## 7. Time Evo­lu­tion

Ab­stract

The evo­lu­tion of sys­tems in time is less im­por­tant in quan­tum me­chan­ics than in clas­si­cal physics, since in quan­tum me­chan­ics so much can be learned from the en­ergy eigen­val­ues and eigen­func­tions. Still, time evo­lu­tion is needed for such im­por­tant phys­i­cal processes as the cre­ation and ab­sorp­tion of light and other ra­di­a­tion. And many other phys­i­cal processes of prac­ti­cal im­por­tance are sim­plest to un­der­stand in terms of clas­si­cal physics. To trans­late a typ­i­cal rough clas­si­cal de­scrip­tion into cor­rect quan­tum me­chan­ics re­quires an un­der­stand­ing of un­steady quan­tum me­chan­ics.

The chap­ter starts with the in­tro­duc­tion of the Schrö­din­ger equa­tion. This equa­tion is as im­por­tant for quan­tum me­chan­ics as New­ton’s sec­ond law is for clas­si­cal me­chan­ics. A for­mal so­lu­tion to the equa­tion can be writ­ten im­me­di­ately down for most sys­tems of in­ter­est.

One di­rect con­se­quence of the Schrö­din­ger equa­tion is en­ergy con­ser­va­tion. Sys­tems that have a def­i­nite value for their en­ergy con­serve that en­ergy in the sim­plest pos­si­ble way: they just do not change at all. They are sta­tion­ary states. Sys­tems that have un­cer­tainty in en­ergy do evolve in a non­triv­ial way. But such sys­tems do still con­serve the prob­a­bil­ity of each of their pos­si­ble en­ergy val­ues.

Of course, the en­ergy of a sys­tem is only con­served if no de­vi­ous ex­ter­nal agent is adding or re­mov­ing en­ergy. In quan­tum me­chan­ics that usu­ally boils down to the con­di­tion that the Hamil­ton­ian must be in­de­pen­dent of time. If there is a nasty ex­ter­nal agent that does mess things up, analy­sis may still be pos­si­ble if that agent is a slow­poke. Since physi­cists do not know how to spell slow­poke, they call this the adi­a­batic ap­prox­i­ma­tion. More pre­cisely, they call it adi­a­batic be­cause they know how to spell adi­a­batic, but not what it means.

The Schrö­din­ger equa­tion is read­ily used to de­scribe the evo­lu­tion of ex­pec­ta­tion val­ues of phys­i­cal quan­ti­ties. This makes it pos­si­ble to show that New­ton’s equa­tions are re­ally an ap­prox­i­ma­tion of quan­tum me­chan­ics valid for macro­scopic sys­tems. It also makes it pos­si­ble to for­mu­late the pop­u­lar en­ergy-time un­cer­tainty re­la­tion­ship.

Next, the Schrö­din­ger equa­tion does not just ex­plain en­ergy con­ser­va­tion. It also ex­plains where other con­ser­va­tion laws such as con­ser­va­tion of lin­ear and an­gu­lar mo­men­tum come from. For ex­am­ple, an­gu­lar mo­men­tum con­ser­va­tion is a di­rect con­se­quence of the fact that space has no pre­ferred di­rec­tion.

It is then shown how these var­i­ous con­ser­va­tion laws can be used to bet­ter un­der­stand the emis­sion of elec­tro­mag­netic ra­di­a­tion by say an hy­dro­gen atom. In par­tic­u­lar, they pro­vide con­di­tions on the emis­sion process that are called se­lec­tion rules.

Next, the Schrö­din­ger equa­tion is used to de­scribe the de­tailed time evo­lu­tion of a sim­ple quan­tum sys­tem. The sys­tem al­ter­nates be­tween two phys­i­cally equiv­a­lent states. That pro­vides a model for how the fun­da­men­tal forces of na­ture arise. It also pro­vides a model for the emis­sion of ra­di­a­tion by an atom or an atomic nu­cleus.

Un­for­tu­nately, the model for emis­sion of ra­di­a­tion turns out to have some prob­lems. These re­quire the con­sid­er­a­tion of quan­tum sys­tems in­volv­ing two states that are not phys­i­cally equiv­a­lent. That analy­sis then fi­nally al­lows a com­pre­hen­sive de­scrip­tion of the in­ter­ac­tion be­tween atoms and the elec­tro­mag­netic field. It turns out that emis­sion of ra­di­a­tion can be stim­u­lated by ra­di­a­tion that al­ready ex­ists. That al­lows for the op­er­a­tion of masers and lasers that dump out macro­scopic amounts of mono­chro­matic, co­her­ent ra­di­a­tion.

The fi­nal sec­tions dis­cuss ex­am­ples of the non­triv­ial evo­lu­tion of sim­ple quan­tum sys­tems with in­fi­nite num­bers of states. Be­fore that can be done, first the so-far ne­glected eigen­func­tions of po­si­tion and lin­ear mo­men­tum must be dis­cussed. Po­si­tion eigen­func­tions turn out to be spikes, while lin­ear mo­men­tum eigen­func­tions turn out to be waves. Par­ti­cles that have sig­nif­i­cant and sus­tained spa­tial lo­cal­iza­tion can be iden­ti­fied as pack­ets of waves. These ideas can be gen­er­al­ized to the mo­tion of con­duc­tion elec­trons in crys­tals.

The mo­tion of such wave pack­ets is then ex­am­ined. If the forces change slowly on quan­tum scales, wave pack­ets move ap­prox­i­mately like clas­si­cal par­ti­cles do. Un­der such con­di­tions, a sim­ple the­ory called the WKB ap­prox­i­ma­tion ap­plies.

If the forces vary more rapidly on quan­tum scales, more weird ef­fects are ob­served. For ex­am­ple, wave pack­ets may be re­pelled by at­trac­tive forces. On the other hand, wave pack­ets can pen­e­trate through bar­ri­ers even though clas­si­cally speak­ing, they do not have enough en­ergy to do so. That is called tun­nel­ing. It is im­por­tant for var­i­ous ap­pli­ca­tions. A sim­ple es­ti­mate for the prob­a­bil­ity that a par­ti­cle will tun­nel through a bar­rier can be ob­tained from the WKB ap­prox­i­ma­tion.

Nor­mally, a wave packet will be par­tially trans­mit­ted and par­tially re­flected by a fi­nite bar­rier. That pro­duces the weird quan­tum sit­u­a­tion that the same par­ti­cle is go­ing in two dif­fer­ent di­rec­tions at the same time. From a more prac­ti­cal point of view, scat­ter­ing par­ti­cles from ob­jects is a pri­mary tech­nique that physi­cists use to ex­am­ine na­ture.

Sub­sec­tions