X. Perturbation Theory

Transcription

1 X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall. I practice, this usually arises because we ca experietally cotrol the iportace of Ĥ ; for exaple, if Ĥ represets the iteractio with a exteral agetic field, we ca cotrol the stregth of this iteractio by varyig the agitude of the field. I this geeral situatio, it is useful to cosider the geeral ailtoia: ( λ ) λ ere, λ is our cotrol paraeter it allows us to isolate the ifluece of Ĥ o the eigevalues ad eigestates of Ĥ ( λ). At the ed of the calculatio, the physical ailtoia will always correspod to λ, but at the iterediate stages λ allows us to collect ters i a eaigful way. Now, give a arbitrary ailtoia, how are we to choose the appropriate referece ailtoia? It is clear that for a give Ĥ we ca choose ay referece we lie by writig: ( λ ) λ( ) ece, if we defie the full ailtoia taes the desired refereceperturbatio for for ay choice of Ĥ! I practice this is coplicated by the fact that differet choices of the referece Ĥ will give differet perturbatio expasios a good choice will give accurate aswers, but a bad choice will give poor results. Thus, i ay situatios, the accurate use of perturbatio theory essetially reduces to the art of choosig a good referece Ĥ. owever, i this course we will assue that we ow the exact eigevalues ad eigestates of Ĥ :. This severely liits our choices of Ĥ, sice at preset we oly ow two exactly solvable ailtoias (the aroic oscillator ad the

2 piecewise costat potetial). ece, every proble we treat will loo lie (aroic oscillator other ters) or (step potetial other ters). It should be stressed that this is ot a ecessary assuptio to apply perturbatio theory; oe ca also forulate perturbative expasios based o approxiately solvable referece ailtoias, but we will ot treat this case i this course. Give that we ow the eigestates ad eigevalues of Ĥ, we ow see to uderstad how Ĥ iflueces these eigevalues ad eigestates. Thus, we are iterested i the solutios of the euatio: λ λ λ λ Now, recall that our physical picture is that Ĥ has a sall ifluece o Ĥ. This ca be eforced by exaiig the behavior of the eigesyste for sall λ. To this ed, we assue that we ca expad the eigestates ad eigevalues i a Taylor series i λ : λ λ λ... ( λ ) λ λ... If we plug these expasios ito the eigevalue euatio, we obtai λ λ λ... ( ) ( ) ( λ λ...)( λ λ...) Now, this euatio ust be true for all λ. Therefore, we ca euate the coefficiets of λ, λ, λ. If we expad the products ad collect ters, we fid: th order ( λ ): st order ( λ ): d order ( λ ): etc The th order relatio is trivially satisfied because are the eigestates of the referece ailtoia. Our tas ow is to obtai ( closed for expressios for the other uows (i.e.,, )... ) i these euatios. We do this by projectig the euatios oto

3 cleverly chose states. For exaple, if we ultiply the st order euatio by o both sides, we obtai: where, i the last lie, we have used the fact that for a orthooral basis. To deal with the Kroecer delta, we treat the ad cases separately. For we obtai: Ad thus, we obtai a expressio for the first order eergy that siply reuires sadwichig the perturbig ailtoia betwee the zeroth order eigestates. I the case, we obtai: What does this last expressio ea? Note that if we wated to expad i ters of the eigestates of Ĥ we would write:. The uows i this expasio are the overlap coefficiets. By copariso with the previous expressio, we see that we ow ow all of these coefficiets i ters of atrix eleets of Ĥ ad the zeroth order eigevalues. The oe exceptio is. To fix this coefficiet, we costrai the or of λ. The oralizatio of λ does ot ifluece the out coe of ay experiet. owever, it is coveiet to eforce iterediate oralizatio:

4 ( λ λ...) λ Note that this is ot the sae as covetioal oralizatio, ad so ( λ) ( λ). owever, if we eforce iterediate oralizatio for all λ, we ca euate the coefficiets of differet powers of λ. Thus, th order ( λ ): th order ( λ, > ): ( ) Thus, i iterediate oralizatio, ow all the overlaps is zero. We therefore ad we ca therefore expad the first order wavefuctio i ters of the zeroth order eigestates:. This expressio iediately gives us a uatitative easure that lets us assess whether Ĥ is really sall : if << ( ) for all, the the first order wavefuctio will be sall. I this case our ualitative picture of perturbatio theory will be correct. Now, i practice the first order eergy correctio is very ofte zero either because of syetry or because Ĥ has bee chose very cleverly. I this case, it becoes ecessary to go to secod order i the λ expasio to deterie how Ĥ affects the eigevalues. That is, it becoes ecessary to copute. To obtai this, we project the λ euatio o to : ece, we obtai a expressio for :

5 As a exaple of how we ca apply this i practice, let s loo at the ailtoia: p ω Or, i reduced uits ( ω ħ ): p First, we idetify Ĥ ad Ĥ : p The zeroth order eergy is just the aroic oscillator eergy: while the first order eergy is ad the secod order eergy is owever, we could have coputed these i aother way. Notice that our full ailtoia is still a aroic oscillator

6 p ~ ω but with the effective freuecy ~ ω. Thus, we ca easily write dow the exact eigevalues of the full ailtoia for this siple case: ~ ( ) ( ω ). If we expad this i a power series i : ( ) ( ) 8 ( )... Coparig this with our expressios for the first ad secod order eergy correctios, we see that... which is exactly what we would expect based o our assuptio that ( λ ) λ λ... Settig λ i the latter expressio iediately yields the forer. Note that this is really just a coveiet way to chec that we have doe all the algebra right; i the geeral case, the exact eigevalues will be uow ad the perturbative results will be our oly guide. Now, the expressios above do ot apply if the eigestate we are iterested i is degeerate with aother eigestate. I this case ad the secod order eergy correctio diverges! This, i tur, is related to the fact that, i geeral, the first order chage i the eergy is ot well defied for a degeerate state: a state () () () a b that is a liear cobiatio of degeerate ' states is also a eigestate with the sae eigevalue. owever the first order eergy chage will typically be differet fro ad '. Because of this abiguity i the first derivative, the secod derivative eds up beig ifiite. Looig at the secod order eergy expressio, the oly way this could possibly be avoided is if is also zero. It ight see very uliely that this would happe but, as we are about to show, for ay uber of degeerate states we ca always ae it so that the atrix eleets of Ĥ are zero betwee differet degeerate states. Assue we have degeerate zeroth order states:

7 For which:,, 3,... j j where the eergy,, is the sae for all j. Of course, ay liear cobiatio of the wat to exploit this abiguity to force is also a eigestate with eergy ad we j to be zero for all j i i j. First ote that, i the basis, is proportioal to the j idetity atrix: ( ) Thus, if we ae a chage of basis (i.e. replace each by soe liear cobiatio of the j ) we fid: T T T T T T Where, i the last step, we have used the fact that a chage of basis is uitary. Thus, we see that is ivariat to a chage of basis that just ixes the degeerate levels. O the other had is ot ivariat to this chage of basis. We therefore trasfor to the particular liear cobiatio of the (call it the ~ basis) that reders diagoal. I this basis we have: ~ ~ ~ ~ ~ ~ ~ ~ where, i the first step, we have used the fact that is (by assuptio) diagoal. Coparig the secod ad third expressios iediately shows that ~ ~ if i j. The ~ basis is j i the correct basis i which to apply perturbatio theory, because the first order correctios are well defied. ssetially, whe the states are degeerate, eve a tiy (ifiitesial) perturbatio j proportioal to causes the eigestates to spotaeously shift to j j j j

8 ~ j ; the origial ailtoia does ot distiguish betwee these states because they are degeerate, but ε breas the degeeracy for ay ε. It is therefore coveiet to wor i the basis fro the outset. Further exaiatio of the atrix represetatio of shows that the proper first order correctio is give by: ~ ( ) ~ j That is, the correct first order eergy is give by the stadard expressio, but usig the ew fuctios ~. Oe ca go beyod this to forula re-write the secod order eergy as: ~ ~. ere, the ~ j j j ~ j states are still eigestates of Ĥ. owever, withi the subspace of degeerate eigestates of Ĥ, the ~ are chose so that the off-diagoal eleets of Ĥ vaish ad oe ca therefore eglect these ters i the suatio. Now, usually, the first order correctio will be eough to brea the degeeracy, i which case the secod order eergy is ot eeded. owever, i extraordiary cases, the secod order correctio ay be eeded. Typically, a physical Ĥ will have ay differet blocs of degeerate eigevalues. To state this pictorially: I practice, oe ust deal with each of the degeerate sub-blocs separately; oe ust fid the ~ that diagoalize Ĥ i each bloc

9 ad copute the secod order eergy correctio for each state i the bloc i tur before ovig o to the ext bloc. Note that aig a chage of basis withi oe bloc (e.g. bloc ) will ot affect the first or secod order eergy for a state i aother bloc (e.g. bloc ). ece, if you are oly iterested i states i, say, bloc, you do ot eed to worry about trasforig the degeerate states i bloc.