­ef­fi­cient.
• Some con­stant.

$\setbox 0=\hbox{${\cal B}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Mag­netic field strength.

$\setbox 0=\hbox{$b$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Re­peat­edly used to in­di­cate the am­pli­tude of the spin-down state
• Re­peat­edly used to in­di­cate the am­pli­tude of the sec­ond state in a two-state sys­tem.
• End point of an in­te­gra­tion in­ter­val.
• The sec­ond of a pair of par­ti­cles.
• Some co­ef­fi­cient.
• Some con­stant.

ba­sis    

A ba­sis is a min­i­mal set of vec­tors or func­tions that you can write all other vec­tors or func­tions in terms of. For ex­am­ple, the unit vec­tors ${\hat\imath}$, ${\hat\jmath}$, and ${\hat k}$ are a ba­sis for nor­mal three-di­men­sional space. Every three-di­men­sional vec­tor can be writ­ten as a lin­ear com­bi­na­tion of the three.

$\setbox 0=\hbox{$C$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• A third ma­trix or op­er­a­tor.
• A va­ri­ety of dif­fer­ent con­stants.

$\POW9,{\circ}$C    

De­grees Centi­grade. A com­monly used tem­per­a­ture scale that has the value $-$273.15 $\POW9,{\circ}$C in­stead of zero when sys­tems are in their ground state. Rec­om­men­da­tion: use de­grees Kelvin (K) in­stead. How­ever, dif­fer­ences in tem­per­a­ture are the same in Centi­grade as in Kelvin.

$\setbox 0=\hbox{$c$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The speed of light, 299 792 458 m/s ex­actly (by de­f­i­n­i­tion of the ve­loc­ity unit).
• Speed of sound.
• Spring con­stant.
• A va­ri­ety of dif­fer­ent con­stants.

Cauchy-Schwartz in­equal­ity    

The Cauchy-Schwartz in­equal­ity de­scribes a lim­i­ta­tion on the mag­ni­tude of in­ner prod­ucts. In par­tic­u­lar, it says that for any $f$ and $g$,

$$\vert\left\langle\vphantom{g}f\hspace{-\nulldelimiterspace... ...hspace{.03em}\right.\!\left\vert\vphantom{g}g\right\rangle }$$

In words, the mag­ni­tude of an in­ner prod­uct $\left\langle\vphantom{g}f\hspace{-\nulldelimiterspace}\hspace{.03em}\right.\!\left\vert\vphantom{f}g\right\rangle$ is at most the mag­ni­tude (i.e. the length or norm) of $f$ times the one of $g$. For ex­am­ple, if $f$ and $g$ are real vec­tors, the in­ner prod­uct is the dot prod­uct and you have $f\cdot{g}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vert f\vert\vert g\vert\cos\theta$, where $\vert f\vert$ is the length of vec­tor $f$ and $\vert g\vert$ the one of $g$, and $\theta$ is the an­gle in be­tween the two vec­tors. Since a co­sine is less than one in mag­ni­tude, the Cauchy-Schwartz in­equal­ity is there­fore true for vec­tors.

But it is true even if $f$ and $g$ are func­tions. To prove it, first rec­og­nize that $\left\langle\vphantom{g}f\hspace{-\nulldelimiterspace}\hspace{.03em}\right.\!\left\vert\vphantom{f}g\right\rangle$ may in gen­eral be a com­plex num­ber, which ac­cord­ing to (2.6) must take the form $e^{{\rm i}\alpha}\vert\left\langle\vphantom{g}f\hspace{-\nulldelimiterspace}\hspace{.03em}\right.\!\left\vert\vphantom{f}g\right\rangle \vert$ where $\alpha$ is some real num­ber whose value is not im­por­tant, and that $\left\langle\vphantom{f}g\hspace{-\nulldelimiterspace}\hspace{.03em}\right.\!\left\vert\vphantom{g}f\right\rangle$ is its com­plex con­ju­gate $e^{-{\rm i}\alpha}\vert\left\langle\vphantom{g}f\hspace{-\nulldelimiterspace}\hspace{.03em}\right.\!\left\vert\vphantom{f}g\right\rangle \vert$. Now, (yes, this is go­ing to be some con­vo­luted rea­son­ing), look at

$$\left\langle\vphantom{f + \lambda e^{-{\rm i}\alpha} g}f +... ...m i}\alpha} g}f + \lambda e^{-{\rm i}\alpha} g\right\rangle$$

where $\lambda$ is any real num­ber. The above dot prod­uct gives the square mag­ni­tude of $f+{\lambda}e^{-{\rm i}\alpha}g$, so it can never be neg­a­tive. But if you mul­ti­ply out, you get

$$\left\langle\vphantom{f}f\hspace{-\nulldelimiterspace}\hsp... ...03em}\right.\!\left\vert\vphantom{g}g\right\rangle \lambda^2$$

and if this qua­dratic form in $\lambda$ is never neg­a­tive, its dis­crim­i­nant must be less or equal to zero:

$$\vert\left\langle\vphantom{g}f\hspace{-\nulldelimiterspace... ...\hspace{.03em}\right.\!\left\vert\vphantom{g}g\right\rangle$$

and tak­ing square roots gives the Cauchy-Schwartz in­equal­ity.

Clas­si­cal    

Can mean any older the­ory. In this work, most of the time it ei­ther means non­quan­tum, or non­rel­a­tivis­tic.

$\setbox 0=\hbox{$\cos$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The co­sine func­tion, a pe­ri­odic func­tion os­cil­lat­ing be­tween 1 and -1 as shown in [41, pp. 40-]. See also sin.

curl    

The curl of a vec­tor $\vec{v}$ is de­fined as $\mathop{\rm curl}\nolimits \vec{v}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\mathop{\rm {rot}}\vec{v}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\nabla$ $\times$ $\vec{v}$.

$\setbox 0=\hbox{$D$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Dif­fer­ence in wave num­ber val­ues.

$\setbox 0=\hbox{$\vec{D}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Prim­i­tive (trans­la­tion) vec­tor of a rec­i­p­ro­cal lat­tice.

$\setbox 0=\hbox{${\cal D}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Den­sity of states.

D    

Of­ten used to in­di­cate a state with two units of or­bital an­gu­lar mo­men­tum.

$\setbox 0=\hbox{$d$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The dis­tance be­tween the pro­tons of a hy­dro­gen mol­e­cule.
• The dis­tance be­tween the atoms or lat­tice points in a crys­tal.
• A con­stant.

$\setbox 0=\hbox{$\vec{d}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Prim­i­tive (trans­la­tion) vec­tor of a crys­tal lat­tice.

$\setbox 0=\hbox{${\rm d}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

In­di­cates a van­ish­ingly small change or in­ter­val of the fol­low­ing vari­able. For ex­am­ple, ${\rm d}{x}$ can be thought of as a small seg­ment of the $x$-axis.

In three di­men­sions, ${\rm d}^3{\skew0\vec r}$ $\vphantom0\raisebox{1.5pt}{$\equiv$}$ ${\rm d}{x}{\rm d}{y}{\rm d}{z}$ is an in­fin­i­tes­i­mal vol­ume el­e­ment. The sym­bol $\int$ means that you sum over all such in­fin­i­tes­i­mal vol­ume el­e­ments.

de­riv­a­tive    

A de­riv­a­tive of a func­tion is the ra­tio of a van­ish­ingly small change in a func­tion di­vided by the van­ish­ingly small change in the in­de­pen­dent vari­able that causes the change in the func­tion. The de­riv­a­tive of $f(x)$ with re­spect to $x$ is writ­ten as ${\rm d}{f}$/${\rm d}{x}$, or also sim­ply as $f'$. Note that the de­riv­a­tive of func­tion $f(x)$ is again a func­tion of $x$: a ra­tio $f'$ can be found at every point $x$. The de­riv­a­tive of a func­tion $f(x,y,z)$ with re­spect to $x$ is writ­ten as $\partial{f}$/$\partial{x}$ to in­di­cate that there are other vari­ables, $y$ and $z$, that do not vary.

de­ter­mi­nant    

The de­ter­mi­nant of a square ma­trix $A$ is a sin­gle num­ber in­di­cated by $\vert A\vert$. If this num­ber is nonzero, $A\vec{v}$ can be any vec­tor $\vec{w}$ for the right choice of $\vec{v}$. Con­versely, if the de­ter­mi­nant is zero, $A\vec{v}$ can only pro­duce a very lim­ited set of vec­tors, though if it can pro­duce a vec­tor $w$, it can do so for mul­ti­ple vec­tors $\vec{v}$.

There is a re­cur­sive al­go­rithm that al­lows you to com­pute de­ter­mi­nants from in­creas­ingly big­ger ma­tri­ces in terms of de­ter­mi­nants of smaller ma­tri­ces. For a 1 $\times$ 1 ma­trix con­sist­ing of a sin­gle num­ber, the de­ter­mi­nant is sim­ply that num­ber:

$$\left\vert a_{11} \right\vert = a_{11}$$

(This de­ter­mi­nant should not be con­fused with the ab­solute value of the num­ber, which is writ­ten the same way. Since you nor­mally do not deal with 1 $\times$ 1 ma­tri­ces, there is nor­mally no con­fu­sion.) For 2 $\times$ 2 ma­tri­ces, the de­ter­mi­nant can be writ­ten in terms of 1 $\times$ 1 de­ter­mi­nants:

$$\left\vert \begin{array}{ll} a_{11} & a_{12} \\ a_{... ... \\ a_{21} & \phantom{a_{22}} \end{array} \right\vert$$

so the de­ter­mi­nant is $a_{11}a_{22}-a_{12}a_{21}$ in short. For 3 $\times$ 3 ma­tri­ces, you have

$$\begin{eqnarray*} \lefteqn{ \left\vert \begin{array}{lll} a_{11} & a_{12... ...a_{31} & a_{32} & \phantom{a_{33}} \end{array} \right\vert \end{eqnarray*}$$

and you al­ready know how to work out those 2 $\times$ 2 de­ter­mi­nants, so you now know how to do 3 $\times$ 3 de­ter­mi­nants. Writ­ten out fully:

$$a_{11}(a_{22}a_{33}-a_{23}a_{32}) -a_{12}(a_{21}a_{33}-a_{23}a_{31}) +a_{13}(a_{21}a_{32}-a_{22}a_{31})$$

For 4 $\times$ 4 de­ter­mi­nants,

$$\begin{eqnarray*} \lefteqn{ \left\vert \begin{array}{llll} a_{11} & a_{1... ...a_{42} & a_{43} & \phantom{a_{44}} \end{array} \right\vert \end{eqnarray*}$$

Etcetera. Note the al­ter­nat­ing sign pat­tern of the terms.

As you might in­fer from the above, com­put­ing a good size de­ter­mi­nant takes a large amount of work. For­tu­nately, it is pos­si­ble to sim­plify the ma­trix to put ze­ros in suit­able lo­ca­tions, and that can cut down the work of find­ing the de­ter­mi­nant greatly. You are al­lowed to use the fol­low­ing ma­nip­u­la­tions with­out se­ri­ously af­fect­ing the com­puted de­ter­mi­nant:

1. You can trans­posethe ma­trix, i.e. change its columns into its rows.
2. You can cre­ate ze­ros in a row by sub­tract­ing a suit­able mul­ti­ple of an­other row.
3. You can also swap rows, as long as you re­mem­ber that each time that you swap two rows, it will flip over the sign of the com­puted de­ter­mi­nant.
4. You can also mul­ti­ply an en­tire row by a con­stant, but that will mul­ti­ply the com­puted de­ter­mi­nant by the same con­stant.

Ap­ply­ing these tricks in a sys­tem­atic way, called “Gauss­ian elim­i­na­tion” or “re­duc­tion to up­per tri­an­gu­lar form”, you can elim­i­nate all ma­trix co­ef­fi­cients $a_{ij}$ for which $j$ is less than $i$, and that makes eval­u­at­ing the de­ter­mi­nant pretty much triv­ial.

div(er­gence)    

The di­ver­gence of a vec­tor $\vec{v}$ is de­fined as $\div\vec{v}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\nabla\cdot\vec{v}$.

$\setbox 0=\hbox{$E$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The to­tal en­ergy. Pos­si­ble val­ues are the eigen­val­ues of the Hamil­ton­ian.
• $E_n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $E_1$/$n^2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-{m_{\rm e}}e^4$/$32\pi^2\epsilon_0^2\hbar^2n^2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-\hbar^2$/$2{m_{\rm e}}a_0^2n^2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12\alpha^2{m_{\rm e}}c^2$/$n^2$ may in­di­cate the non­rel­a­tivis­tic (Bohr) en­ergy lev­els of the hy­dro­gen atom. The ground state en­ergy $E_1$ equals -13.605 692 5 eV. This does not in­clude rel­a­tivis­tic and pro­ton mo­tion cor­rec­tions.
• In­ter­nal en­ergy of a sub­stance.

$\setbox 0=\hbox{${\cal E}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Elec­tric field strength.

$\setbox 0=\hbox{$e$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The ba­sis for the nat­ural log­a­rithms. Equal to 2.718 281 828 459... This num­ber pro­duces the ex­po­nen­tial func­tion $e^x$, or $\exp(x)$, or in words $e$ to the power $x$, whose de­riv­a­tive with re­spect to $x$ is again $e^x$. If $a$ is a con­stant, then the de­riv­a­tive of $e^{ax}$ is $ae^{ax}$. Also, if $a$ is an or­di­nary real num­ber, then $e^{{{\rm i}}a}$ is a com­plex num­ber with mag­ni­tude 1.
• The mag­ni­tude of the charge of an elec­tron or pro­ton, equal to 1.602 176 57 10$\POW9,{-19}$ C.
• Emis­siv­ity of elec­tro­mag­netic ra­di­a­tion.
• Of­ten used to in­di­cate a unit vec­tor.
• A su­per­script $e$ may in­di­cate a sin­gle-elec­tron quan­tity.
• Spe­cific in­ter­nal en­ergy of a sub­stance.

e    

May in­di­cate

• Sub­script e may in­di­cate an elec­tron.

$\setbox 0=\hbox{$e^{{{\rm i}}ax}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

As­sum­ing that $a$ is an or­di­nary real num­ber, and $x$ a real vari­able, $e^{{{\rm i}}ax}$ is a com­plex func­tion of mag­ni­tude one. The de­riv­a­tive of $e^{{{\rm i}}ax}$ with re­spect to $x$ is ${{\rm i}}ae^{{{\rm i}}ax}$

eigen­vec­tor    

A con­cept from lin­ear al­ge­bra. A vec­tor $\vec{v}$ is an eigen­vec­tor of a ma­trix $A$ if $\vec{v}$ is nonzero and $A\vec{v}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\lambda\vec{v}$ for some num­ber $\lambda$ called the cor­re­spond­ing eigen­value.

The ba­sic quan­tum me­chan­ics sec­tion of this book avoids lin­ear al­ge­bra com­pletely, and the ad­vanced part al­most com­pletely. The few ex­cep­tions are al­most all two-di­men­sional ma­trix eigen­value prob­lems. In case you did not have any lin­ear al­ge­bra, here is the so­lu­tion: the two-di­men­sional ma­trix eigen­value prob­lem

$$\left(\begin{array}{cc} a_{11}&a_{12} \ a_{21}&a_{22} \end{array}\right) \vec v = \lambda \vec v$$

has eigen­val­ues that are the two roots of the qua­dratic equa­tion

$$\lambda^2 - (a_{11}+a_{22})\lambda + a_{11}a_{22}-a_{12}a_{21} = 0$$

The cor­re­spond­ing eigen­vec­tors are

$$\vec v_1 = \left(\begin{array}{c} a_{12} \ \lambda_1-a_... ...begin{array}{c} \lambda_2-a_{22} \ a_{21}\end{array}\right)$$

On oc­ca­sion you may have to swap $\lambda_1$ and $\lambda_2$ to use these for­mu­lae. If $\lambda_1$ and $\lambda_2$ are equal, there might not be two eigen­vec­tors that are not mul­ti­ples of each other; then the ma­trix is called de­fec­tive. How­ever, Her­mit­ian ma­tri­ces are never de­fec­tive.

See also ma­trix” and “de­ter­mi­nant.

eV    

The elec­tron volt, a com­monly used unit of en­ergy. Its value is equal to 1.602 176 57 10$\POW9,{-19}$ J.

ex­po­nen­tial func­tion    

A func­tion of the form $e^{\ldots}$, also writ­ten as $\exp(\ldots)$. See func­tion and $e$.

$\setbox 0=\hbox{$F$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The force in New­ton­ian me­chan­ics. Equal to the neg­a­tive gra­di­ent of the po­ten­tial. Quan­tum me­chan­ics is for­mu­lated in terms of po­ten­tials, not forces.
• The anti-de­riv­a­tive of some func­tion $f$.
• Some func­tion.
• Helmholtz free en­ergy.

$\setbox 0=\hbox{${\cal F}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Fock op­er­a­tor.

$\setbox 0=\hbox{$f$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• A generic func­tion.
• A generic vec­tor.
• A frac­tion.
• The res­o­nance fac­tor.
• Spe­cific Helmholtz free en­ergy.
• Fre­quency.

func­tion    

A math­e­mat­i­cal ob­ject that as­so­ciates val­ues with other val­ues. A func­tion $f(x)$ as­so­ciates every value of $x$ with a value $f$. For ex­am­ple, the func­tion $f(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $x^2$ as­so­ciates $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 with $f$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12$ with $f$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac14$, $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 with $f$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2 with $f$ $\vphantom0\raisebox{1.5pt}{$=$}$ 4, $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 3 with $f$ $\vphantom0\raisebox{1.5pt}{$=$}$ 9, and more gen­er­ally, any ar­bi­trary value of $x$ with the square of that value $x^2$. Sim­i­larly, func­tion $f(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $x^3$ as­so­ciates any ar­bi­trary $x$ with its cube $x^3$, $f(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sin(x)$ as­so­ciates any ar­bi­trary $x$ with the sine of that value, etcetera.

One way of think­ing of a func­tion is as a pro­ce­dure that al­lows you, when­ever given a num­ber, to com­pute an­other num­ber.

A wave func­tion $\Psi(x,y,z)$ as­so­ciates each spa­tial po­si­tion $(x,y,z)$ with a wave func­tion value. Go­ing be­yond math­e­mat­ics, its square mag­ni­tude as­so­ciates any spa­tial po­si­tion with the rel­a­tive prob­a­bil­ity of find­ing the par­ti­cle near there.

func­tional    

A func­tional as­so­ciates en­tire func­tions with sin­gle num­bers. For ex­am­ple, the ex­pec­ta­tion en­ergy is math­e­mat­i­cally a func­tional: it as­so­ciates any ar­bi­trary wave func­tion with a num­ber: the value of the ex­pec­ta­tion en­ergy if physics is de­scribed by that wave func­tion.

$\setbox 0=\hbox{$G$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Gibbs free en­ergy.
• New­ton’s con­stant of grav­i­ta­tion, 6.673 8 10$\POW9,{-11}$ m$\POW9,{3}$/kg s$\POW9,{2}$.

$\setbox 0=\hbox{$g$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• A sec­ond generic func­tion or a sec­ond generic vec­tor.
• The strength of grav­ity, by de­f­i­n­i­tion equal to 9.806 65 m/s$\POW9,{2}$ ex­actly un­der stan­dard con­di­tions on the sur­face of the earth.
• The g-fac­tor, a nondi­men­sional con­stant that in­di­cates the gy­ro­mag­netic ra­tio rel­a­tive to charge and mass. For the elec­tron $g_e$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$2.002 319 304 361 5. For the pro­ton $g_p$ $\vphantom0\raisebox{1.5pt}{$=$}$ 5.585 694 71. For the neu­tron, based on the mass and charge of the pro­ton, $g_n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$3.826 085 5.
• Spe­cific Gibbs free en­ergy/chem­i­cal po­ten­tial.

Gauss' The­o­rem    

This the­o­rem, also called di­ver­gence the­o­rem or Gauss-Os­tro­grad­sky the­o­rem, says that for a con­tin­u­ously dif­fer­en­tiable vec­tor $\vec{v}$,

$$\int_V \nabla \cdot \vec v { \rm d}V = \int_A \vec v \cdot \vec n { \rm d}A$$

where the first in­te­gral is over the vol­ume of an ar­bi­trary re­gion and the sec­ond in­te­gral is over all the sur­face area of that re­gion; $\vec{n}$ is at each point found as the unit vec­tor that is nor­mal to the sur­face at that point.

grad(ient)    

The gra­di­ent of a scalar $f$ is de­fined as $\mathop{\rm grad}\nolimits {f}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\nabla{f}$.

$\setbox 0=\hbox{$H$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The Hamil­ton­ian, or to­tal en­ergy, op­er­a­tor. Its eigen­val­ues are in­di­cated by $E$.
• $H_n$ stands for the $n$-th or­der Her­mite poly­no­mial.
• En­thalpy.

$\setbox 0=\hbox{$h$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The orig­i­nal Planck con­stant $h$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\pi\hbar$.
• $h_n$ is a one-di­men­sional har­monic os­cil­la­tor eigen­func­tion.
• Sin­gle-elec­tron Hamil­ton­ian.
• Spe­cific en­thalpy.

$\setbox 0=\hbox{$\hbar$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The re­duced Planck con­stant, equal to 1.054 571 73 10$\POW9,{-34}$ J s. A mea­sure of the un­cer­tainty of na­ture in quan­tum me­chan­ics. Mul­ti­ply by $2\pi$ to get the orig­i­nal Planck con­stant $h$. For nu­clear physics, a fre­quently help­ful value is $\hbar{c}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 197.326 972 MeV fm.

hy­per­sphere    

A hy­per­sphere is the gen­er­al­iza­tion of the nor­mal three-di­men­sion­al sphere to $n$-di­men­sion­al space. A sphere of ra­dius $R$ in three-di­men­sion­al space con­sists of all points sat­is­fy­ing

$$r_1^2 + r_2^2 + r_3^2 \mathrel{\raisebox{-.7pt}{$\leqslant$}}R^2$$

where $r_1$, $r_2$, and $r_3$ are Carte­sian co­or­di­nates with ori­gin at the cen­ter of the sphere. Sim­i­larly a hy­per­sphere in $n$-di­men­sion­al space is de­fined as all points sat­is­fy­ing

$$r_1^2 + r_2^2 + \ldots + r_n^2 \mathrel{\raisebox{-.7pt}{$\leqslant$}}R^2$$

So a two-di­men­sion­al hy­per­sphere of ra­dius $R$ is re­ally just a cir­cle of ra­dius $R$. A one-di­men­sion­al hy­per­sphere is re­ally just the line seg­ment $\vphantom{0}\raisebox{1.5pt}{$-$}$$R$ $\raisebox{-.3pt}{$\leqslant$}$ $x$ $\raisebox{-.3pt}{$\leqslant$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$$R$.

The vol­ume” ${\cal V}_n$ and sur­face “area $A_n$ of an $n$-di­men­sion­al hy­per­sphere is given by

$${\cal V}_n = C_n R^n \qquad A_n = n C_n R^{n-1}$$

$$C_n = \left\{ \begin{array}{l} \strut(2\pi)^{n/2}/2 \t... ...imes n \quad \mbox{if $n$ is odd} \end{array} \right.$$

(This is read­ily de­rived re­cur­sively. For a sphere of unit ra­dius, note that the $n$-di­men­sion­al vol­ume is an in­te­gra­tion of $n{-}1$-di­men­sion­al vol­umes with re­spect to $r_1$. Then reno­tate $r_1$ as $\sin\phi$ and look up the re­sult­ing in­te­gral in a ta­ble book. The for­mula for the area fol­lows be­cause ${\cal V}=\int{A}{\rm d}{r}$ where $r$ is the dis­tance from the ori­gin.) In three di­men­sions, $C_3=4\pi/3$ ac­cord­ing to the above for­mula. That makes the three-di­men­sion­al vol­ume $4\pi{R}^3$$\raisebox{.5pt}{$/$}$​3 equal to the ac­tual vol­ume of the sphere, and the three-di­men­sion­al area $4{\pi}R^2$ equal to the ac­tual sur­face area. On the other hand in two di­men­sions, $C_2=\pi$. That makes the two-di­men­sion­al vol­ume ${\pi}R^2$ re­ally the area of the cir­cle. Sim­i­larly the two-di­men­sion­al sur­face area $2{\pi}R$ is re­ally the perime­ter of the cir­cle. In one di­men­sions $C_1=2$ and the vol­ume $2R$ is re­ally the length of the in­ter­val, and the area 2 is re­ally its num­ber of end points.

Of­ten the in­fin­i­tes­i­mal $n$-di­men­sion­al vol­ume el­e­ment ${\rm d}^n{\skew0\vec r}$ is needed. This is the in­fin­i­tes­i­mal in­te­gra­tion el­e­ment for in­te­gra­tion over all co­or­di­nates. It is:

$${\rm d}^n{\skew0\vec r}= {\rm d}r_1 {\rm d}r_2 \ldots {\rm d}r_n = {\rm d}A_n {\rm d}r$$

Specif­i­cally, in two di­men­sions:

$${\rm d}^2{\skew0\vec r}= {\rm d}r_1 {\rm d}r_2 = {\rm d}x {\rm d}y = (r { \rm d}\theta) {\rm d}r = {\rm d}A_2 {\rm d}r$$

while in three di­men­sions:

$${\rm d}^3{\skew0\vec r}= {\rm d}r_1 {\rm d}r_2 {\rm d}r_3 ... ... { \rm d}\theta {\rm d}\phi) {\rm d}r = {\rm d}A_3 {\rm d}r$$

The ex­pres­sions in paren­the­ses are ${\rm d}{A}_2$ in po­lar co­or­di­nates, re­spec­tively ${\rm d}{A}_3$ in spher­i­cal co­or­di­nates.

$\setbox 0=\hbox{$I$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The num­ber of elec­trons or par­ti­cles.
• Elec­tri­cal cur­rent.
• Unit ma­trix or op­er­a­tor, which does not do any­thing. See ma­trix.
• $I_{\rm A}$ is Avo­gadro’s num­ber, 6.022 141 3 10$\POW9,{26}$ par­ti­cles per kmol. (More stan­dard sym­bols are $N_{\rm {A}}$ or $L$, but they are in­com­pat­i­ble with the gen­eral no­ta­tions in this book.)

$\setbox 0=\hbox{$\Im$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The imag­i­nary part of a com­plex num­ber. If $c$ $\vphantom0\raisebox{1.5pt}{$=$}$ $c_r+{{\rm i}}c_i$ with $c_r$ and $c_i$ real num­bers, then $\Im(c)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $c_i$. Note that $c-c^*$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2{\rm i}\Im(c)$.

$\setbox 0=\hbox{${\cal I}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• ${\cal I}$ is ra­di­a­tion en­ergy in­ten­sity.
• ${\cal I}_{\rm {R}}$ is mo­ment of in­er­tia.

$\setbox 0=\hbox{$i$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The num­ber of a par­ti­cle.
• A sum­ma­tion in­dex.
• A generic in­dex or counter.

Not to be con­fused with ${\rm i}$.

$\setbox 0=\hbox{${\hat\imath}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The unit vec­tor in the $x$-di­rec­tion.

$\setbox 0=\hbox{${\rm i}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The stan­dard square root of mi­nus one: ${\rm i}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{-1}$, ${\rm i}^2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$1, 1/${\rm i}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-{\rm i}$, ${\rm i}^*$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-{\rm i}$.

in­dex no­ta­tion    

A more con­cise and pow­er­ful way of writ­ing vec­tor and ma­trix com­po­nents by us­ing a nu­mer­i­cal in­dex to in­di­cate the com­po­nents. For Carte­sian co­or­di­nates, you might num­ber the co­or­di­nates $x$ as 1, $y$ as 2, and $z$ as 3. In that case, a sum like $v_x+v_y+v_z$ can be more con­cisely writ­ten as $\sum_i{v}_i$. And a state­ment like $v_x$ $\raisebox{.2pt}{$\ne$}$ 0, $v_y$ $\raisebox{.2pt}{$\ne$}$ 0, $v_z$ $\raisebox{.2pt}{$\ne$}$ 0 can be more com­pactly writ­ten as $v_i$ $\raisebox{.2pt}{$\ne$}$ 0. To re­ally see how it sim­pli­fies the no­ta­tions, have a look at the ma­trix en­try. (And that one shows only 2 by 2 ma­tri­ces. Just imag­ine 100 by 100 ma­tri­ces.)

iff    

Em­phatic if. Should be read as if and only if.

in­te­ger    

In­te­ger num­bers are the whole num­bers: $\ldots,-2,-1,0,1,2,3,4,\ldots$.

in­verse    

(Of ma­tri­ces or op­er­a­tors.) If an op­er­a­tor $A$ con­verts a vec­tor or func­tion $f$ into a vec­tor or func­tion $g$, then the in­verse of the op­er­a­tor $A^{-1}$ con­verts $g$ back into $f$. For ex­am­ple, the op­er­a­tor 2 con­verts vec­tors or func­tions into two times them­selves, and its in­verse op­er­a­tor $\frac12$ con­verts these back into the orig­i­nals. Some op­er­a­tors do not have in­verses. For ex­am­ple, the op­er­a­tor 0 con­verts all vec­tors or func­tions into zero. But given zero, there is no way to fig­ure out what func­tion or vec­tor it came from; the in­verse op­er­a­tor does not ex­ist.

ir­ro­ta­tional    

A vec­tor $\vec{v}$ is ir­ro­ta­tional if its curl $\nabla$ $\times$ $\vec{v}$ is zero.

iso    

Means equal” or “con­stant.

• Isen­thalpic: con­stant en­thalpy.
• Isen­tropic: con­stant en­tropy. This is a process that is both adi­a­batic and re­versible.
• Iso­baric: con­stant pres­sure.
• Iso­choric: con­stant (spe­cific) vol­ume.
• Isospin: you don’t want to know.
• Isother­mal: con­stant tem­per­a­ture.

iso­lated    

An iso­lated sys­tem is one that does not in­ter­act with its sur­round­ings in any way. No heat is trans­fered with the sur­round­ings, no work is done on or by the sur­round­ings.

$\setbox 0=\hbox{$J$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• To­tal an­gu­lar mo­men­tum.
• Num­ber of nu­clei in a quan­tum com­pu­ta­tion of elec­tronic struc­ture.

$\setbox 0=\hbox{$j$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The az­imuthal quan­tum num­ber of to­tal an­gu­lar mo­men­tum, in­clud­ing both or­bital and spin con­tri­bu­tions.
• $\vec{j}$ is elec­tric cur­rent den­sity.
• The num­ber of a nu­cleus in a quan­tum com­pu­ta­tion.
• A sum­ma­tion in­dex.
• A generic in­dex or counter.

$\setbox 0=\hbox{${\hat\jmath}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The unit vec­tor in the $y$-di­rec­tion.

$\setbox 0=\hbox{$K$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• An ex­change in­te­gral in Hartree-Fock.
• Max­i­mum wave num­ber value.

$\setbox 0=\hbox{${\mathscr K}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Thom­son (Kelvin) co­ef­fi­cient.

K    

May in­di­cate:

• The atomic states or or­bitals with the­o­ret­i­cal Bohr en­ergy $E_1$
• De­grees Kelvin.

$\setbox 0=\hbox{$k$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• A wave num­ber. A wave num­ber is a mea­sure for how fast a pe­ri­odic func­tion os­cil­lates with vari­a­tions in spa­tial po­si­tion. In quan­tum me­chan­ics, $k$ is nor­mally de­fined as $\sqrt{2m(E-V)}$/$\hbar$. The vec­tor ${\vec k}$ is not to be con­fused with the unit vec­tor in the $z$-di­rec­tion ${\hat k}$.
• A generic sum­ma­tion in­dex.

$\setbox 0=\hbox{${\hat k}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The unit vec­tor in the $z$-di­rec­tion.

$\setbox 0=\hbox{$k_{\rm B}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Boltz­mann con­stant. Equal to 1.380 649 10$\POW9,{-23}$ J/K. Re­lates ab­solute tem­per­a­ture to a typ­i­cal unit of heat mo­tion en­ergy.

kmol    

A kilo mole refers to 6.022 141 3 10$\POW9,{26}$ atoms or mol­e­cules. The weight of this many par­ti­cles is about the num­ber of pro­tons and neu­trons in the atom nu­cleus/mol­e­cule nu­clei. So a kmol of hy­dro­gen atoms has a mass of about 1 kg, and a kmol of hy­dro­gen mol­e­cules about 2 kg. A kmol of he­lium atoms has a mass of about 4 kg, since he­lium has two pro­tons and two neu­trons in its nu­cleus. These num­bers are not very ac­cu­rate, not just be­cause the elec­tron masses are ig­nored, and the free neu­tron and pro­ton masses are some­what dif­fer­ent, but also be­cause of rel­a­tiv­ity ef­fects that cause ac­tual nu­clear masses to de­vi­ate from the sum of the free pro­ton and neu­tron masses.

$\setbox 0=\hbox{$L$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• An­gu­lar mo­men­tum.
• Or­bital an­gu­lar mo­men­tum.

$\setbox 0=\hbox{${\cal L}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

La­grangian.

L    

The atomic states or or­bitals with the­o­ret­i­cal Bohr en­ergy $E_2$

$\setbox 0=\hbox{$l$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The az­imuthal quan­tum num­ber of an­gu­lar mo­men­tum.
• The az­imuthal quan­tum num­ber of or­bital an­gu­lar mo­men­tum. Here $s$ is used for spin, and $j$ for com­bined an­gu­lar mo­men­tum.)
• A generic sum­ma­tion in­dex.

$\setbox 0=\hbox{$\ell$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The typ­i­cal length in the har­monic os­cil­la­tor prob­lem.
• The di­men­sions of a solid block (with sub­scripts).
• A length.
• Mul­ti­pole level in tran­si­tions.

$\setbox 0=\hbox{$\pounds $}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

La­grangian den­sity. This is best un­der­stood in the UK.

$\setbox 0=\hbox{$\lim$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

In­di­cates the fi­nal re­sult of an ap­proach­ing process. $\lim_{\varepsilon\to0}$ in­di­cates for prac­ti­cal pur­poses the value of the fol­low­ing ex­pres­sion when $\varepsilon$ is ex­tremely small.

lin­ear com­bi­na­tion    

A very generic con­cept in­di­cat­ing sums of ob­jects times co­ef­fi­cients. For ex­am­ple, a po­si­tion vec­tor ${\skew0\vec r}$ in ba­sic physics is the lin­ear com­bi­na­tion $x{\hat\imath}+y{\hat\jmath}+z{\hat k}$ with the ob­jects the unit vec­tors ${\hat\imath}$, ${\hat\jmath}$, and ${\hat k}$ and the co­ef­fi­cients the po­si­tion co­or­di­nates $x$, $y$, and $z$. A lin­ear com­bi­na­tion of a set of func­tions $f_1(x),f_2(x),f_3(x),\ldots,f_n(x)$ would be the func­tion

$$c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) + \ldots c_n f_n(x)$$

where $c_1,c_2,c_3,\ldots,c_n$ are con­stants, i.e. in­de­pen­dent of $x$.

lin­ear de­pen­dence    

A set of vec­tors or func­tions is lin­early de­pen­dent if at least one of the set can be ex­pressed in terms of the oth­ers. Con­sider the ex­am­ple of a set of func­tions $f_1(x),f_2(x),\ldots,f_n(x)$. This set is lin­early de­pen­dent if

$$c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) + \ldots c_n f_n(x) = 0$$

where at least one of the con­stants $c_1,c_2,c_2,\ldots,c_n$ is nonzero. To see why, sup­pose that say $c_2$ is nonzero. Then you can di­vide by $c_2$ and re­arrange to get

$$f_2(x) = - \frac{c_1}{c_2} f_1(x) - \frac{c_3}{c_2} f_3(x) - \ldots - \frac{c_n}{c_2} f_n(x)$$

That ex­presses $f_2(x)$ in terms of the other func­tions.

lin­ear in­de­pen­dence    

A set of vec­tors or func­tions is lin­early in­de­pen­dent if none of the set can be ex­pressed in terms of the oth­ers. Con­sider the ex­am­ple of a set of func­tions $f_1(x),f_2(x),\ldots,f_n(x)$. This set is lin­early in­de­pen­dent if

$$c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) + \ldots c_n f_n(x) = 0$$

only if every one of the con­stants $c_1,c_2,c_3,\ldots,c_n$ is zero. To see why, as­sume that say $f_2(x)$ could be ex­pressed in terms of the oth­ers,

$$f_2(x) = C_1 f_1(x) + C_3 f_3(x) + \ldots + C_n f_n(x)$$

Then tak­ing $c_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, $c_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-C_1$, $c_3$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-C_3$, ...$c_n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-C_n$, the con­di­tion above would be vi­o­lated. So $f_2$ can­not be ex­pressed in terms of the oth­ers.

$\setbox 0=\hbox{$M$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Mol­e­c­u­lar mass. See mol­e­c­u­lar mass.
• Fig­ure of merit.

$\setbox 0=\hbox{${\cal M}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Mir­ror op­er­a­tor.

M    

The atomic states or or­bitals with the­o­ret­i­cal Bohr en­ergy $E_3$

$\setbox 0=\hbox{$m$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Mass.
  • $m_{\rm e}$: elec­tron mass. Equal to 9.109 382 9 10$\POW9,{-31}$ kg. The rest mass en­ergy is 0.510 998 93 MeV.
  • $m_{\rm p}$: pro­ton mass. Equal to 1.672 621 78 10$\POW9,{-27}$ kg. The rest mass en­ergy is 938.272 013 MeV.
  • $m_{\rm n}$: neu­tron mass. Equal to 1.674 927 10$\POW9,{-27}$ kg. The rest mass en­ergy is 939.565 561 MeV.
  • Al­pha par­ti­cle: 6.644 656 8 10$\POW9,{-27}$ kg or 3 727.379 24 MeV.
    Deuteron: 3.343 583 5 10$\POW9,{-27}$ kg or 1 875.612 86 MeV.
    He­lion: 5.006 412 3 10$\POW9,{-27}$ kg or 2 808.391 482 MeV.
  • $m_{\rm {u}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.660 538 92 10$\POW9,{-27}$ kg is the atomic mass con­stant.
  • $m$: generic par­ti­cle mass.
• The mag­netic quan­tum num­ber of an­gu­lar mo­men­tum. The type odf an­gu­lar mo­men­tum may be in­di­cated by a sub­script $l$ for or­bital, $s$ for spin, or $j$ for net (or­bital plus spin).
• Num­ber of a sin­gle-elec­tron wave func­tion.
• Num­ber of rows in a ma­trix.
• A generic sum­ma­tion in­dex or generic in­te­ger.

ma­trix    

A ta­ble of num­bers.

As a sim­ple ex­am­ple, a two-di­men­sional (or $2\times2$) ma­trix $A$ is a ta­ble of four num­bers called $a_{11}$, $a_{12}$, $a_{21}$, and $a_{22}$:

$$A = \left( \begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right)$$

un­like a two-di­men­sional vec­tor $\vec{v}$, which would con­sist of only two num­bers $v_1$ and $v_2$ arranged in a col­umn:

$$\vec v = \left( \begin{array}{l} v_1 \\ v_2 \end{array} \right)$$

(Such a vec­tor can be seen as a rec­tan­gu­lar ma­trix of size 2 $\times$ 1, but let’s not get into that.) (Note that in quan­tum me­chan­ics, if a vec­tor is writ­ten as a col­umn, con­sid­ered the nor­mal case, it is called a ket vec­tor. If the com­plex con­ju­gates of its num­bers are writ­ten as a row, it is called a bra vec­tor.)

In in­dex no­ta­tion, a ma­trix $A$ is a set of num­bers, or co­ef­fi­cients, $\{a_{ij}\}$ in­dexed by two in­dices. The first in­dex $i$ is the row num­ber at which the co­ef­fi­cient $\{a_{ij}\}$ is found in ma­trix $A$, and the sec­ond in­dex $j$ is the col­umn num­ber. In in­dex no­ta­tion, a ma­trix turns a vec­tor $\vec{v}$ into an­other vec­tor $\vec{w}=A\vec{v}$ ac­cord­ing to the recipe

$$w_i = \sum_{\mbox{{\scriptsize all }}j} a_{ij} v_j \quad \mbox{for all $i$}$$

where $v_j$ stands for “the $j$-th com­po­nent of vec­tor $\vec{v}$,” and $w_i$ for “the $i$-th com­po­nent of vec­tor $\vec{w}$.”

As an ex­am­ple, the prod­uct of $A$ and $\vec{v}$ above is by de­f­i­n­i­tion

$$\left( \begin{array}{ll} a_{11} & a_{12} \\ a_{21} ... ...2} v_2 \\ a_{21} v_1 + a_{22} v_2 \end{array} \right)$$

which is just an­other two-di­men­sional ket vec­tor.

Note that in ma­trix mul­ti­pli­ca­tions, like in the ex­am­ple above, in geo­met­ric terms you take dot prod­ucts be­tween the rows of the first fac­tor and the columns of the sec­ond fac­tor.

To mul­ti­ply two ma­tri­ces to­gether, just think of the columns of the sec­ond ma­trix as sep­a­rate vec­tors. For ex­am­ple, to mul­ti­ply two $2\times2$ ma­tri­ces $A$ and $B$ to­gether:

$$\left( \begin{array}{ll} a_{11} & a_{12} \\ a_{21} ... ...{21} & a_{21} b_{12} + a_{22} b_{22} \end{array} \right)$$

which is an­other two-di­men­sional ma­trix.

(Note that you can­not nor­mally swap the or­der of ma­tric mul­ti­pli­ca­tion. The ma­trix $BA$ is dif­fer­ent from ma­trix $AB$. In the spe­cial case that $AB$ and $BA$ are the same and $A$ and $B$ have com­plete sets of eigen­vec­tors, then they have a com­mon com­plete set of eigen­vec­tors, {D.18}.)

In in­dex no­ta­tion, if $C=AB$, then each co­ef­fi­cient $c_{ij}$ of ma­trix $C$ is given in terms of the co­ef­fi­cients of $A$ and $B$ as

$$c_{ij} = \sum_k a_{ik} b_{kj}$$

Note that the in­dex $k$ that you sum over is the sec­ond of $A$ but the first of $B$. In short, you sum over “neigh­bor­ing in­dices.” Since you sum over all $k$, the re­sult does not de­pend on $k$.

The zero ma­trix, usu­ally called $Z$, is like the num­ber zero; it does not change a ma­trix it is added to. And it turns what­ever it is mul­ti­plied with into zero. A zero ma­trix has every co­ef­fi­cient zero. For ex­am­ple, in two di­men­sions:

$$Z = \left( \begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array} \right)$$

A unit, or iden­tity, ma­trix, usu­ally called $I$, is the equiv­a­lent of the num­ber one for ma­tri­ces; it does not change the vec­tor or ma­trix it is mul­ti­plied with. A unit ma­trix is one on its main di­ag­o­nal $i=j$ and zero else­where. The 2 by 2 unit ma­trix is:

$$I = \left( \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array} \right)$$

More gen­er­ally the co­ef­fi­cients, $\{\delta_{ij}\}$, of a unit ma­trix are one if $i$ $\vphantom0\raisebox{1.5pt}{$=$}$ $j$ and zero oth­er­wise.

The trans­pose of a ma­trix $A$, $A^{\rm {T}}$, is what you get if you swap the two in­dices. Graph­i­cally, it turns its rows into its columns and vice versa. The ad­joint or Her­mit­ian ad­joint ma­trix $A^\dagger$ is what you get if you both swap the two in­dices in a ma­trix $A$ and then take the com­plex con­ju­gate of every co­ef­fi­cient. If you want to take a ma­trix to the other side of an in­ner prod­uct, you will need to change it to its Her­mit­ian ad­joint. Her­mit­ian ma­tri­cesare equal to their Her­mit­ian ad­joint, so this does noth­ing for them.

The in­verse of a ma­trix $A$, $A^{-1}$ is a ma­trix so that $A^{-1}A$ equals the iden­tity ma­trix $I$. That is much like the in­verse of a sim­ple num­ber times that num­ber gives one. And, just like the num­ber zero has no in­verse, a ma­trix with zero de­ter­mi­nant has no in­verse. Oth­er­wise, you can swap the or­der; $AA^{-1}$ equals the unit ma­trix too. (For num­bers this is triv­ial, for ma­tri­ces you need to look a bit closer to un­der­stand why it is true.)

See also de­ter­mi­nant and eigen­vec­tor.

met­ric pre­fixes    

In the met­ric sys­tem, the pre­fixes Y, Z, E, P, T, G, M, and k stand for 10$\POW9,{i}$ with $i$ $\vphantom0\raisebox{1.5pt}{$=$}$ 24, 21, 18, 15, 12, 9, 6, and 3, re­spec­tively. Sim­i­larly, d, c, m, $\mu$, n, p, f, a, z, y stand for 10$\POW9,{-i}$ with $i$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, 2, 3, 6, 9, 12, 15, 18, 21, and 24 re­spec­tively. For ex­am­ple, 1 ns is 10$\POW9,{-9}$ sec­onds. Eng­lish let­ter u is of­ten used as in­stead of greek $\mu$. Names cor­re­spond­ing to the men­tioned pre­fixes Y-k are yotta, zetta, exa, peta, tera, giga, mega, kilo, and cor­re­spond­ing to d-y are deci, centi, milli, mi­cro, nano, pico, femto, atto, zepto, and yocto.

mol­e­c­u­lar mass    

Typ­i­cal ther­mo­dy­nam­ics books for en­gi­neers tab­u­late val­ues of the mol­e­c­u­lar mass, as a nondi­men­sional num­ber. The bot­tom line first: these num­bers should have been called the mo­lar mass of the sub­stance, for the nat­u­rally oc­cur­ring iso­tope ra­tio on earth. And they should have been given units of kg/kmol. That is how you use these num­bers in ac­tual com­pu­ta­tions. So just ig­nore the fact that what these books re­ally tab­u­late is of­fi­cially called the rel­a­tive mol­e­c­u­lar mass for the nat­ural iso­tope ra­tio.

Don’t blame these text­books too much for mak­ing a mess of things. Physi­cists have his­tor­i­cally bandied about a zil­lion dif­fer­ent names for what is es­sen­tially a sin­gle num­ber. Like mol­e­c­u­lar mass, “rel­a­tive mol­e­c­u­lar mass,” mol­e­c­u­lar weight, “atomic mass,” rel­a­tive atomic mass, “atomic weight,” mo­lar mass, “rel­a­tive mo­lar mass,” etcetera are ba­si­cally all the same thing.

All of these have val­ues that equal the mass of a mol­e­cule rel­a­tive to a ref­er­ence value for a sin­gle nu­cleon. So these value are about equal to the num­ber of nu­cle­ons (pro­tons and neu­trons) in the nu­clei of a sin­gle mol­e­cule. (For an iso­tope ra­tio, that be­comes the av­er­age num­ber of nu­cle­ons. Do note that nu­clei are suf­fi­ciently rel­a­tivis­tic that a pro­ton or neu­tron can be no­tice­ably heav­ier in one nu­cleus than an­other, and that neu­trons are a bit heav­ier than pro­tons even in iso­la­tion.) The of­fi­cial ref­er­ence nu­cleon weight is de­fined based on the most com­mon car­bon iso­tope car­bon-12. Since car­bon-12 has 6 pro­tons plus 6 neu­trons, the ref­er­ence nu­cleon weight is taken to be one twelfth of the car­bon-12 atomic weight. That is called the uni­fied atomic mass unit (u) or Dal­ton (Da). The atomic mass unit (amu) is an older vir­tu­ally iden­ti­cal unit, but physi­cists and chemists could never quite agree on what its value was. No kid­ding.

If you want to be po­lit­i­cally cor­rect, the deal is as fol­lows. Mol­e­c­u­lar mass is just what the term says, the mass of a mol­e­cule, in mass units. (I found zero ev­i­dence in ei­ther the IU­PAC Gold Book or NIST SP811 for the claim of Wikipedia that it must al­ways be ex­pressed in u.) Mo­lar mass is just what the words says, the mass of a mole. Of­fi­cial SI units are kg/mol, but you will find it in g/mol, equiv­a­lent to kg/kmol. (You can­not ex­pect enough brains from in­ter­na­tional com­mit­tees to re­al­ize that if you de­fine the kg and not the g as unit of mass, then it would be a smart idea to also de­fine kmol in­stead of mol as unit of par­ti­cle count.) Sim­ply ig­nore rel­a­tive atomic and mol­e­c­u­lar masses, you do not care about them. (I found zero ev­i­dence in ei­ther the IU­PAC Gold Book or NIST SP811 for the claims of Wikipedia that the mol­e­c­u­lar mass can­not be an av­er­age over iso­topes or that the mo­lar mass must be for a nat­ural iso­tope ra­tio. In fact, NIST uses mo­lar mass of car­bon-12 and specif­i­cally in­cludes the pos­si­bil­ity of an av­er­age in the rel­a­tive mol­e­c­u­lar mass.)

See also the atomic mass con­stant $m_{\rm {u}}$.

$\setbox 0=\hbox{$N$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Num­ber of states.
• Num­ber of sin­gle-par­ti­cle states.
• Num­ber of neu­trons in a nu­cleus.

N    

May in­di­cate

• The atomic states or or­bitals with the­o­ret­i­cal Bohr en­ergy $E_4$.
• Sub­script N in­di­cates a nu­cleus.

$\setbox 0=\hbox{$n$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The prin­ci­pal quan­tum num­ber for hy­dro­gen atom en­ergy eigen­func­tions.
• A quan­tum num­ber for har­monic os­cil­la­tor en­ergy eigen­func­tions.
• Num­ber of a sin­gle-elec­tron or sin­gle-par­ti­cle wave func­tion.
• Generic sum­ma­tion in­dex over en­ergy eigen­func­tions.
• Generic sum­ma­tion in­dex over other eigen­func­tions.
• In­te­ger fac­tor in Fourier wave num­bers.
• Prob­a­bil­ity den­sity.
• Num­ber of columns in a ma­trix.
• A generic sum­ma­tion in­dex or generic in­te­ger.
• A nat­ural num­ber.
• $n_s$ is the num­ber of spin states.

and maybe some other stuff.

n    

May in­di­cate

• A sub­script n may in­di­cate a neu­tron.

nat­ural    

Nat­ural num­bers are the num­bers: $1,2,3,4,\ldots$.

nor­mal    

A nor­mal op­er­a­tor or ma­trix is one that has or­tho­nor­mal eigen­func­tions or eigen­vec­tors. Since eigen­vec­tors are not or­tho­nor­mal in gen­eral, a nor­mal op­er­a­tor or ma­trix is ab­nor­mal! An­other ex­am­ple of a highly con­fus­ing term. Such a ma­trix should have been called or­tho­di­ag­o­nal­iz­able or some­thing of the kind. To be fair, the au­thor is not aware of any physi­cists be­ing in­volved in this par­tic­u­lar term; it may be the math­e­mati­cians that are to blame here.

For an op­er­a­tor or ma­trix $A$ to be nor­mal, it must com­mute with its Her­mit­ian ad­joint, $[A,A^\dagger]$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Her­mit­ian ma­tri­ces are nor­mal since they are equal to their Her­mit­ian ad­joint. Skew-Her­mit­ian ma­tri­ces are nor­mal since they are equal to the neg­a­tive of their Her­mit­ian ad­joint. Uni­tary ma­tri­ces are nor­mal be­cause they are the in­verse of their Her­mit­ian ad­joint.

O    

May in­di­cate the ori­gin of the co­or­di­nate sys­tem.

op­po­site    

The op­po­site of a num­ber $a$ is $-a$. In other words, it is the ad­di­tive in­verse.

$\setbox 0=\hbox{$P$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The lin­ear mo­men­tum eigen­func­tion.
• A power se­ries so­lu­tion.
• Prob­a­bil­ity.
• Pres­sure.
• Her­mit­ian part of an an­ni­hi­la­tion op­er­a­tor.

$\setbox 0=\hbox{${\cal P}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Par­ti­cle ex­change op­er­a­tor. Ex­changes the po­si­tions and spins of two iden­ti­cal par­ti­cles.

$\setbox 0=\hbox{${\mathscr P}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Peltier co­ef­fi­cient.

P    

Of­ten used to in­di­cate a state with one unit of or­bital an­gu­lar mo­men­tum.

$\setbox 0=\hbox{$p$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Lin­ear mo­men­tum.
• Lin­ear mo­men­tum in the $x$-di­rec­tion.
• In­te­gra­tion vari­able with units of lin­ear mo­men­tum.

p    

May in­di­cate

• An en­ergy state with or­bital az­imuthal quan­tum num­ber $l$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.
• A su­per­script p may in­di­cate a sin­gle-par­ti­cle quan­tity.
• A sub­script p may in­di­cate a pe­ri­odic func­tion.
• A sub­script p may in­di­cate a pro­ton.

per­pen­dic­u­lar bi­sec­tor    

For two given points $P$ and $Q$, the per­pen­dic­u­lar bi­sec­tor con­sists of all points $R$ that are equally far from $P$ as they are from $Q$. In two di­men­sions, the per­pen­dic­u­lar bi­sec­tor is the line that passes through the point ex­actly half way in be­tween $P$ and $Q$, and that is or­thog­o­nal to the line con­nect­ing $P$ and $Q$. In three di­men­sions, the per­pen­dic­u­lar bi­sec­tor is the plane that passes through the point ex­actly half way in be­tween $P$ and $Q$, and that is or­thog­o­nal to the line con­nect­ing $P$ and $Q$. In vec­tor no­ta­tion, the per­pen­dic­u­lar bi­sec­tor of points $P$ and $Q$ is all points $R$ whose ra­dius vec­tor ${\skew0\vec r}$ sat­is­fies the equa­tion:

$$({\skew0\vec r}-{\skew0\vec r}_P)\cdot({\skew0\vec r}_Q-{\... ..._Q-{\skew0\vec r}_P)\cdot({\skew0\vec r}_Q-{\skew0\vec r}_P)$$

(Note that the halfway point ${\skew0\vec r}-{\skew0\vec r}_P$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\textstyle\frac{1}{2}}({\skew0\vec r}_Q-{\skew0\vec r}_P)$ is in­cluded in this for­mula, as is the half way point plus any vec­tor that is nor­mal to $({\skew0\vec r}_Q-{\skew0\vec r}_P)$.)

phase an­gle    

Any com­plex num­ber can be writ­ten in po­lar form as $c$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vert c\vert e^{{\rm i}\alpha}$ where both the mag­ni­tude $\vert c\vert$ and the phase an­gle $\alpha$ are real num­bers. Note that when the phase an­gle varies from zero to $2\pi$, the com­plex num­ber $c$ varies from pos­i­tive real to pos­i­tive imag­i­nary to neg­a­tive real to neg­a­tive imag­i­nary and back to pos­i­tive real. When the com­plex num­ber is plot­ted in the com­plex plane, the phase an­gle is the di­rec­tion of the num­ber rel­a­tive to the ori­gin. The phase an­gle $\alpha$ is of­ten called the ar­gu­ment, but so is about every­thing else in math­e­mat­ics, so that is not very help­ful.

In com­plex time-de­pen­dent waves of the form $e^{{\rm i}({\omega}t-\phi)}$, and its real equiv­a­lent $\cos({\omega}t-\phi)$, the phase an­gle $\phi$ gives the an­gu­lar ar­gu­ment of the wave at time zero.

pho­ton    

Unit of elec­tro­mag­netic ra­di­a­tion (which in­cludes light, x-rays, mi­crowaves, etcetera). A pho­ton has a en­ergy $\hbar\omega$, where $\omega$ is its an­gu­lar fre­quency, and a wave length $2{\pi}c$/$\omega$ where $c$ is the speed of light.

po­ten­tial    

In or­der to op­ti­mize con­fu­sion, pretty much every­thing in physics that is scalar is called po­ten­tial. Po­ten­tial en­ergy is rou­tinely con­cisely re­ferred to as po­ten­tial. It is the en­ergy that a par­ti­cle can pick up from a force field by chang­ing its po­si­tion. It is in Joule. But an elec­tric po­ten­tial is taken to be per unit charge, which gives it units of volts. Then there are ther­mo­dy­namic po­ten­tials like the chem­i­cal po­ten­tial.

$\setbox 0=\hbox{$p_x$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Lin­ear mo­men­tum in the $x$-di­rec­tion. (In the one-di­men­sional cases at the end of the un­steady evo­lu­tion chap­ter, the $x$ sub­script is omit­ted.) Com­po­nents in the $y$- and $z$-di­rec­tions are $p_y$ and $p_z$. Clas­si­cal New­ton­ian physics has $p_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mu$ where $m$ is the mass and $u$ the ve­loc­ity in the $x$-di­rec­tion. In quan­tum me­chan­ics, the pos­si­ble val­ues of $p_x$ are the eigen­val­ues of the op­er­a­tor ${\widehat p}_x$ which equals $\hbar\partial$/${\rm i}\partial{x}$. (But which be­comes canon­i­cal mo­men­tum in a mag­netic field.)

$\setbox 0=\hbox{$Q$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate

• Num­ber of en­ergy eigen­func­tions of a sys­tem of par­ti­cles.
• Anti-Her­mit­ian part of an an­ni­hi­la­tion op­er­a­tor di­vided by ${\rm i}$.
• Heat flow or heat.
• Charge.
• Elec­tric quadru­pole mo­ment.
• En­ergy re­lease.

$\setbox 0=\hbox{$q$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Charge.
• Heat flux den­sity.
• The num­ber of an en­ergy eigen­func­tion of a sys­tem of par­ti­cles.
• Generic in­dex.

$\setbox 0=\hbox{$R$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Ideal gas con­stant.
• Tran­si­tion rate.
• Nu­clear ra­dius.
• Re­flec­tion co­ef­fi­cient.
• Some ra­dius or typ­i­cal ra­dius (like in the Yukawa po­ten­tial).
• Some func­tion of $r$ to be de­ter­mined.
• Some func­tion of $(x,y,z)$ to be de­ter­mined.
• $R_{nl}$ is a hy­dro­gen ra­dial wave func­tion.
• $R_{\rm {u}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 8.314 462 kJ/kmol K is the uni­ver­sal gas con­stant. It is the equiv­a­lent of Boltz­mann's con­stant, but for a kmol in­stead of a sin­gle atom or mol­e­cule.

$\setbox 0=\hbox{${\cal R}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Ro­ta­tion op­er­a­tor.

$\setbox 0=\hbox{$\Re$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The real part of a com­plex num­ber. If $c$ $\vphantom0\raisebox{1.5pt}{$=$}$ $c_r+{{\rm i}}c_i$ with $c_r$ and $c_i$ real num­bers, then $\Re(c)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $c_r$. Note that $c+c^*$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\Re(c)$.

$\setbox 0=\hbox{$r$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The ra­dial dis­tance from the cho­sen ori­gin of the co­or­di­nate sys­tem.
• $r_i$ typ­i­cally in­di­cates the $i$-th Carte­sian com­po­nent of the ra­dius vec­tor ${\skew0\vec r}$.
• Some ra­tio.

$\setbox 0=\hbox{${\skew0\vec r}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The po­si­tion vec­tor. In Carte­sian co­or­di­nates $(x,y,z)$ or $x{\hat\imath}+y{\hat\jmath}+z{\hat k}$. In spher­i­cal co­or­di­nates $r\hat\imath_r$. Its three Carte­sian com­po­nents may be in­di­cated by $r_1,r_2,r_3$ or by $x,y,z$ or by $x_1,x_2,x_3$.

rec­i­p­ro­cal    

The rec­i­p­ro­cal of a num­ber $a$ is 1/$a$. In other words, it is the mul­ti­plica­tive in­verse.

rel­a­tiv­ity    

The spe­cial the­ory of rel­a­tiv­ity ac­counts for the ex­per­i­men­tal ob­ser­va­tion that the speed of light $c$ is the same in all lo­cal co­or­di­nate sys­tems. It nec­es­sar­ily drops the ba­sic con­cepts of ab­solute time and length that were cor­ner stones in New­ton­ian physics.

Al­bert Ein­stein should be cred­ited with the bold­ness to squarely face up to the un­avoid­able where oth­ers wa­vered. How­ever, he should also be cred­ited for the bold­ness of swip­ing the ba­sic ideas from Lorentz and Poin­caré with­out giv­ing them proper, or any, credit. The ev­i­dence is very strong he was aware of both works, and his var­i­ous ar­gu­ments are al­most car­bon copies of those of Poin­caré, but in his pa­per it looks like it all came from Ein­stein, with the ex­is­tence of the ear­lier works not men­tioned. (Note that the gen­eral the­ory of rel­a­tiv­ity, which is of no in­ter­est to this book, is al­most surely prop­erly cred­ited to Ein­stein. But he was a lot less hun­gry then.)

Rel­a­tiv­ity im­plies that a length seen by an ob­server mov­ing at a speed $v$ is shorter than the one seen by a sta­tion­ary ob­server by a fac­tor $\sqrt{1-(v/c)^2}$ as­sum­ing the length is in the di­rec­tion of mo­tion. This is called Lorentz-Fitzger­ald con­trac­tion. It makes galac­tic travel some­what more con­ceiv­able be­cause the size of the galaxy will con­tract for an as­tro­naut in a rocket ship mov­ing close to the speed of light. Rel­a­tiv­ity also im­plies that the time that an event takes seems to be slower by a fac­tor $1/\sqrt{1-(v/c)^2}$ if the event is seen by an ob­server in mo­tion com­pared to the lo­ca­tion where the event oc­curs. That is called time di­la­tion. Some high-en­ergy par­ti­cles gen­er­ated in space move so fast that they reach the sur­face of the earth though this takes much more time than the par­ti­cles would last at rest in a lab­o­ra­tory. The de­cay time in­creases be­cause of the mo­tion of the par­ti­cles. (Of course, as far as the par­ti­cles them­selves see it, the dis­tance to travel is a lot shorter than it seems to be to earth. For them, it is a mat­ter of length con­trac­tion.)

The fol­low­ing for­mu­lae give the rel­a­tivis­tic mass, mo­men­tum, and ki­netic en­ergy of a par­ti­cle in mo­tion:

$$m= \frac{m_0}{\sqrt{1-(v/c)^2}} \qquad p = m v \qquad T = mc^2 - m_0c^2$$

where $m_0$ is the rest mass of the par­ti­cle, i.e.  the mass as mea­sured by an ob­server to whom the par­ti­cle seems at rest. The for­mula for ki­netic en­ergy re­flects the fact that even if a par­ti­cle is at rest, it still has an amount of build-in en­ergy equal to $m_0c^2$ left. The to­tal en­ergy of a par­ti­cle in empty space, be­ing ki­netic and rest mass en­ergy, is given by

$$E = m c^2 = \sqrt{(m_0c^2)^2 + c^2p^2}$$

as can be ver­i­fied by sub­sti­tut­ing in the ex­pres­sion for the mo­men­tum, in terms of the rest mass, and then tak­ing both terms in­side the square root un­der a com­mon de­nom­i­na­tor. For small lin­ear mo­men­tum $p$, this can be ap­prox­i­mated as $\frac12m_0v^2$.

Rel­a­tiv­ity seemed quite a dra­matic de­par­ture of New­ton­ian physics when it de­vel­oped. Then quan­tum me­chan­ics started to emerge...

rot    

The rot of a vec­tor $\vec{v}$ is de­fined as $\mathop{\rm curl}\nolimits \vec{v}$ $\vphantom0\raisebox{1.5pt}{$\equiv$}$ $\mathop{\rm {rot}}\vec{v}$ $\vphantom0\raisebox{1.5pt}{$\equiv$}$ $\nabla$ $\times$ $\vec{v}$.

$\setbox 0=\hbox{$S$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Num­ber of states per unit vol­ume.
• Num­ber of states at a given en­ergy level.
• Spin an­gu­lar mo­men­tum (as an al­ter­na­tive to us­ing $L$ or $J$ for generic an­gu­lar mo­men­tum.)
• En­tropy.
• $S_{12}$ is a fac­tor in the so-called ten­sor po­ten­tial of nu­cle­ons.

$\setbox 0=\hbox{${\cal S}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The ac­tion in­te­gral of La­grangian me­chan­ics, {A.1}

$\setbox 0=\hbox{${\mathscr S}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

See­beck co­ef­fi­cient.

S    

Of­ten used to in­di­cate a state of zero or­bital an­gu­lar mo­men­tum.

$\setbox 0=\hbox{$s$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Spin value of a par­ti­cle. Equals $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ for elec­trons, pro­tons, and neu­trons, is also half an odd nat­ural num­ber for other fermi­ons, and is a non­neg­a­tive in­te­ger for bosons. It is the az­imuthal quan­tum num­ber $l$ due to spin.
• Spe­cific en­tropy.
• As an in­dex, shelf num­ber.

s    

May in­di­cate:

• An en­ergy state with or­bital az­imuthal quan­tum num­ber $l$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Such a state is spher­i­cally sym­met­ric.

scalar    

A quan­tity that is not a vec­tor, a quan­tity that is just a sin­gle num­ber.

$\setbox 0=\hbox{$\sin$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The sine func­tion, a pe­ri­odic func­tion os­cil­lat­ing be­tween 1 and -1 as shown in [41, pp. 40-]. Good to re­mem­ber: $\cos^2\alpha+\sin^2\alpha$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 and $\sin2\alpha$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\sin\alpha\cos\alpha$ and $\cos2\alpha$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\cos^2\alpha-\sin^2\alpha$.

so­le­noidal    

A vec­tor $\vec{v}$ is so­le­noidal if its di­ver­gence $\nabla\cdot\vec{v}$ is zero.

spec­trum    

In this book, a spec­trum nor­mally means a plot of en­ergy lev­els along the ver­ti­cal axis. Of­ten, the hor­i­zon­tal co­or­di­nate is used to in­di­cate a sec­ond vari­able, such as the den­sity of states or the par­ti­cle ve­loc­ity.

For light (pho­tons), a spec­trum can be ob­tained ex­per­i­men­tally by send­ing the light through a prism. This sep­a­rates the col­ors in the light, and each color means a par­tic­u­lar en­ergy of the pho­tons.

The word spec­trum is also of­ten used in a more gen­eral math­e­mat­i­cal sense, but not in this book as far as I can re­mem­ber.

spher­i­cal co­or­di­nates    

The spher­i­cal co­or­di­nates $r$, $\theta$, and $\phi$ of an ar­bi­trary point P are de­fined as

Fig­ure N.3: Spher­i­cal co­or­di­nates of an ar­bi­trary point P.

In Carte­sian co­or­di­nates, the unit vec­tors in the $x$, $y$, and $z$ di­rec­tions are called ${\hat\imath}$, ${\hat\jmath}$, and ${\hat k}$. Sim­i­larly, in spher­i­cal co­or­di­nates, the unit vec­tors in the $r$, $\theta$, and $\phi$ di­rec­tions are called ${\hat\imath}_r$, ${\hat\imath}_\theta$, and ${\hat\imath}_\phi$. Here, say, the $\theta$ di­rec­tion is de­fined as the di­rec­tion of the change in po­si­tion if you in­crease $\theta$ by an in­fin­i­tes­i­mally small amount while keep­ing $r$ and $\varphi$ the same. Note there­fore in par­tic­u­lar that the di­rec­tion of ${\hat\imath}_r$ is the same as that of ${\skew0\vec r}$; ra­di­ally out­ward.

An ar­bi­trary vec­tor $\vec{v}$ can be de­com­posed in com­po­nents $v_r$, $v_\theta$, and $v_\phi$ along these unit vec­tors. In par­tic­u­lar

$$\vec v \equiv v_r {\hat\imath}_r + v_\theta {\hat\imath}_\theta + v_\phi {\hat\imath}_\phi$$

Re­call from cal­cu­lus that in spher­i­cal co­or­di­nates, a vol­ume in­te­gral of an ar­bi­trary func­tion $f$ takes the form

$$\int f { \rm d}^3{\skew0\vec r}= \int\int\int f r^2 \sin\theta { \rm d}r {\rm d}\theta {\rm d}\phi$$

In other words, the vol­ume el­e­ment in spher­i­cal co­or­di­nates is

$${\rm d}V = {\rm d}^3 {\skew0\vec r}= r^2 \sin\theta { \rm d}r {\rm d}\theta {\rm d}\phi$$

Of­ten it is con­ve­nient of think of vol­ume in­te­gra­tions as a two-step process: first per­form an in­te­gra­tion over the an­gu­lar co­or­di­nates $\theta$ and $\phi$. Phys­i­cally, that in­te­grates over spher­i­cal sur­faces. Then per­form an in­te­gra­tion over $r$ to in­te­grate all the spher­i­cal sur­faces to­gether. The com­bined in­fin­i­tes­i­mal an­gu­lar in­te­gra­tion el­e­ment

$${\rm d}\Omega = \sin\theta {\rm d}\theta {\rm d}\phi$$

is called the in­fin­i­tes­i­mal solid an­gle ${\rm d}\Omega$. In two-di­men­sional po­lar co­or­di­nates $r$ and $\theta$, the equiv­a­lent would be the in­fin­i­tes­i­mal po­lar an­gle ${\rm d}\theta$. Re­call that ${\rm d}\theta$, (in proper ra­di­ans of course), equals the ar­clength of an in­fin­i­tes­i­mal part of the cir­cle of in­te­gra­tion di­vided by the cir­cle ra­dius. Sim­i­larly ${\rm d}\Omega$ is the sur­face of an in­fin­i­tes­i­mal part of the sphere of in­te­gra­tion di­vided by the square sphere ra­dius.

See the $\nabla$ en­try for the gra­di­ent op­er­a­tor and Lapla­cian in spher­i­cal co­or­di­nates.

Stokes' The­o­rem    

This the­o­rem, first de­rived by Kelvin and first pub­lished by some­one else I can­not re­call, says that for any rea­son­ably smoothly vary­ing vec­tor $\vec{v}$,

$$\int_A \left(\nabla \times \vec v\right) { \rm d}A = \oint \vec v \cdot {\rm d}\vec r$$

where the first in­te­gral is over any smooth sur­face area $A$ and the sec­ond in­te­gral is over the edge of that sur­face. How did Stokes get his name on it? He tor­tured his stu­dents with it, that’s how!

One im­por­tant con­se­quence of the Stokes the­o­rem is for vec­tor fields $\vec{v}$ that are ir­ro­ta­tional, i.e. that have $\nabla$ $\times$ $\vec{v}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Such fields can be writ­ten as

$$\vec v = \nabla f \qquad f({\skew0\vec r}) \equiv \int_{... ...erline{\skew0\vec r}})\cdot{\rm d}{\underline{\skew0\vec r}}$$

Here ${\skew0\vec r}_{\rm {ref}}$ is the po­si­tion of an ar­bi­trar­ily cho­sen ref­er­ence point, usu­ally the ori­gin. The rea­son the field $\vec{v}$ can be writ­ten this way is the Stokes the­o­rem. Be­cause of the the­o­rem, it does not make a dif­fer­ence along which path from ${\skew0\vec r}_{\rm {ref}}$ to ${\skew0\vec r}$ you in­te­grate. (Any two paths give the same an­swer, as long as $\vec{v}$ is ir­ro­ta­tional every­where in be­tween the paths.) So the de­f­i­n­i­tion of $f$ is un­am­bigu­ous. And you can ver­ify that the par­tial de­riv­a­tives of $f$ give the com­po­nents of $\vec{v}$ by ap­proach­ing the fi­nal po­si­tion ${\skew0\vec r}$ in the in­te­gra­tion from the cor­re­spond­ing di­rec­tion.

sym­me­try    

A sym­me­try is an op­er­a­tion un­der which an ob­ject does not change. For ex­am­ple, a hu­man face is al­most, but not com­pletely, mir­ror sym­met­ric: it looks al­most the same in a mir­ror as when seen di­rectly. The elec­tri­cal field of a sin­gle point charge is spher­i­cally sym­met­ric; it looks the same from what­ever an­gle you look at it, just like a sphere does. A sim­ple smooth glass (like a glass of wa­ter) is cylin­dri­cally sym­met­ric; it looks the same what­ever way you ro­tate it around its ver­ti­cal axis.

$\setbox 0=\hbox{$T$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Ab­solute tem­per­a­ture. The ab­solute tem­per­a­ture in de­grees K equals the tem­per­a­ture in centi­grade plus 273.15. When the ab­solute tem­per­a­ture is zero, (i.e. at $-$273.15 $\POW9,{\circ}$C), na­ture is in the state of low­est pos­si­ble en­ergy.
• Ki­netic en­ergy. A hat in­di­cates the as­so­ci­ated op­er­a­tor. The op­er­a­tor is given by the Lapla­cian times $-\hbar^2$/$2m$.
• Isospin. A hat in­di­cates the as­so­ci­ated op­er­a­tor. A vec­tor sym­bol or sub­script dis­tin­guishes it from ki­netic en­ergy.
• Tesla. The unit of mag­netic field strength, kg/C-s.

$\setbox 0=\hbox{${\cal T}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Trans­la­tion op­er­a­tor that trans­lates a wave func­tion through space. The amount of trans­la­tion is usu­ally in­di­cated by a sub­script.

$\setbox 0=\hbox{$t$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Time.
• $t_t$ is the quan­tum num­ber of square isospin.

tem­per­a­ture    

A mea­sure of the heat mo­tion of the par­ti­cles mak­ing up macro­scopic ob­jects. At ab­solute zero tem­per­a­ture, the par­ti­cles are in the ground state of low­est pos­si­ble en­ergy.

triple prod­uct    

A prod­uct of three vec­tors. There are two dif­fer­ent ver­sions:

• The scalar triple prod­uct $\vec{a}\cdot(\vec{b}\times\vec{c})$. In in­dex no­ta­tion,
  
  $$\vec a \cdot (\vec b\times\vec c) = \sum_i a_i(b_{{\over... ... b_{{\overline{\overline{\imath}}}} c_{{\overline{\imath}}})$$
  
  
  where ${\overline{\imath}}$ is the in­dex fol­low­ing $i$ in the se­quence 123123..., and ${\overline{\overline{\imath}}}$ the one pre­ced­ing it. This triple prod­uct equals the de­ter­mi­nant $\vert\vec{a}\vec{b}\vec{c}\vert$ formed with the three vec­tors. Geo­met­ri­cally, it is plus or mi­nus the vol­ume of the par­al­lelepiped that has vec­tors $\vec{a}$, $\vec{b}$, and $\vec{c}$ as edges. Ei­ther way, as long as the vec­tors are nor­mal vec­tors and not op­er­a­tors,
  
  $$\vec a \cdot (\vec b\times\vec c) = \vec b \cdot (\vec c\times\vec a) = \vec c \cdot (\vec a\times\vec b)$$
  
  
  and you can change the two sides of the dot prod­uct with­out chang­ing the triple prod­uct, and/or you can change the sides in the vec­to­r­ial prod­uct with a change of sign. If any of the vec­tors is an op­er­a­tor, use the in­dex no­ta­tion ex­pres­sion to work it out.
• The vec­to­r­ial triple prod­uct $\vec{a}$ $\times$ $(\vec{b}\times\vec{c})$. In in­dex no­ta­tion, com­po­nent num­ber $i$ of this triple prod­uct is
  
  $$a_{{\overline{\imath}}} (b_ic_{{\overline{\imath}}}-b_{{\o... ...verline{\imath}}}}c_i-b_ic_{{\overline{\overline{\imath}}}})$$
  
  
  which may be rewrit­ten as
  
  $$a_ib_ic_i + a_{{\overline{\imath}}}b_ic_{{\overline{\imath... ...e{\overline{\imath}}}} b_{{\overline{\overline{\imath}}}}c_i$$
  
  
  In par­tic­u­lar, as long as the vec­tors are nor­mal ones,
  
  $$\vec a \times(\vec b\times\vec c) = (\vec a\cdot\vec c)\vec b - (\vec a\cdot\vec b)\vec c$$
  
  

$\setbox 0=\hbox{$U$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• A uni­tary op­er­a­tor, in other words one that does not change the mag­ni­tude of the wave func­tion.
• Of­ten used for en­ergy, though not in this book.

$\setbox 0=\hbox{${\cal U}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

The time shift op­er­a­tor: ${\cal U}(\tau,t)$ changes the wave func­tion $\Psi(\ldots;t)$ into $\Psi(\ldots;t+\tau)$. If the Hamil­ton­ian is in­de­pen­dent of time

$${\cal U}(\tau,t) = {\cal U}_\tau = e^{-{\rm i}H \tau/\hbar}$$

$\setbox 0=\hbox{$u$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The first ve­loc­ity com­po­nent in a Carte­sian co­or­di­nate sys­tem.
• A com­plex co­or­di­nate in the de­riva­tion of spher­i­cal har­mon­ics.
• An in­te­gra­tion vari­able.

u    

May in­di­cate the atomic mass con­stant, equiv­a­lent to 1.660 538 92 10$\POW9,{-27}$ kg or 931.494 06 MeV/$c^2$.

$\setbox 0=\hbox{$V$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The po­ten­tial en­ergy. $V$ is used in­ter­change­ably for the nu­mer­i­cal val­ues of the po­ten­tial en­ergy and for the op­er­a­tor that cor­re­sponds to mul­ti­ply­ing by $V$. In other words, $\widehat{V}$ is sim­ply writ­ten as $V$.

$\setbox 0=\hbox{${\cal V}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Vol­ume.

$\setbox 0=\hbox{$v$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The sec­ond ve­loc­ity com­po­nent in a Carte­sian co­or­di­nate sys­tem.
• Mag­ni­tude of a ve­loc­ity (speed).
• $v$ is spe­cific vol­ume.
• A com­plex co­or­di­nate in the de­riva­tion of spher­i­cal har­mon­ics.
• As $v^{\rm ee}$, a sin­gle elec­tron pair po­ten­tial.

$\setbox 0=\hbox{$\vec{v}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Ve­loc­ity vec­tor.
• Generic vec­tor.
• Sum­ma­tion in­dex of a lat­tice po­ten­tial.

vec­tor    

Sim­ply put, a list of num­bers. A vec­tor $\vec{v}$ in in­dex no­ta­tion is a set of num­bers $\{v_i\}$ in­dexed by an in­dex $i$. In nor­mal three-di­men­sional Carte­sian space, $i$ takes the val­ues 1, 2, and 3, mak­ing the vec­tor a list of three num­bers, $v_1$, $v_2$, and $v_3$. These num­bers are called the three com­po­nents of $\vec{v}$. The list of num­bers can be vi­su­al­ized as a col­umn, and is then called a ket vec­tor, or as a row, in which case it is called a bra vec­tor. This con­ven­tion in­di­cates how mul­ti­pli­ca­tion should be con­ducted with them. A bra times a ket pro­duces a sin­gle num­ber, the dot prod­uct or in­ner prod­uct of the vec­tors:

$$\left(1,3,5\right)\left(\begin{array}{c}7\ 11\ 13\end{array}\right) = 1\;7 + 3\;11 + 5\;13 = 105$$

To turn a ket into a bra for pur­poses of tak­ing in­ner prod­ucts, write the com­plex con­ju­gates of its com­po­nents as a row.

For­mal de­f­i­n­i­tions of vec­tors vary, but real math­e­mati­cians will tell you that vec­tors are ob­jects that can be ma­nip­u­lated in cer­tain ways (ad­di­tion and mul­ti­pli­ca­tion by a scalar). Some physi­cists de­fine vec­tors as ob­jects that trans­form in a cer­tain way un­der co­or­di­nate trans­for­ma­tion (one-di­men­sion­al ten­sors); that is not the same thing.

vec­to­r­ial prod­uct    

An vec­to­r­ial prod­uct, or cross prod­uct is a prod­uct of vec­tors that pro­duces an­other vec­tor. If

$$\vec c=\vec a\times\vec b,$$

it means in in­dex no­ta­tion that the $i$-th com­po­nent of vec­tor $\vec{c}$ is

$$c_i = a_{{\overline{\imath}}} b_{{\overline{\overline{\ima... ... - a_{{\overline{\overline{\imath}}}}b_{{\overline{\imath}}}$$

where ${\overline{\imath}}$ is the in­dex fol­low­ing $i$ in the se­quence 123123..., and ${\overline{\overline{\imath}}}$ the one pre­ced­ing it. For ex­am­ple, $c_1$ will equal $a_2b_3-a_3b_2$.

W    

May in­di­cate:

• Watt, the SI unit of power.
• The W$\POW9,{\pm}$ are the charged car­ri­ers of the weak force. See also Z$\POW9,{0}$.
• W.u. stands for Weis­skopf unit, a sim­ple de­cay ball­park for gamma de­cay.

$\setbox 0=\hbox{$w$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• The third ve­loc­ity com­po­nent in a Carte­sian co­or­di­nate sys­tem.
• Weight fac­tor.

$\setbox 0=\hbox{$\vec{w}$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Generic vec­tor.

$\setbox 0=\hbox{$X$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Used in this book to in­di­cate a func­tion of $x$ to be de­ter­mined.

$\setbox 0=\hbox{$x$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• First co­or­di­nate in a Carte­sian co­or­di­nate sys­tem.
• A generic ar­gu­ment of a func­tion.
• An un­known value.

$\setbox 0=\hbox{$Y$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Used in this book to in­di­cate a func­tion of $y$ to be de­ter­mined.

$\setbox 0=\hbox{$Y_l^m$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

Spher­i­cal har­monic. Eigen­func­tion of both an­gu­lar mo­men­tum in the $z$-di­rec­tion and of to­tal square an­gu­lar mo­men­tum.

$\setbox 0=\hbox{$y$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Sec­ond co­or­di­nate in a Carte­sian co­or­di­nate sys­tem.
• A sec­ond generic ar­gu­ment of a func­tion.
• A sec­ond un­known value.

$\setbox 0=\hbox{$Z$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Atomic num­ber (num­ber of pro­tons in the nu­cleus).
• Num­ber of par­ti­cles.
• Par­ti­tion func­tion.
• The Z$\POW9,{0}$ is the un­charged car­rier of the weak force. See also W$\POW9,{\pm}$.
• Used in this book to in­di­cate a func­tion of $z$ to be de­ter­mined.

$\setbox 0=\hbox{$z$}\kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0\kern-.025em\raise.0433em\box0$    

May in­di­cate:

• Third co­or­di­nate in a Carte­sian co­or­di­nate sys­tem.
• A third generic ar­gu­ment of a func­tion.
• A third un­known value.