3.3 The Op­er­a­tors of Quan­tum Me­chan­ics

The nu­mer­i­cal quan­ti­ties that the old New­ton­ian physics uses, (po­si­tion, mo­men­tum, en­ergy, ...), are just shad­ows of what re­ally de­scribes na­ture: op­er­a­tors. The op­er­a­tors de­scribed in this sec­tion are the key to quan­tum me­chan­ics.

As the first ex­am­ple, while a math­e­mat­i­cally pre­cise value of the po­si­tion $x$ of a par­ti­cle never ex­ists, in­stead there is an $x$-​po­si­tion op­er­a­tor ${\widehat x}$. It turns the wave func­tion $\Psi$ into $x\Psi$:

$$\Psi(x,y,z,t) \quad \begin{picture}(100,10) \put(50,12){... ...t(0,2){\vector(1,0){100}} \end{picture} \quad x \Psi(x,y,z,t)$$ (3.3)

The op­er­a­tors ${\widehat y}$ and ${\widehat z}$ are de­fined sim­i­larly as ${\widehat x}$.

In­stead of a lin­ear mo­men­tum $p_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mu$, there is an $x$-​mo­men­tum op­er­a­tor

$$\fbox{$\displaystyle {\widehat p}_x = \frac{\hbar}{{\rm i}} \frac{\partial}{\partial x} $}$$ (3.4)

that turns $\Psi$ into its $x$-​de­riv­a­tive:

$$\Psi(x,y,z,t) \quad \begin{picture}(100,25) \put(50,21){... ...00}} \end{picture} \quad \frac{\hbar}{{\rm i}}\Psi_x(x,y,z,t)$$ (3.5)

The con­stant $\hbar$ is called “Planck's con­stant.” (Or rather, it is Planck's orig­i­nal con­stant $h$ di­vided by $2\pi$.) If it would have been zero, all these trou­bles with quan­tum me­chan­ics would not oc­cur. The blobs would be­come points. Un­for­tu­nately, $\hbar$ is very small, but nonzero. It is about 10$\POW9,{-34}$ kg m$\POW9,{2}$/s.

The fac­tor ${\rm i}$ in ${\widehat p}_x$ makes it a Her­mit­ian op­er­a­tor (a proof of that is in de­riva­tion {D.9}). All op­er­a­tors re­flect­ing macro­scopic phys­i­cal quan­ti­ties are Her­mit­ian.

The op­er­a­tors ${\widehat p}_y$ and ${\widehat p}_z$ are de­fined sim­i­larly as ${\widehat p}_x$:

$$\fbox{$\displaystyle {\widehat p}_y = \frac{\hbar}{{\rm i}... ...t p}_z = \frac{\hbar}{{\rm i}} \frac{\partial}{\partial z} $}$$ (3.6)

The ki­netic en­ergy op­er­a­tor ${\widehat T}$ is:

$${\widehat T}= \frac{{\widehat p}_x^2 + {\widehat p}_y^2 + {\widehat p}_z^2}{2 m}$$ (3.7)

Its shadow is the New­ton­ian no­tion that the ki­netic en­ergy equals:

$$T = \frac12 m \left( u^2 + v^2 + w^2 \right) = \frac{(mu)^2 + (mv)^2 + (mw)^2}{2m}$$

This is an ex­am­ple of the “New­ton­ian anal­ogy”: the re­la­tion­ships be­tween the dif­fer­ent op­er­a­tors in quan­tum me­chan­ics are in gen­eral the same as those be­tween the cor­re­spond­ing nu­mer­i­cal val­ues in New­ton­ian physics. But since the mo­men­tum op­er­a­tors are gra­di­ents, the ac­tual ki­netic en­ergy op­er­a­tor is, from the mo­men­tum op­er­a­tors above:

$${\widehat T}= - \frac{\hbar^2}{2m} \left( \frac{\partial^... ...^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right).$$ (3.8)

Math­e­mati­cians call the set of sec­ond or­der de­riv­a­tive op­er­a­tors in the ki­netic en­ergy op­er­a­tor the Lapla­cian, and in­di­cate it by $\nabla^2$:

$$\fbox{$\displaystyle \nabla^2 \equiv \frac{\partial^2}{\p... ...artial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} $}$$ (3.9)

In those terms, the ki­netic en­ergy op­er­a­tor can be writ­ten more con­cisely as:

$$\fbox{$\displaystyle {\widehat T}= - \frac{\hbar^2}{2m} \nabla^2 $}$$ (3.10)

Fol­low­ing the New­ton­ian anal­ogy once more, the to­tal en­ergy op­er­a­tor, in­di­cated by $H$, is the the sum of the ki­netic en­ergy op­er­a­tor above and the po­ten­tial en­ergy op­er­a­tor $V(x,y,z,t)$:

$$\fbox{$\displaystyle H = -\frac{\hbar^2}{2m} \nabla^2 + V $}$$ (3.11)

This to­tal en­ergy op­er­a­tor $H$ is called the Hamil­ton­ian and it is very im­por­tant. Its eigen­val­ues are in­di­cated by $E$ (for en­ergy), for ex­am­ple $E_1$, $E_2$, $E_3$, ...with:

$$H \psi_n = E_n \psi_n \quad\mbox{for } n = 1, 2, 3, ...$$ (3.12)

where $\psi_n$ is eigen­func­tion num­ber $n$ of the Hamil­ton­ian.

It is seen later that in many cases a more elab­o­rate num­ber­ing of the eigen­val­ues and eigen­vec­tors of the Hamil­ton­ian is de­sir­able in­stead of us­ing a sin­gle counter $n$. For ex­am­ple, for the elec­tron of the hy­dro­gen atom, there is more than one eigen­func­tion for each dif­fer­ent eigen­value $E_n$, and ad­di­tional coun­ters $l$ and $m$ are used to dis­tin­guish them. It is usu­ally best to solve the eigen­value prob­lem first and de­cide on how to num­ber the so­lu­tions af­ter­wards.

(It is also im­por­tant to re­mem­ber that in the lit­er­a­ture, the Hamil­ton­ian eigen­value prob­lem is com­monly re­ferred to as the time-in­de­pen­dent Schrö­din­ger equa­tion. How­ever, this book prefers to re­serve the term Schrö­din­ger equa­tion for the un­steady evo­lu­tion of the wave func­tion.)

Key Points

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Phys­i­cal quan­ti­ties cor­re­spond to op­er­a­tors in quan­tum me­chan­ics.

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Ex­pres­sions for var­i­ous im­por­tant op­er­a­tors were given.

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Ki­netic en­ergy is in terms of the so-called Lapla­cian op­er­a­tor.

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The im­por­tant to­tal en­ergy op­er­a­tor, (ki­netic plus po­ten­tial en­ergy,) is called the Hamil­ton­ian.