4. Sin­gle-Par­ti­cle Sys­tems


In this chap­ter, the ma­chin­ery to deal with sin­gle par­ti­cles is worked out, cul­mi­nat­ing in the vi­tal so­lu­tions for the hy­dro­gen atom and hy­dro­gen mol­e­c­u­lar ion.

The first sec­tion cov­ers the har­monic os­cil­la­tor. This vi­brat­ing sys­tem is a sim­ple model for such sys­tems as an atom in a trap, crys­tal vi­bra­tions, and elec­tro­mag­netic waves.

Next, be­fore the hy­dro­gen atom can be dis­cussed, first the quan­tum me­chan­ics of an­gu­lar mo­men­tum needs to be cov­ered. Just like you need an­gu­lar mo­men­tum to solve the mo­tion of a planet around the sun in clas­si­cal physics, so do you need an­gu­lar mo­men­tum for the mo­tion of an elec­tron around a nu­cleus in quan­tum me­chan­ics. The eigen­val­ues of an­gu­lar mo­men­tum and their quan­tum num­bers are crit­i­cally im­por­tant for many other rea­sons be­sides the hy­dro­gen atom.

Af­ter an­gu­lar mo­men­tum, the hy­dro­gen atom can be dis­cussed. The so­lu­tion is messy, but fun­da­men­tally not much dif­fer­ent from that of the par­ti­cle in the pipe or the har­monic os­cil­la­tor of the pre­vi­ous chap­ter.

The hy­dro­gen atom is the ma­jor step to­wards ex­plain­ing heav­ier atoms and then chem­i­cal bonds. One rather un­usual chem­i­cal bond can al­ready be dis­cussed in this chap­ter: that of a ion­ized hy­dro­gen mol­e­cule. A hy­dro­gen mol­e­c­u­lar ion has only one elec­tron.

But the hy­dro­gen mol­e­c­u­lar ion can­not read­ily be solved ex­actly, even if the mo­tion of the nu­clei is ig­nored. So an ap­prox­i­mate method will be used. Be­fore this can be done, how­ever, a prob­lem must be ad­dressed. The hy­dro­gen mol­e­c­u­lar ion ground state is de­fined to be the state of low­est en­ergy. But an ap­prox­i­mate ground state is not an ex­act en­ergy eigen­func­tion and has un­cer­tain en­ergy. So how should the term low­est en­ergy be de­fined for the ap­prox­i­ma­tion?

To an­swer that, be­fore tack­ling the mol­e­c­u­lar ion, first sys­tems with un­cer­tainty in a vari­able of in­ter­est are dis­cussed. The ex­pec­ta­tion value of a vari­able will be de­fined to be the av­er­age of the eigen­val­ues, weighted by their prob­a­bil­ity. The stan­dard de­vi­a­tion will be de­fined as a mea­sure of how much un­cer­tainty there is to that ex­pec­ta­tion value.

With a pre­cise math­e­mat­i­cal de­f­i­n­i­tion of un­cer­tainty, the ob­vi­ous next ques­tion is whether two dif­fer­ent vari­ables can be cer­tain at the same time. The com­mu­ta­tor of the two op­er­a­tors will be in­tro­duced to an­swer it. That then al­lows the Heisen­berg un­cer­tainty re­la­tion­ship to be for­mu­lated. Not only can po­si­tion and lin­ear mo­men­tum not be cer­tain at the same time; a spe­cific equa­tion can be writ­ten down for how big the un­cer­tainty must be, at the very least.

With the math­e­mat­i­cal ma­chin­ery of un­cer­tainty de­fined, the hy­dro­gen mol­e­c­u­lar ion is solved last.