Sub­sec­tions

### 9.1 The Vari­a­tional Method

Solv­ing the equa­tions of quan­tum me­chan­ics is typ­i­cally dif­fi­cult, so ap­prox­i­ma­tions must usu­ally be made. One very ef­fec­tive tool for find­ing ap­prox­i­mate so­lu­tions is the vari­a­tional prin­ci­ple. This sec­tion gives some of the ba­sic ideas, in­clud­ing ways to ap­ply it best.

#### 9.1.1 Ba­sic vari­a­tional state­ment

Find­ing the state of a phys­i­cal sys­tem in quan­tum me­chan­ics means find­ing the wave func­tion that de­scribes it. For ex­am­ple, at suf­fi­ciently low tem­per­a­tures, phys­i­cal sys­tems will be de­scribed by the ground state wave func­tion. The prob­lem is that if there are more than a cou­ple of par­ti­cles in the sys­tem, the wave func­tion is a very high-di­men­sional func­tion. It is far too com­plex to be crunched out us­ing brute force on any cur­rent com­puter.

How­ever, the ex­pec­ta­tion value of en­ergy is just a sim­ple sin­gle num­ber for any given wave func­tion. It is de­fined as

where is the Hamil­ton­ian of the sys­tem. The key ob­ser­va­tion on which the vari­a­tional method is based is that the ground state is the state among all al­low­able wave func­tions that has the low­est ex­pec­ta­tion value of en­ergy:
 (9.1)

That means that if you would find for all pos­si­ble sys­tem wave func­tions, you would be able to pick out the ground state sim­ply as the state that has the low­est value.

Of course, find­ing the ex­pec­ta­tion value of the en­ergy for all pos­si­ble wave func­tions is still an im­pos­si­ble task. But you may be able to guess a generic type of wave func­tion that you would ex­pect to be able to ap­prox­i­mate the ground state well, un­der suit­able con­di­tions. Nor­mally, suit­able con­di­tions means that the ap­prox­i­ma­tion will be good only if var­i­ous pa­ra­me­ters ap­pear­ing in the ap­prox­i­mate wave func­tion are well cho­sen.

That then leaves you with the much smaller task of find­ing good val­ues for this lim­ited set of pa­ra­me­ters. Here the key idea is:

 (9.2)

Fol­low­ing that idea, what you do is ad­just the pa­ra­me­ters val­ues so that you get the low­est pos­si­ble value of the ex­pec­ta­tion en­ergy for your type of ap­prox­i­mate wave func­tion. The true ground state wave func­tion al­ways has the low­est pos­si­ble en­ergy, so the lower you make your ap­prox­i­mate en­ergy, the closer that en­ergy is to the ex­act value.

So this pro­ce­dure gives you the best pos­si­ble ap­prox­i­ma­tion to the true en­ergy, and en­ergy is usu­ally the key quan­tity in quan­tum me­chan­ics. In ad­di­tion you know for sure that the true en­ergy must be lower than your ap­prox­i­ma­tion, which is also of­ten very use­ful in­for­ma­tion.

The vari­a­tional method as de­scribed above has al­ready been used ear­lier in this book to find an ap­prox­i­mate ground state for the hy­dro­gen mol­e­c­u­lar ion, chap­ter 4.6, and for the hy­dro­gen mol­e­cule, chap­ter 5.2. It will also be used to find an ap­prox­i­mate ground state for the he­lium atom, {A.38.2}. The method works quite well even for the crude ap­prox­i­mate wave func­tions used in those ex­am­ples.

To be sure, it is not at all ob­vi­ous that get­ting the best en­ergy will also pro­duce the best wave func­tion. Af­ter all, best is a some­what tricky term for a com­plex ob­ject like a wave func­tion. To take an ex­am­ple from an­other field, surely you would not ar­gue that the best sprinter in the world must also be the best per­son in the world.

But in this case, your wave func­tion will in fact be close to the ex­act wave func­tion if you man­age to get close enough to the ex­act en­ergy. More pre­cisely, as­sum­ing that the ground state is unique, the closer your en­ergy gets to the ex­act en­ergy, the closer your wave func­tion gets to the ex­act wave func­tion. One way of think­ing about it is to note that your ap­prox­i­mate wave func­tion is al­ways a com­bi­na­tion of the de­sired ex­act ground state plus pol­lut­ing amounts of higher en­ergy states. By min­i­miz­ing the en­ergy, in some sense you min­i­mize the amount of these pol­lut­ing higher en­ergy states. The math­e­mat­ics of that idea is ex­plored in more de­tail in ad­den­dum {A.7}.

And there are other ben­e­fits to specif­i­cally get­ting the en­ergy as ac­cu­rate as pos­si­ble. One prob­lem is of­ten to fig­ure out whether a sys­tem is bound. For ex­am­ple, can you add an­other elec­tron to a hy­dro­gen atom and have that elec­tron at least weakly bound? The an­swer is not ob­vi­ous. But if us­ing a suit­able ap­prox­i­mate so­lu­tion, you man­age to show that the ap­prox­i­mate en­ergy of the bound sys­tem is less than that of hav­ing the ad­di­tional elec­tron at in­fin­ity, then you have proved that the bound state ex­ist. De­spite the fact that your so­lu­tion has er­rors. The rea­son is that, by de­f­i­n­i­tion, the ground state must have lower en­ergy than your ap­prox­i­mate wave func­tion. So the ground state is even more tightly bound to­gether than your ap­prox­i­mate wave func­tion says.

An­other rea­son to specif­i­cally get­ting the en­ergy as ac­cu­rate as pos­si­ble is that en­ergy val­ues are di­rectly re­lated to how fast sys­tems evolve in time when not in the ground state, chap­ter 7.

For the above rea­sons, it is also great that the er­rors in en­ergy turn out to be un­ex­pect­edly small in a vari­a­tional pro­ce­dure, when com­pared to the er­rors in the guessed wave func­tion, {A.7}.

To get the sec­ond low­est en­ergy state, you could search for the low­est en­ergy among all wave func­tions or­thog­o­nal to the ground state. But since you would not know the ex­act ground state, you would need to use your ap­prox­i­mate one in­stead. That would in­volve some er­ror, and it is no longer sure that the true sec­ond-low­est en­ergy level is no higher than what you com­pute, but any­way. The supris­ing ac­cu­racy in en­ergy will still ap­ply.

If you want to get truly ac­cu­rate re­sults in a vari­a­tional method, in gen­eral you will need to in­crease the num­ber of pa­ra­me­ters. The mol­e­c­u­lar ex­am­ple so­lu­tions were based on the atomic ground states, and you could con­sider adding some ex­cited states to the mix. In gen­eral, a pro­ce­dure us­ing ap­pro­pri­ate guessed func­tions is called a Rayleigh-Ritz method. Al­ter­na­tively, you could just chop space up into lit­tle pieces, or el­e­ments, and use a sim­ple poly­no­mial within each piece. That is called a fi­nite-el­e­ment method. In ei­ther case, you end up with a fi­nite, but rel­a­tively large num­ber of un­knowns; the pa­ra­me­ters and co­ef­fi­cients of the func­tions, or the co­ef­fi­cients of the poly­no­mi­als.

#### 9.1.2 Dif­fer­en­tial form of the state­ment

You might by now won­der about the wis­dom of try­ing to find the min­i­mum en­ergy by search­ing through the count­less pos­si­ble com­bi­na­tions of a lot of pa­ra­me­ters. Brute-force search worked fine for the hy­dro­gen mol­e­cule ex­am­ples since they re­ally only de­pended non­triv­ially on the dis­tance be­tween the nu­clei. But if you add some more pa­ra­me­ters for bet­ter ac­cu­racy, you quickly get into trou­ble. Semi-an­a­lyt­i­cal ap­proaches like Hartree-Fock even leave whole func­tions un­spec­i­fied. In that case, sim­ply put, every sin­gle func­tion value is an un­known pa­ra­me­ter, and a func­tion has in­fi­nitely many of them. You would be search­ing in an in­fi­nite-di­men­sion­al space, and might search for­ever.

Usu­ally it is a much bet­ter idea to write some equa­tions for the min­i­mum en­ergy first. From cal­cu­lus, you know that if you want to find the min­i­mum of a func­tion, the so­phis­ti­cated way to do it is to note that the de­riv­a­tives of the func­tion must be zero at the min­i­mum. Less rig­or­ously, but a lot more in­tu­itive, at the min­i­mum of a func­tion the changes in the func­tion due to small changes in the vari­ables that it de­pends on must be zero. Math­e­mati­cians may not like that, since the word small has no rig­or­ous mean­ing. But un­less you mis­use your small quan­ti­ties, you can al­ways con­vert your re­sults us­ing them to rig­or­ous math­e­mat­ics af­ter the fact.

In the sim­plest pos­si­ble ex­am­ple of a func­tion of one vari­able , a rig­or­ous math­e­mati­cian would say that at a min­i­mum, the de­riv­a­tive must be zero. But a physi­cists may not like that, for if you say de­riv­a­tive, you must say with re­spect to what vari­able; you must say what is as well as what is. There is of­ten more than one pos­si­ble choice for , with none pre­ferred un­der all cir­cum­stances. So a typ­i­cal physi­cist would say that the change in due to a small change in what­ever vari­able it de­pends on must be zero. It is the same thing, since for a small enough change in the vari­able, , so that if is zero, then so is . (Math­e­mat­i­cally more ac­cu­rately, if be­comes small enough, be­comes zero com­pared to .) If there is more than one in­de­pen­dent vari­able that the func­tion de­pends on, then the de­riv­a­tives be­come par­tial de­riv­a­tives, be­comes , and spec­i­fy­ing the pre­cise de­riv­a­tives would be­come much messier still.

In vari­a­tional pro­ce­dures, it is com­mon to use in­stead of or for the small change in . This book will do so too.

So in quan­tum me­chan­ics, the fact that the ex­pec­ta­tion en­ergy must be min­i­mal in the ground state can be writ­ten as:

 (9.3)

The changes must be ac­cept­able; you can­not al­low that the changed wave func­tion is no longer nor­mal­ized. Also, if there are bound­ary con­di­tions, the changed wave func­tion should still sat­isfy them. (There may be ex­cep­tions per­mit­ted to the lat­ter un­der some con­di­tions, but these will be ig­nored here.) So, in gen­eral you have con­strained min­i­miza­tion; you can­not make your changes com­pletely ar­bi­trary.

#### 9.1.3 Us­ing La­grangian mul­ti­pli­ers

As an ex­am­ple of how the vari­a­tional for­mu­la­tion of the pre­vi­ous sub­sec­tion can be ap­plied an­a­lyt­i­cally, and how it can also de­scribe eigen­states of higher en­ergy, this sub­sec­tion will work out a very ba­sic ex­am­ple. The idea is to fig­ure out what you get if you truly zero the changes in the ex­pec­ta­tion value of en­ergy over all ac­cept­able wave func­tions . (In­stead of just over all pos­si­ble ver­sions of a nu­mer­i­cal ap­prox­i­ma­tion, say.) It will il­lus­trate how the La­grangian mul­ti­plier method can deal with the con­straints.

The dif­fer­en­tial state­ment is:

But ac­cept­able is not a math­e­mat­i­cal con­cept. What does it mean? Well, if it is as­sumed that there are no bound­ary con­di­tions, (like the har­monic os­cil­la­tor, but un­like the par­ti­cle in a pipe,) then ac­cept­able just means that the wave func­tion must re­main nor­mal­ized un­der the change. So the change in must be zero, and you can write more specif­i­cally:

But how do you crunch a state­ment like that down math­e­mat­i­cally? Well, there is a very im­por­tant math­e­mat­i­cal trick to sim­plify this. In­stead of rig­or­ously try­ing to en­force that the changed wave func­tion is still nor­mal­ized, just al­low any change in wave func­tion. But add penalty points to the change in ex­pec­ta­tion en­ergy if the change in wave func­tion goes out of al­lowed bounds:

Here is the penalty fac­tor. Such penalty fac­tors are called “La­grangian mul­ti­pli­ers” af­ter a fa­mous math­e­mati­cian who prob­a­bly watched a lot of soc­cer. For a change in wave func­tion that does not go out of bounds, the sec­ond term is zero, so noth­ing changes. And if the change does go out of bounds, the sec­ond term will can­cel any re­sult­ing er­ro­neous gain or de­crease in ex­pec­ta­tion en­ergy, {D.48}, as­sum­ing that the penalty fac­tor is care­fully tuned. Note that the penalty fac­tor must be real be­cause the other two quan­ti­ties in the equa­tion above are changes in real func­tions.

You do not, how­ever, have to ex­plic­itly tune the penalty fac­tor your­self. All you need to know is that a proper one ex­ists. In ac­tual ap­pli­ca­tion, all you do in ad­di­tion to en­sur­ing that the pe­nal­ized change in ex­pec­ta­tion en­ergy is zero is en­sure that at least the unchanged wave func­tion is nor­mal­ized. It is re­ally a mat­ter of count­ing equa­tions ver­sus un­knowns. Com­pared to sim­ply set­ting the change in ex­pec­ta­tion en­ergy to zero with no con­straints on the wave func­tion, one ad­di­tional un­known has been added, the penalty fac­tor. And quite gen­er­ally, if you add one more un­known to a sys­tem of equa­tions, you need one more equa­tion to still have a unique so­lu­tion. As the one-more equa­tion, use the nor­mal­iza­tion con­di­tion. With enough equa­tions to solve, you will get the cor­rect so­lu­tion, which means that the im­plied value of the penalty fac­tor should be OK too.

So what does this vari­a­tional state­ment now pro­duce? Writ­ing out the dif­fer­ences ex­plic­itly, you must have

Mul­ti­ply­ing out, can­cel­ing equal terms and ig­nor­ing terms that are qua­drat­i­cally small in , you get

Re­mark­ably, you can throw away the sec­ond of each pair of in­ner prod­ucts in the ex­pres­sion above. To see why, re­mem­ber that you can al­low any change you want, in­clud­ing the you are now look­ing at times . If you plug that into the above equa­tion and di­vide the en­tire thing by to get rid of the added fac­tors again, you get

The two ad­di­tional mi­nus signs arise be­cause an comes out of the left side of an in­ner prod­uct as , but out of the right side as . Av­er­ag­ing this equa­tion with the orig­i­nal above it has the ef­fect of throw­ing away the sec­ond of each pair of in­ner prod­ucts in the orig­i­nal equa­tion.

You can now com­bine the re­main­ing two terms into one in­ner prod­uct with on the left:

If this is to be zero for any change , then the right hand side of the in­ner prod­uct must un­avoid­ably be zero. For ex­am­ple, just take equal to a small num­ber times the right hand side, you will get times the square norm of the right hand side, and that can only be zero if the right hand side is. So 0, or

So you see that you have re­cov­ered the Hamil­ton­ian eigen­value prob­lem from the re­quire­ment that the vari­a­tion of the ex­pec­ta­tion en­ergy is zero. Un­avoid­ably then, will have to be an en­ergy eigen­value . It of­ten hap­pens that La­grangian mul­ti­pli­ers have a phys­i­cal mean­ing be­yond be­ing merely penalty fac­tors. But note that there is no re­quire­ment for this to be the ground state. Any en­ergy eigen­state would sat­isfy the equa­tion; the vari­a­tional prin­ci­ple works for them all.

In­deed, you may re­mem­ber from cal­cu­lus that the de­riv­a­tives of a func­tion may be zero at more than one point. For ex­am­ple, a func­tion might also have a max­i­mum, or lo­cal min­ima and max­ima, or sta­tion­ary points where the func­tion is nei­ther a max­i­mum nor a min­i­mum, but the de­riv­a­tives are zero any­way. This sort of thing hap­pens here too: the ground state is the state of low­est pos­si­ble en­ergy, but there will be other states for which is zero, and these will cor­re­spond to en­ergy eigen­states of higher en­ergy, {D.49}.