5. Mul­ti­ple-Par­ti­cle Sys­tems


So far, only wave func­tions for sin­gle par­ti­cles have been dis­cussed. This chap­ter ex­plains how the ideas gen­er­al­ize to more par­ti­cles. The ba­sic idea is sim­ple: you just keep adding more and more ar­gu­ments to your wave func­tion.

That sim­ple idea will im­me­di­ately be used to de­rive a so­lu­tion for the hy­dro­gen mol­e­cule. The chem­i­cal bond that keeps the mol­e­cule to­gether is a two-elec­tron one. It in­volves shar­ing the two elec­trons in a very weird way that can only be de­scribed in quan­tum terms.

Now it turns out that usu­ally chem­i­cal bonds in­volve the shar­ing of two elec­trons like in the hy­dro­gen mol­e­cule, not just one as in the hy­dro­gen mol­e­c­u­lar ion. To un­der­stand the rea­son, sim­ple ap­prox­i­mate sys­tems will be ex­am­ined that have no more than two dif­fer­ent states. It will then be seen that shar­ing low­ers the en­ergy due to twi­light terms. These are usu­ally more ef­fec­tive for two-elec­tron bonds than for sin­gle elec­tron-ones.

Be­fore sys­tems with more than two elec­trons can be dis­cussed, a dif­fer­ent is­sue must be ad­dressed first. Elec­trons, as well as most other quan­tum par­ti­cles, have in­trin­sic an­gu­lar mo­men­tum called spin. It is quan­tized much like or­bital an­gu­lar mo­men­tum. Elec­trons can ei­ther have spin an­gu­lar mo­men­tum $\frac12\hbar$ or $-\frac12\hbar$ in a given di­rec­tion. It is said that the elec­tron has spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em
/\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$. Pho­tons can have an­gu­lar mo­men­tum $\hbar$, 0, or $\vphantom{0}\raisebox{1.5pt}{$-$}$$\hbar$ in a given di­rec­tion and have spin 1. Par­ti­cles with half-in­te­ger spin like elec­trons are called fermi­ons. Par­ti­cles with in­te­ger spin like pho­tons are called bosons.

For quan­tum me­chan­ics there are two con­se­quences. First, it means that spin must be added to the wave func­tion as an un­cer­tain quan­tity in ad­di­tion to po­si­tion. That can be done in var­i­ous equiv­a­lent ways. Sec­ond, it turns out that there are re­quire­ments on the wave func­tion de­pend­ing on whether par­ti­cles are bosons or fermi­ons. In par­tic­u­lar, wave func­tions must stay the same if two iden­ti­cal bosons, say two pho­tons, are in­ter­changed. Wave func­tions must change sign when any two elec­trons, or any other two iden­ti­cal fermi­ons, are in­ter­changed.

This so-called an­ti­sym­metriza­tion re­quire­ment is usu­ally not such a big deal for two elec­tron sys­tems. Two elec­trons can sat­isfy the re­quire­ment by as­sum­ing a suit­able com­bined spin state. How­ever, for more than two elec­trons, the ef­fects of the an­ti­sym­metriza­tion re­quire­ment are dra­matic. They de­ter­mine the very na­ture of the chem­i­cal el­e­ments be­yond he­lium. With­out the an­ti­sym­metriza­tion re­quire­ments on the elec­trons, chem­istry would be some­thing com­pletely dif­fer­ent. And there­fore, so would all of na­ture be. Be­fore that can be prop­erly un­der­stood, first a bet­ter look is needed at the ways in which the sym­metriza­tion re­quire­ments can be sat­is­fied. It is then seen that the re­quire­ment for fermi­ons can be for­mu­lated as the so-called Pauli ex­clu­sion prin­ci­ple. The prin­ci­ple says that any num­ber $I$ of iden­ti­cal fermi­ons must oc­cupy $I$ dif­fer­ent quan­tum states. Fermi­ons are ex­cluded from en­ter­ing the same quan­tum state.

At that point, the atoms heav­ier than hy­dro­gen can be prop­erly dis­cussed. It can also be ex­plained why atoms pre­vent each other from com­ing too close. Fi­nally, the de­rived quan­tum prop­er­ties of the atoms are used to de­scribe the var­i­ous types of chem­i­cal bonds.