Sub­sec­tions

5.5 Mul­ti­ple-Par­ti­cle Sys­tems In­clud­ing Spin

Spin will turn out to have a ma­jor ef­fect on how quan­tum par­ti­cles be­have. There­fore, quan­tum me­chan­ics as dis­cussed so far must be gen­er­al­ized to in­clude spin. Just like there is a prob­a­bil­ity that a par­ti­cle is at some po­si­tion , there is the ad­di­tional prob­a­bil­ity that it has spin an­gu­lar mo­men­tum in an ar­bi­trar­ily cho­sen -​di­rec­tion and this must be in­cluded in the wave func­tion. This sec­tion dis­cusses how.

5.5.1 Wave func­tion for a sin­gle par­ti­cle with spin

The first ques­tion is how spin should be in­cluded in the wave func­tion of a sin­gle par­ti­cle. If spin is ig­nored, a sin­gle par­ti­cle has a wave func­tion , de­pend­ing on po­si­tion and on time . Now, the spin is just some other scalar vari­able that de­scribes the par­ti­cle, in that re­spect no dif­fer­ent from say the -​po­si­tion of the par­ti­cle. The “every pos­si­ble com­bi­na­tion” idea of al­low­ing every pos­si­ble com­bi­na­tion of states to have its own prob­a­bil­ity in­di­cates that needs to be added to the list of vari­ables. So the com­plete wave func­tion of the par­ti­cle can be writ­ten out fully as:

 (5.16)

The value of gives the prob­a­bil­ity of find­ing the par­ti­cle within a vicin­ity of and with spin an­gu­lar mo­men­tum in the -​di­rec­tion .

But note that there is a big dif­fer­ence be­tween the spin co­or­di­nate and the po­si­tion co­or­di­nates: while the po­si­tion vari­ables can take on any value, the val­ues of are highly lim­ited. In par­tic­u­lar, for the elec­tron, pro­ton, and neu­tron, can only be or , noth­ing else. You do not re­ally have a full axis, just two points.

As a re­sult, there are other mean­ing­ful ways of writ­ing the wave func­tion. The full wave func­tion can be thought of as con­sist­ing of two parts and that only de­pend on po­si­tion:

 (5.17)

These two parts can in turn be thought of as be­ing the com­po­nents of a two-di­men­sion­al vec­tor that only de­pends on po­si­tion:

Re­mark­ably, Dirac found that the wave func­tion for par­ti­cles like elec­trons has to be a vec­tor, if it is as­sumed that the rel­a­tivis­tic equa­tions take a guessed sim­ple and beau­ti­ful form, like the Schrö­din­ger and all other ba­sic equa­tions of physics are sim­ple and beau­ti­ful. Just like rel­a­tiv­ity re­veals that par­ti­cles should have build-in en­ergy, it also re­veals that par­ti­cles like elec­trons have build-in an­gu­lar mo­men­tum. A de­scrip­tion of the Dirac equa­tion is in chap­ter 12.12 if you are cu­ri­ous.

The two-di­men­sion­al vec­tor is called a “spinor” to in­di­cate that its com­po­nents do not change like those of or­di­nary phys­i­cal vec­tors when the co­or­di­nate sys­tem is ro­tated. (How they do change is of no im­por­tance here, but will even­tu­ally be de­scribed in de­riva­tion {D.68}.) The spinor can also be writ­ten in terms of a mag­ni­tude times a unit vec­tor:

This book will just use the scalar wave func­tion ; not a vec­tor one. But it is of­ten con­ve­nient to write the scalar wave func­tion in a form equiv­a­lent to the vec­tor one:

 (5.18)

The square mag­ni­tude of func­tion gives the prob­a­bil­ity of find­ing the par­ti­cle near a po­si­tion with spin-up. That of gives the prob­a­bil­ity of find­ing it with spin-down. The spin-up func­tion and the spin-down func­tion are in some sense the equiv­a­lent of the unit vec­tors and in nor­mal vec­tor analy­sis; they have by de­f­i­n­i­tion the fol­low­ing val­ues:

The func­tion ar­gu­ments will usu­ally be left away for con­cise­ness, so that

is the way the wave func­tion of, say, an elec­tron will nor­mally be writ­ten out.

Key Points
Spin must be in­cluded as an in­de­pen­dent vari­able in the wave func­tion of a par­ti­cle with spin.

Usu­ally, the wave func­tion of a sin­gle par­ti­cle with spin will be writ­ten as

where de­ter­mines the prob­a­bil­ity of find­ing the par­ti­cle near a given lo­ca­tion with spin up, and the one for find­ing it spin down.

The func­tions and have the val­ues

and rep­re­sent the pure spin-up, re­spec­tively spin-down states.

5.5.1 Re­view Ques­tions
1.

What is the nor­mal­iza­tion re­quire­ment of the wave func­tion of a spin par­ti­cle in terms of and ?

5.5.2 In­ner prod­ucts in­clud­ing spin

In­ner prod­ucts are im­por­tant: they are needed for find­ing nor­mal­iza­tion fac­tors, ex­pec­ta­tion val­ues, un­cer­tainty, ap­prox­i­mate ground states, etcetera. The ad­di­tional spin co­or­di­nates add a new twist, since there is no way to in­te­grate over the few dis­crete points on the spin axis. In­stead, you must sum over these points.

As an ex­am­ple, the in­ner prod­uct of two ar­bi­trary elec­tron wave func­tions and is

or writ­ing out the two-term sum,

The in­di­vid­ual fac­tors in the in­te­grals are by de­f­i­n­i­tion the spin-up com­po­nents and and the spin down com­po­nents and of the wave func­tions, so:

In other words, the in­ner prod­uct with spin eval­u­ates as

 (5.19)

It is spin-up com­po­nents to­gether and spin-down com­po­nents to­gether.

An­other way of look­ing at this, or maybe re­mem­ber­ing it, is to note that the spin states are an or­tho­nor­mal pair,

 (5.20)

as can be ver­i­fied di­rectly from the de­f­i­n­i­tions of those func­tions as given in the pre­vi­ous sub­sec­tion. Then you can think of an in­ner prod­uct with spin as mul­ti­ply­ing out as:

Key Points
In in­ner prod­ucts, you must sum over the spin states.

For spin par­ti­cles:

which is spin-up com­po­nents to­gether plus spin-down com­po­nents to­gether.

The spin-up and spin-down states and are an or­tho­nor­mal pair.

5.5.2 Re­view Ques­tions
1.

Show that the nor­mal­iza­tion re­quire­ment for the wave func­tion of a spin par­ti­cle in terms of and re­quires its norm to be one.

2.

As­sume that and are nor­mal­ized spa­tial wave func­tions. Now show that a com­bi­na­tion of the two like is a nor­mal­ized wave func­tion with spin.

5.5.3 Com­mu­ta­tors in­clud­ing spin

There is no known in­ter­nal phys­i­cal mech­a­nism that gives rise to spin like there is for or­bital an­gu­lar mo­men­tum. For­tu­nately, this lack of de­tailed in­for­ma­tion about spin is to a con­sid­er­able amount made less of an is­sue by knowl­edge about its com­mu­ta­tors.

In par­tic­u­lar, physi­cists have con­cluded that spin com­po­nents sat­isfy the same com­mu­ta­tion re­la­tions as the com­po­nents of or­bital an­gu­lar mo­men­tum:

 (5.21)

These equa­tions are called the “fun­da­men­tal com­mu­ta­tion re­la­tions.” As will be shown in chap­ter 12, a large amount of in­for­ma­tion about spin can be teased from them.

Fur­ther, spin op­er­a­tors com­mute with all func­tions of the spa­tial co­or­di­nates and with all spa­tial op­er­a­tors, in­clud­ing po­si­tion, lin­ear mo­men­tum, and or­bital an­gu­lar mo­men­tum. The rea­son why can be un­der­stood from the given de­scrip­tion of the wave func­tion with spin. First of all, the square spin op­er­a­tor just mul­ti­plies the en­tire wave func­tion by the con­stant , and every­thing com­mutes with a con­stant. And the op­er­a­tor of spin in an ar­bi­trary -​di­rec­tion com­mutes with spa­tial func­tions and op­er­a­tors in much the same way that an op­er­a­tor like com­mutes with func­tions de­pend­ing on and with . The -​com­po­nent of spin cor­re­sponds to an ad­di­tional axis sep­a­rate from the , , and ones, and only af­fects the vari­a­tion in this ad­di­tional di­rec­tion. For ex­am­ple, for a par­ti­cle with spin one half, mul­ti­plies the spin-up part of the wave func­tion by the con­stant and by . Spa­tial func­tions and op­er­a­tors com­mute with these con­stants for both and hence com­mute with for the en­tire wave func­tion. Since the -​di­rec­tion is ar­bi­trary, this com­mu­ta­tion ap­plies for any spin com­po­nent.

Key Points
While a de­tailed mech­a­nism of spin is miss­ing, com­mu­ta­tors with spin can be eval­u­ated.

The com­po­nents of spin sat­isfy the same mu­tual com­mu­ta­tion re­la­tions as the com­po­nents of or­bital an­gu­lar mo­men­tum.

Spin com­mutes with spa­tial func­tions and op­er­a­tors.

5.5.3 Re­view Ques­tions
1.

Are not some com­mu­ta­tors miss­ing from the fun­da­men­tal com­mu­ta­tion re­la­tion­ship? For ex­am­ple, what is the com­mu­ta­tor ?

5.5.4 Wave func­tion for mul­ti­ple par­ti­cles with spin

The ex­ten­sion of the ideas of the pre­vi­ous sub­sec­tions to­wards mul­ti­ple par­ti­cles is straight­for­ward. For two par­ti­cles, such as the two elec­trons of the hy­dro­gen mol­e­cule, the full wave func­tion fol­lows from the every pos­si­ble com­bi­na­tion idea as

 (5.22)

The value of gives the prob­a­bil­ity of si­mul­ta­ne­ously find­ing par­ti­cle 1 within a vicin­ity of with spin an­gu­lar mo­men­tum in the -​di­rec­tion , and par­ti­cle 2 within a vicin­ity of with spin an­gu­lar mo­men­tum in the -​di­rec­tion .

Re­strict­ing the at­ten­tion again to spin par­ti­cles like elec­trons, pro­tons and neu­trons, there are now four pos­si­ble spin states at any given point, with cor­re­spond­ing spa­tial wave func­tions

 (5.23)

For ex­am­ple, gives the prob­a­bil­ity of find­ing par­ti­cle 1 within a vicin­ity of with spin up, and par­ti­cle 2 within a vicin­ity of with spin down.

The wave func­tion can be writ­ten us­ing purely spa­tial func­tions and purely spin func­tions as

As you might guess from this multi-line dis­play, usu­ally this will be writ­ten more con­cisely as

by leav­ing out the ar­gu­ments of the spa­tial and spin func­tions. The un­der­stand­ing is that the first of each pair of ar­rows refers to par­ti­cle 1 and the sec­ond to par­ti­cle 2.

The in­ner prod­uct now eval­u­ates as

This can be writ­ten in terms of the purely spa­tial com­po­nents as
 (5.24)

It re­flects the fact that the four spin ba­sis states , , , and are an or­tho­nor­mal quar­tet.

Key Points
The wave func­tion of a sin­gle par­ti­cle with spin gen­er­al­izes in a straight­for­ward way to mul­ti­ple par­ti­cles with spin.

The wave func­tion of two spin par­ti­cles can be writ­ten in terms of spa­tial com­po­nents mul­ti­ply­ing pure spin states as

where the first ar­row of each pair refers to par­ti­cle 1 and the sec­ond to par­ti­cle 2.

In terms of spa­tial com­po­nents, the in­ner prod­uct eval­u­ates as in­ner prod­ucts of match­ing spin com­po­nents:

The four spin ba­sis states , , , and are an or­tho­nor­mal quar­tet.

5.5.4 Re­view Ques­tions
1.

As an ex­am­ple of the or­tho­nor­mal­ity of the two-par­ti­cle spin states, ver­ify that is zero, so that and are in­deed or­thog­o­nal. Do so by ex­plic­itly writ­ing out the sums over and .

2.

A more con­cise way of un­der­stand­ing the or­tho­nor­mal­ity of the two-par­ti­cle spin states is to note that an in­ner prod­uct like equals , where the first in­ner prod­uct refers to the spin states of par­ti­cle 1 and the sec­ond to those of par­ti­cle 2. The first in­ner prod­uct is zero be­cause of the or­thog­o­nal­ity of and , mak­ing zero too.

To check this ar­gu­ment, write out the sums over and for and ver­ify that it is in­deed the same as the writ­ten out sum for given in the an­swer for the pre­vi­ous ques­tion.

The un­der­ly­ing math­e­mat­i­cal prin­ci­ple is that sums of prod­ucts can be fac­tored into sep­a­rate sums as in:

This is sim­i­lar to the ob­ser­va­tion in cal­cu­lus that in­te­grals of prod­ucts can be fac­tored into sep­a­rate in­te­grals:

5.5.5 Ex­am­ple: the hy­dro­gen mol­e­cule

As an ex­am­ple, this sec­tion con­sid­ers the ground state of the hy­dro­gen mol­e­cule. It was found in sec­tion 5.2 that the ground state elec­tron wave func­tion must be of the ap­prox­i­mate form

where was the elec­tron ground state of the left hy­dro­gen atom, and the one of the right one; was just a nor­mal­iza­tion con­stant. This so­lu­tion ex­cluded all con­sid­er­a­tion of spin.

In­clud­ing spin, the ground state wave func­tion must be of the gen­eral form

As you might guess, in the ground state, each of the four spa­tial func­tions , , , and must be pro­por­tional to the no-spin so­lu­tion above. Any­thing else would have more than the low­est pos­si­ble en­ergy, {D.24}.

So the ap­prox­i­mate ground state in­clud­ing spin must take the form

 (5.25)

where , , , and are con­stants.

Key Points
The elec­tron wave func­tion for the hy­dro­gen mol­e­cule de­rived pre­vi­ously ig­nored spin.

In the full elec­tron wave func­tion, each spa­tial com­po­nent must sep­a­rately be pro­por­tional to .

5.5.5 Re­view Ques­tions
1.

Show that the nor­mal­iza­tion re­quire­ment for means that

5.5.6 Triplet and sin­glet states

In the case of two par­ti­cles with spin , it is of­ten more con­ve­nient to use slightly dif­fer­ent ba­sis states to de­scribe the spin states than the four ar­row com­bi­na­tions , , , and . The more con­ve­nient ba­sis states can be writ­ten in ket no­ta­tion, and they are:

 (5.26)

A state has net spin , giv­ing a net square an­gu­lar mo­men­tum , and has net an­gu­lar mo­men­tum in the -​di­rec­tion . For ex­am­ple, if the two par­ti­cles are in the state , the net square an­gu­lar mo­men­tum is , and their net an­gu­lar mo­men­tum in the -​di­rec­tion is .

The and states can be writ­ten as

This shows that while they have zero an­gu­lar mo­men­tum in the -​di­rec­tion; they do not have a value for the net spin: they have a 50/50 prob­a­bil­ity of net spin 1 and net spin 0. A con­se­quence is that and can­not be writ­ten in ket no­ta­tion; there is no value for . (Re­lated to that, these states also do not have a def­i­nite value for the dot prod­uct of the two spins, {A.10}.

In­ci­den­tally, note that com­po­nents of an­gu­lar mo­men­tum sim­ply add up, as the New­ton­ian anal­ogy sug­gests. For ex­am­ple, for , the spin an­gu­lar mo­men­tum of the first elec­tron adds to the of the sec­ond elec­tron to pro­duce zero. But New­ton­ian analy­sis does not al­low square an­gu­lar mo­menta to be added to­gether, and nei­ther does quan­tum me­chan­ics. In fact, it is quite a messy ex­er­cise to ac­tu­ally prove that the triplet and sin­glet states have the net spin val­ues claimed above. (See chap­ter 12 if you want to see how it is done.)

The spin states and that ap­ply for a sin­gle spin- par­ti­cle are of­ten re­ferred to as the “dou­blet” states, since there are two of them.

Key Points
The set of spin states , , , and are of­ten bet­ter re­placed by the triplet and sin­glet states , , , and .

The triplet and sin­glet states have def­i­nite val­ues for the net square spin.

5.5.6 Re­view Ques­tions
1.

Like the states , , , and ; the triplet and sin­glet states are an or­tho­nor­mal quar­tet. For ex­am­ple, check that the in­ner prod­uct of and is zero.