11.2 Sin­gle-Par­ti­cle ver­sus Sys­tem States

The pur­pose of this sec­tion is to de­scribe the generic form of the en­ergy eigen­func­tions of a sys­tem of weakly in­ter­act­ing par­ti­cles.

The to­tal num­ber of par­ti­cles will be in­di­cated by $I$. If the in­ter­ac­tions be­tween the $I$ par­ti­cles are ig­nored, any en­ergy eigen­func­tion of the com­plete sys­tem of $I$ par­ti­cles can be writ­ten in terms of sin­gle-par­ti­cle en­ergy eigen­func­tions $\pp1/{\skew0\vec r}//z/,\pp2/{\skew0\vec r}//z/,\ldots$.

The ba­sic case is that of non­in­ter­act­ing par­ti­cles in a box, like dis­cussed in chap­ter 6.2. For such par­ti­cles the sin­gle-par­ti­cle eigen­func­tions take the spa­tial form

$$\pp{n}//// = \sqrt{\frac{8}{\ell_x\ell_y\ell_z}} \sin(k_xx)\sin(k_yy)\sin(k_zz)$$

where $k_x$, $k_y$, and $k_z$ are con­stants, called the wave num­ber com­po­nents. Dif­fer­ent val­ues for these con­stants cor­re­spond to dif­fer­ent sin­gle-par­ti­cle eigen­func­tions, with sin­gle-par­ti­cle en­ergy

$${\vphantom' E}^{\rm p}_n = \frac{\hbar^2}{2m} (k_x^2 + k_y^2 + k_z^2) = \frac{\hbar^2}{2m} k^2$$

The sin­gle-par­ti­cle en­ergy eigen­func­tions will in this chap­ter be num­bered as $n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $1,2,3,\ldots,N$. Higher val­ues of in­dex $n$ cor­re­spond to eigen­func­tions of equal or higher en­ergy ${\vphantom' E}^{\rm p}_n$.

The sin­gle-par­ti­cle eigen­func­tions do not al­ways cor­re­spond to a par­ti­cle in a box. For ex­am­ple, par­ti­cles caught in a mag­netic trap, like in the Bose-Ein­stein con­den­sa­tion ex­per­i­ments of 1995, might be bet­ter de­scribed us­ing har­monic os­cil­la­tor eigen­func­tions. Or the par­ti­cles might be re­stricted to move in a lower-di­men­sion­al space. But a lot of the for­mu­lae you can find in lit­er­a­ture and in this chap­ter are in fact de­rived as­sum­ing the sim­plest case of non­in­ter­act­ing par­ti­cles in a roomy box.

The de­tails of the sin­gle-par­ti­cle en­ergy eigen­func­tions are not re­ally that im­por­tant in this chap­ter. What is more in­ter­est­ing are the en­ergy eigen­func­tions $\psi^{\rm S}_q$ of com­plete sys­tems of par­ti­cles. It will be as­sumed that these sys­tem eigen­func­tions are num­bered us­ing a counter $q$, but the way they are num­bered also does not re­ally make a dif­fer­ence to the analy­sis.

As long as the in­ter­ac­tions be­tween the par­ti­cles are weak, en­ergy eigen­func­tions of the com­plete sys­tem can be found as prod­ucts of the sin­gle-par­ti­cle ones. As an im­por­tant ex­am­ple, at ab­solute zero tem­per­a­ture, all par­ti­cles will be in the sin­gle-par­ti­cle ground state $\pp1////$, and the sys­tem will be in its ground state

$$\psi^{\rm S}_1 = \pp1/{\skew0\vec r}_1//z1/ \pp1/{\skew0\v... ... \pp1/{\skew0\vec r}_5//z5/ \ldots \pp1/{\skew0\vec r}_I//zI/$$

where $I$ is the to­tal num­ber of par­ti­cles in the sys­tem. This does as­sume that the sin­gle-par­ti­cle ground state en­ergy ${\vphantom' E}^{\rm p}_1$ is not de­gen­er­ate. More im­por­tantly, it as­sumes that the $I$ par­ti­cles are not iden­ti­cal fermi­ons. Ac­cord­ing to the ex­clu­sion prin­ci­ple, at most one fermion can go into a sin­gle-par­ti­cle state. (For spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ fermi­ons like elec­trons, two can go into a sin­gle spa­tial state, one in the spin-up ver­sion, and the other in the spin-down one.)

Sta­tis­ti­cal ther­mo­dy­nam­ics, in any case, is much more in­ter­ested in tem­per­a­tures that are not zero. Then the sys­tem will not be in the ground state, but in some com­bi­na­tion of sys­tem eigen­func­tions of higher en­ergy. As a com­pletely ar­bi­trary ex­am­ple of such a sys­tem eigen­func­tion, take the fol­low­ing one, de­scrib­ing $I$ $\vphantom0\raisebox{1.5pt}{$=$}$ 36 dif­fer­ent par­ti­cles:

$$\psi^{\rm S}_q = \pp24/{\skew0\vec r}_1//z1/ \pp4/{\skew0\... ...6/{\skew0\vec r}_5//z5/ \ldots \pp54/{\skew0\vec r}_{36}//z36/$$

This sys­tem eigen­func­tion has an en­ergy that is the sum of the 36 sin­gle-par­ti­cle eigen­state en­er­gies in­volved:

$${\vphantom' E}^{\rm S}_q = {\vphantom' E}^{\rm p}_{24} + {\... ...\vphantom' E}^{\rm p}_6 + \ldots + {\vphantom' E}^{\rm p}_{54}$$

Fig­ure 11.1: Graph­i­cal de­pic­tion of an ar­bi­trary sys­tem en­ergy eigen­func­tion for 36 dis­tin­guish­able par­ti­cles.

To un­der­stand the ar­gu­ments in this chap­ter, it is es­sen­tial to vi­su­al­ize the sys­tem en­ergy eigen­func­tions as in fig­ure 11.1. In this fig­ure the sin­gle-par­ti­cle states are shown as boxes, and the par­ti­cles that are in those par­tic­u­lar sin­gle-par­ti­cle states are shown in­side the boxes. In the ex­am­ple, par­ti­cle 1 is in­side the $\pp24////$ box, par­ti­cle 2 is in­side the $\pp4////$ one, etcetera. It is just the re­verse from the math­e­mat­i­cal ex­pres­sion above: the math­e­mat­i­cal ex­pres­sion shows for each par­ti­cle in turn what the sin­gle-par­ti­cle eigen­state of that par­ti­cle is. The fig­ure shows for each type of sin­gle-par­ti­cle eigen­state in turn what par­ti­cles are in that eigen­state.

To sim­plify the analy­sis, in the fig­ure sin­gle-par­ti­cle eigen­states of about the same en­ergy have been grouped to­gether on shelves. (As a con­se­quence, a sub­script to a sin­gle-par­ti­cle en­ergy ${\vphantom' E}^{\rm p}$ may re­fer to ei­ther a sin­gle-par­ti­cle eigen­func­tion num­ber $n$ or to a shelf num­ber $s$, de­pend­ing on con­text.) The num­ber of sin­gle-par­ti­cle states on a shelf is in­tended to roughly sim­u­late the den­sity of states of the par­ti­cles in a box as de­scribed in chap­ter 6.3. The larger the en­ergy, the more sin­gle-par­ti­cle states there are at that en­ergy; it in­creases like the square root of the en­ergy. This may not be true for other sit­u­a­tions, such as when the par­ti­cles are con­fined to a lower-di­men­sion­al space, com­pare chap­ter 6.12. Var­i­ous for­mu­lae given here and in lit­er­a­ture may need to be ad­justed then.

Of course, in nor­mal non­nano ap­pli­ca­tions, the num­ber of par­ti­cles will be as­tro­nom­i­cally larger than 36 par­ti­cles; the ex­am­ple is just a small il­lus­tra­tion. Even a mil­limol of par­ti­cles means on the or­der of 10$\POW9,{20}$ par­ti­cles. And un­less the tem­per­a­ture is in­cred­i­bly low, those par­ti­cles will ex­tend to many more sin­gle-par­ti­cle states than the few shown in the fig­ure.

Next, note that you are not go­ing to have some­thing like 10$\POW9,{20}$ dif­fer­ent types of par­ti­cles. In­stead they are more likely to all be he­lium atoms, or all elec­trons or so. If their wave func­tions over­lap non­triv­ially, that makes a big dif­fer­ence be­cause of the sym­metriza­tion re­quire­ments of the sys­tem wave func­tion.

Con­sider first the case that the $I$ par­ti­cles are all iden­ti­cal bosons, like plain he­lium atoms. In that case the wave func­tion must be sym­met­ric, un­changed, un­der the ex­change of any two of the bosons, and the ex­am­ple wave func­tion above is not. If, for ex­am­ple, par­ti­cles 2 and 5 are ex­changed, it turns the ex­am­ple wave func­tion from

$$\psi^{\rm S}_q = \pp24/{\skew0\vec r}_1//z1/ \pp4/{\skew0\... ...6/{\skew0\vec r}_5//z5/ \ldots \pp54/{\skew0\vec r}_{36}//z36/$$

into

$$\psi^{\rm S}_{{\underline q}} = \pp24/{\skew0\vec r}_1//z1... ...4/{\skew0\vec r}_5//z5/ \ldots \pp54/{\skew0\vec r}_{36}//z36/$$

and that is sim­ply a dif­fer­ent wave func­tion, be­cause the states are dif­fer­ent, in­de­pen­dent func­tions. In terms of the pic­to­r­ial rep­re­sen­ta­tion fig­ure 11.1, swap­ping the num­bers 2” and “5 in the par­ti­cles changes the pic­ture.

Fig­ure 11.2: Graph­i­cal de­pic­tion of an ar­bi­trary sys­tem en­ergy eigen­func­tion for 36 iden­ti­cal bosons.

As chap­ter 5.7 ex­plained, to elim­i­nate the prob­lem that ex­chang­ing par­ti­cles 2 and 5 changes the wave func­tion, the orig­i­nal and ex­changed wave func­tions must be com­bined to­gether. And to elim­i­nate the prob­lem for any two par­ti­cles, all wave func­tions that can be ob­tained by merely swap­ping num­bers must be com­bined to­gether equally into a sin­gle wave func­tion mul­ti­plied by a sin­gle un­de­ter­mined co­ef­fi­cient. In terms of fig­ure 11.1, we need to com­bine the wave func­tions with all pos­si­ble per­mu­ta­tions of the num­bers in­side the par­ti­cles into one. And if all per­mu­ta­tions of the num­bers are equally in­cluded, then those num­bers no longer add any non­triv­ial ad­di­tional in­for­ma­tion; they may as well be left out. That makes the pic­to­r­ial rep­re­sen­ta­tion of an ex­am­ple sys­tem wave func­tion for iden­ti­cal bosons as shown in fig­ure 11.2.

Fig­ure 11.3: Graph­i­cal de­pic­tion of an ar­bi­trary sys­tem en­ergy eigen­func­tion for 33 iden­ti­cal fermi­ons.

For iden­ti­cal fermi­ons, the sit­u­a­tion is sim­i­lar, ex­cept that the dif­fer­ent wave func­tions must be com­bined with equal or op­po­site sign, de­pend­ing on whether it takes an odd or even num­ber of par­ti­cle swaps to turn one into the other. And such wave func­tions only ex­ist if the $I$ sin­gle-par­ti­cle wave func­tions in­volved are all dif­fer­ent. That is the Pauli ex­clu­sion prin­ci­ple. The pic­to­r­ial rep­re­sen­ta­tion fig­ure 11.2 for bosons is to­tally un­ac­cept­able for fermi­ons since it uses many of the sin­gle-par­ti­cle states for more than one par­ti­cle. There can be at most one fermion in each type of sin­gle-par­ti­cle state. An ex­am­ple of a wave func­tion that is ac­cept­able for a sys­tem of iden­ti­cal fermi­ons is shown in fig­ure 11.3.

Look­ing at the ex­am­ple pic­to­r­ial rep­re­sen­ta­tions for sys­tems of bosons and fermi­ons, it may not be sur­pris­ing that such par­ti­cles are of­ten called “in­dis­tin­guish­able.“ Of course, in clas­si­cal quan­tum me­chan­ics, there is still an elec­tron 1, an elec­tron 2, etcetera; they are math­e­mat­i­cally dis­tin­guished. Still, it is con­ve­nient to use the term dis­tin­guish­able for par­ti­cles for which the sym­metriza­tion re­quire­ments can be ig­nored.

The prime ex­am­ple is the atoms of an ideal gas in a box; al­most by de­f­i­n­i­tion, the in­ter­ac­tions be­tween such atoms are neg­li­gi­ble. And that al­lows the quan­tum re­sults to be re­ferred back to the well-un­der­stood prop­er­ties of ideal gases ob­tained in clas­si­cal physics. Prob­a­bly you would like to see all re­sults fol­low nat­u­rally from quan­tum me­chan­ics, not clas­si­cal physics, and that would be very nice in­deed. But it would be very hard to fol­low up on. As Baier­lein [4, p. 109] notes, real-life physics adopts whichever the­o­ret­i­cal ap­proach of­fers the eas­i­est cal­cu­la­tion or the most in­sight. This book’s ap­proach re­ally is to for­mu­late as much as pos­si­ble in terms of the quan­tum-me­chan­i­cal ideas dis­cussed here. But do be aware that it is a much more messy world when you go out there.