( ) Time-Independent Perturbation Theory. Michael Fowler 2/16/06

Transcription

1 Tie-Idepedet Perturbatio Theory Michael Fowler /6/6 Itroductio If a ato (ot ecessarily i its groud state) is placed i a exteral electric field, the eergy levels shift, ad the wave fuctios are distorted This is called the Stark effect The ew eergy levels ad wave fuctios could i priciple be foud by writig dow a coplete Hailtoia, icludig the exteral field, ad fidig the eigekets This actually ca be doe i oe case: the hydroge ato, but eve there, if the exteral field is sall copared with the electric field iside the ato (which is billios of volts per eter) it is easier to copute the chages i the eergy levels ad wave fuctios with a schee of successive correctios to the zero-field values This ethod, tered perturbatio theory, is the sigle ost iportat ethod of solvig probles i quatu echaics, ad is widely used i atoic physics, codesed atter ad particle physics It should be oted that there are probles which caot be solved usig perturbatio theory, eve whe the perturbatio is very weak, although such probles are the exceptio rather tha the rule Oe such case is the oe-diesioal proble of free particles perturbed by a localized potetial of stregth λ As we foud earlier i the course, switchig o a arbitrarily weak attractive potetial causes the k free particle wave fuctio to drop below the cotiuu of plae wave eergies ad becoe a localized boud state with bidig eergy of order λ However, chagig the sig of λ to give a repulsive potetial there is o boud state, the lowest eergy plae wave state stays at eergy zero Therefore the eergy shift o switchig o the perturbatio caot be represeted as a power series i λ, the stregth of the perturbatio This particular difficulty does ot i geeral occur i three diesios, where arbitrarily weak potetials do ot give boud states except for certai ay-body probles (like the Cooper pair proble) where the exclusio priciple reduces the effective diesioality of the available states The Perturbatio Series We begi with a Hailtoia H havig kow eigekets ad eigeeergies: H The task is to fid how these eigekets ad eigeeergies chage if a sall ter field, for exaple) is added to the Hailtoia, so: ( ) H H H (a exteral That is to say, o switchig o H,,

2 The basic assuptio i perturbatio theory is that H is sufficietly sall that the leadig correctios are the sae order of agitude as H itself, ad the true eergies ca be better ad better approxiated by a successive series of correctios, each of order H / H copared with the previous oe The strategy, the, is to expad the true wave fuctio ad correspodig eigeeergy as series i / H H, ad ters of the sae order of H H These series are the fed ito ( ) agitude i H / H o the two sides are set equal The equatios thus geerated are solved oe by oe to give progressively ore accurate results To ake it easier to idetify ters of the sae order i H / H o the two sides of the equatio, it is coveiet to itroduce a diesioless paraeter λ which always goes with H, ad the expad as power series i λ, λ λ, etc The ket, ultiplied by λ is therefore of order ( H H ) / This λ is purely a bookkeepig device: we will set it equal to whe we are through! It s just there to keep track of the orders of agitudes of the various ters Puttig the series expasios for, i we have ( λ ) H H ( H λh )( λ λ ) ( λ λ )( λ λ ) We re ow ready to atch the two sides ter by ter i powers of λ The zeroth-order ter, of course, just gives back H First-Order Ters Matchig the ters liear i λ o both sides: H H This equatio is the key to fidig the first-order chage i eergy of both sides with : Takig the ier product H H,

3 3 the usig H, ad, we fid H Takig ow λ, we have established that the first-order chage i the eergy of a state resultig fro addig a perturbig ter H to the Hailtoia is just the expectatio value of H i that state For exaple, we ca estiate the groud state eergy of the heliu ato by treatig the electrostatic repulsio betwee the electros as a perturbatio The zeroth-order groud state has the two (opposite spi) electros i the groud state hydroge-ato wave fuctio (scaled for the doublig of uclear charge) The first-order eergy correctio expectatio value e / r for this groud state wave fuctio is the give by coputig the The geeral expressio for the first-order chage i the wave fuctio is foud by takig the ier product of the first-order equatio with the bra,, H H The last ter is zero, sice, ad i the first ter H, so H ad therefore the wave fuctio correct to first order is: H The Secod-Order ergy Ter To fid the secod-order correctio to the eergy, it is ecessary to atch the secod-order ters i ( H )( ) ( )( λh λ λ λ λ λ λ ) givig: H H

4 4 Takig the ier product with yields: H H The leadig ters o the two sides cacel as before What about the ter, ad both is to say, parallel to, are oralized,? Sice i leadig order that is pure iagiary That just eas that if to this order has a copoet this order) as, that copoet has a sall pure iagiary aplitude, ad ca be writte (to iα e kets by redefiig the phase of, ad with that redefiitio directio, we ca therefore drop the ter, with α sall But the phase factor ca be eliiated So the secod-order correctio to the eergy is: has o copoet i the H H H H Selectio Rules Perturbatio theory ivolves evaluatig atrix eleets of operators Very ofte, ay of the atrix eleets i a su are zero obvious tests are parity ad the Wiger-ckart theore These are exaples of selectio rules: tests to fid if a atrix eleet ay be ozero The Quadratic Stark ffect Whe a hydroge ato i its groud state is placed i a electric field, the electro cloud ad the proto are pulled differet ways, a electric dipole fors, ad the overall eergy is lowered The perturbig Hailtoia fro the electric field is H ez e rcosθ, where is the electric field stregth, the field is i the z-directio, the electro charge e is egative We shall deote the uperturbed eigeeergies of the hydroge ato by particular we deote the groud state eergy by l /, so i The first-order correctio to the groud state eergy e z, where / r/ a ψ ( r) e 3 π a

5 5 This first-order ter is zero sice there are equal cotributios fro positive ad egative z The secod-order ter is l ez ; l, where we are ow usig l to deote the uperturbed hydroge ato wave fuctios, ad here the / (i Rydbergs) are the uperturbed eergies Most of the ters i this ifiite series are zero the selectio rules help get rid of the as follows: sice e z is the copoet of a spherical vector ad is a zero agular oetu state, it follows fro the Wiger-ckart theore that l ca oly be This reduces the secod-order su over states to: ez This is still ot easy to evaluate, but a upper boud ca be foud be observig that, so < ez ezl lez ; l, where we have teporarily restored the full su over, l,, that is, we ve put back all the zero ters The reaso for this seeig backward step is that, havig take the eergy-differece deoiator outside the su, we ca eve iclude i the l su (it s aother zero ter) ad i fact we ca eve iclude the plae-wave (ioized) states as well as the boud states, sice the plae waves all have eergy greater tha zero At this poit, the su becoes a su over all states, ad therefore just becoes the uit operator, l,, so < l,, l l I, ( e z) For the groud state hydroge wave fuctio, z a, e / a, /4, so

6 < e a a ( ) ( 3 ) 4 e /a Furtherore, sice all the ters i the series for boud o : are egative, the first ter sets a lower > ez 8 3 This ca be evaluated i straightforward fashio to fid > 55 a 3 So, eve though we have ot actually evaluated the secod-order correctio to the eergy explicitly, we have it bracketed betwee two values, the lower oe beig ore tha half the upper oe Other igeious ethods have bee developed (see Shakar or Sakurai) to fid that 9 3 the true aswer is a, but i fact the whole proble ca be solved exactly usig 4 parabolic coordiates Degeerate Perturbatio Theory: Distorted -D Haroic Oscillator The above aalysis works fie as log as the successive ters i the perturbatio theory for a coverget series A ecessary coditio is that the atrix eleets of the perturbig Hailtoia ust be saller tha the correspodig eergy level differeces of the origial Hailtoia If H has differet states with the sae eergy, i other words degeerate eergy levels, ad the perturbatio has ozero atrix eleets betwee these degeerate levels, the obviously the theory breaks dow To see just how it breaks dow, ad how to fix it, we cosider the two-diesioal siple haroic oscillator: p x py H ω x ( y ) Recall that for the oe-diesioal siple haroic oscillator the groud state wave fuctio is /4 /4 ωx / ω ξ / ω e π π e with ξ ω x, ad /4 ω ξ π e ξ / The two-diesioal oscillator is siply a product of two oe-diesioal oscillators, so, ω writig η y, the groud state is ω π states, eergy ω above the groud state, are / e ξ ( η ) /, ad the two (degeerate) ext

7 7 / / ω ( ξ η )/ ω ( ξ η )/, ξe,, ηe π π Suppose ow we add a sall perturbatio withα a sall paraeter H αω x y, Notice that H, H,, H,, so accordig to aïve perturbatio theory, there is o first-order correctio to the eergies of these states However, o goig to secod-order i the eergy correctio, the theory breaks dow The atrix eleet, H, is ozero, but the two states,,, have the sae eergy! This gives a ifiite ter i the series for Yet we kow that a sall ter of this type will ot wreck a two-diesioal siple haroic oscillator, so what is wrog with our approach? It is helpful to plot the origial haroic oscillator potetial ω ( x y ) together with the perturbig potetial αω xy The first of course has circular syetry, the secod has axes i the directios x ± y, clibig ost steeply fro the origi alog x y, fallig ost rapidly i the directios x y If we cobie the two potetials ito a sigle quadratic for, the origial circles of costat potetial becoe ellipses, with their axes aliged alog x ± y The proble arises eve i the classical two-diesioal oscillator: picture a ball rollig backwards ad forwards i a sooth saucer, a circular bowl Now iagie the saucer is ade slightly elliptical The ball will still roll backwards ad forwards through the ceter if it is released alog oe of the axes of the ellipse, although with differet periods, as the axes differ i steepess However, if it is released at a poit off the axes, it will describe a coplex path resolvable ito copoets i the two axis directios havig differet periods For the quatu oscillator as for the classical oe, as soo as the perturbatio is itroduced, the eigekets are i the directio of the ew elliptic axes This is a large chage fro the origial x ad y axes, ad defiitely ot proportioal to the sall paraeter α But the origial uperturbed proble had circular syetry, ad there was o particular reaso to choose the x ad y axes as we did If we had istead chose as our origial axes the lies x ± y, the kets would ot have udergoe large chages o switchig o the perturbatio The resolutio of the proble is ow clear: before switchig o the perturbatio, choose a set of basis kets i a degeerate subspace such that the perturbatio is diagoal i that subspace I fact, for the siple haroic oscillator exaple above, the proble ca be solved exactly:

8 8 x y x y ω ( x y ) αω xy ω ( α) ( α) ad it is clear that, despite the results of aïve first-order theory, there is ideed a first order shift i the eergy levels, ( ) ω ω ± α ω ± α / The Liear Stark ffect The hydroge ato, like the two-diesioal haroic oscillator discussed above, has a odegeerate groud state but degeeracy i its lowest excited states Specifically, there are four states, all havig eergy /4 Ryd : ψ / r, r/a ( r) e 3 3π a a ψ r r/a ( r, θφ, ) e cos, 3 3π a a θ ψ r r/a i ( r, θφ, ) e si 3 3 a a θe ± φ ± π Perturbig this syste with a electric field i the z-directio, H ez e rcosθ, ote first that aïve perturbatio theory predicts o first-order shift i ay of these eergy levels However, to secod order, there is a ozero atrix eleet betwee two degeerate levels H All the other atrix eleets betwee these basis kets i the four-diesioal degeerate subspace are zero, so the oly diagoalizatio ecessary is withi the twodiesioal degeerate subspace spaed by,, where / / with H Δ Δ Δ H r rcosθ e 3π a a a 3 ea r/ a e r drs 3 iθdθdφ

9 Diagoalizig eergy shifts 9 H withi this subspace, the, the ew basis states are ( ± ) / with ±Δ, liear i the perturbig electric field The states ± are ot chaged by the presece of the field to this approxiatio, so the coplete eergy ap of the states i the electric field has two states at the origial eergy of -/4Ryd, oe state oved up fro that eergy by Δ, ad oe dow by Δ Notice that the ew eigestates ( ± )/ are ot eigestates of the parity operator a sketch of their wave fuctios reveals that i fact they have ovaishig electric dipole oet μ, ideed this is the reaso for the eergy shift, ±Δ 3ea μ