Chapter 5 APPROXIMATION METHODS FOR STATIONARY STATES

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1 Chapter 5 APPROIMATION METHODS FOR STATIONARY STATES As we have see, the task of predicitig the evolutio of a isolated quatum mechaical ca be reduced to the solutio of a appropriate eigevalue equatio ivolvig the Hamiltoia of the system. Ufortuately, oly a small umber of quatum mechaical systems are ameable to a exact solutio. Moreover, eve whe a exact solutio to the eigevalue problem is available, it is ofte useful to uderstad the behavior of the system i the presece of weak exteral fields that my be imposed i order to probe the structure of its statioary states. I these situatios a approximate method is required for calculatig the eigestates of the Hamiltoia i the presece of a perturbatio that reders a exact solutio uteable. There are two geeral approaches commoly take to solve problems of this sort. The first, referred to as the variatioal method, is most useful i obtaiig iformatio about the groud state of the system, while the secod, geerally referred to as time-idepedet perturbatio theory, is applicable to ay set of discrete levels ad is ot ecessarily restricted to the solutio of the eergy eigevalue problem, but ca be applied to ay observable with a discrete spectrum. 5.1 The Variatioal Method Let H be a time-idepedet observable (e.g., the Hamiltoia) for a physical system havig (for coveiece) a discrete spectrum. The ormalized eigestates { φ i} of H each satisfy the eigevalue equatio H φ i = E φ i (5.1) where for coveiece i what follows we assume that the eigevalues ad correspodig eigestates have bee ordered, so that E 0 E 1 E. (5.) Uder these circumstaces, if ψi is a arbitrary ormalized state of the system it is straightforward to prove the followig simple form of the variatioal theorem: the mea value of H with respect to a arbitrary ormalized state ψi is ecessarily greater tha the actual groud state eergy (i.e., lowest eigevalue) of H, i.e., The proof follows almost trivially upo usig the expasio hhi ψ = hψ H ψi E 0. (5.3) H = φ ie hφ (5.4) of H i its ow eigestates to express the mea value of iterest i the form hhi ψ = hψ φ ie hφ ψi = ψ E, (5.5)

2 134 Approximatio Methods for Statioary States ad the otig that each term i the sum is itself bouded, i.e., ψ E ψ E 0, so that ψ E ψ E 0 = E 0 (5.6) where we have used the assumed ormalizatio hψ ψi = P ψ =1of the otherwise arbitrary state ψi. Note that the equality holds oly if ψi is actually proportioal to the groud state of H. Thus, the variatioal theorem proved above states that the groud state miimizes the mea value of H take with respect to the ormalized states of the space. This has iterestig implicatios. It meas, for example, that oe could simply choose radom vectors i the state space of the system ad evaluate the mea value of H with respect to each. The smallest value obtaied the gives a upper boud for the groud state eergy of the system. By cotiuig this radom, or Mote Carlo, search it is possible, i priciple, to get systematically better (i.e., lower) estimates of the exact groud state eergy. It is also possible to prove a stroger statemet that icludes the simple boud give above as a special case: the mea value of H is actually statioary i the eighborhood of each of its eigestates φi. This fact, which is a more complete ad precise statemet of the variatioal theorem, is compactly expressed i the laguage of the calculus of variatios through the relatio δhhi φ =0. (5.7) To see what this meas physically, let φi be a ormalizable state of the system about which we cosider a family of kets φ(λ)i = φi + λ ηi (5.8) that differ by a small amout from the origial state φi, where ηi is a fixed but arbitrary ormalizeable state ad λ is a real parameter allowig us to parameterize the small but arbitrary variatios δ φi = φ(λ)i φi = λ ηi (5.9) of iterest about the ket φi = φ(0)i. Let us ow deote by ε(λ) =hhi λ = hφ(λ) H φ(λ)i hφ(λ) φ(λ)i (5.10) the mea value of H with respect to the varied state φ(λ)i, iwhichwehaveicluded the ormalizatio i the deomiator so that we do ot have to worry about costraiig the variatio to ormalized states. With these defiitios, the, we wish to prove the followig: the state φi is a eigestate of H if ad oly if, for arbitrary ηi, ε =0. (5.11) λ λ=0 To prove the statemet we first compute the derivative of ε(λ) usig the chai rule, i.e., itroducig the otatio φ 0 i = φ(λ)i λ = ηi hφ 0 = hφ(λ) λ = hη (5.1)

3 The Variatioal Method 135 we have (sice H is idepedet of λ) ε = hφ0 H φi + hφ H φ0 i λ λ=0 hφ φi hφ φi hφ H φi hφ φi hφ 0 φi + hφ φ 0 i. (5.13) This ca be multiplied through by hφ φi ad the idetity φ 0 i = ηi used to obtai the relatio hφ φi ε = hη H φi + hφ H ηi hφ H φi [hη φi + hφ ηi]. (5.14) λ λ=0 hφ φi Now deote by E the mea value of H with respect to the uvaried state, i.e., set E = ε(0) = hφ H φi (5.15) hφ φi ad write E [hη φi + hφ ηi] =hη E φi + hφ E ηi toputtheaboveexpressioitheform hφ φi ε = hη (H E) φi + hφ (H E) ηi. (5.16) λ λ=0 We ow ote that if φi is a actual eigestate of H its eigevalue must be equal to E = ε(0), i which case the right had side of the last expressio vaishes (idepedet of the state ηi). Sice φi is ozero, we coclude that the derivative i (5.11) ad (5.16) vaishes for arbitrary variatios δ φi = λ ηi about ay eigestate φi of H. To prove the coverse we ote that, if the derivative of ε(λ) with respect to λ does ideed vaish for arbitrary kets ηi, the it must do so for ay particular ket we choose; for example, if we pick ηi =(H E) φi. (5.17) the, (5.16) above reduces to the relatio 0=hη (H E) φi + hφ (H E) ηi = hφ (H E) φi. (5.18) Because, by assumptio, H is Hermitia ad E real we ca ow iterpret this last equatio as tellig us that hφ (H E) φi = (H E) φi =0, (5.19) which meas that the vector (H E) φi must vaish, ad that φi is therefore a eigestate of H with eigevalue E wheever the derivative (5.11) vaishes for arbitrary variatios δ φi = λ ηi, completig the proof. I practice, use of this priciple is referred to as the variatioal method, the basic steps of which we eumerate below: 1. Choose a appropriate family { φ(α)i} of ormalized trial kets which deped parameterically o a set of variables α = {α 1, α,, α }, referred to as variatioal parameters.. Calculate the mea value as a fuctio of the parameters α. hh(α)i = hφ(α) H φ(α)i (5.0) 3. Miimize E(α) =hh(α)i with respect to the variatioal parameters by fidig the values α 0 for which hhi =0 i =1,,,. (5.1) α i α=α0

4 136 Approximatio Methods for Statioary States The value E(α 0 ) so obtaied is the variatioal estimate of the groud state eergy with respect to this family of trial kets, ad the correspodig state φ(α 0 )i provides the correspodig variatioal approximatio to the groud state. It should be oted that if the family { φ(α)i} of trial kets actually cotais the groud state (or ay excited state), the variatioal priciple shows that the techique described above will fid it ad the correspodig eergy exactly. This ca sometimes be exploited. For example, if symmetry properties of the groud state are kow (parity, agular mometum, etc.) it is ofte possible to choose a family of trial kets that are orthogoal to the exact groud state of the system. I this situatio, the variatioal method will the yield a upper boud to the eergy of the lowest lyig excited state of the system that is ot orthogoal to the family of trial kets employed. We also ote, i passig, that our derivatio of the variatioal priciple shows that a approximatio to the groud state φ 0 i that is correct to order ε (i.e., φ(ε)i = φ 0 i + ε ηi) will yield a estimate of the groud state eergy E 0 which is correct to order ε. This follows from the fact that i a expasio E (ε) =E 0 + ε E ε! ε=0 + ε E ε + (5.) ε=0 of the mea eergy about that of the actual groud state, the liear term vaishes due to the statioarity coditio derived above. This explais the ofte observed pheomeo that a rather poor approximatio to the eigestate ca yield a relatively good estimate of the groud state eergy. A particulalry useful applicatio of the variatioal method ivolves what is referred to as the Rayleigh-Ritz method, which overcomes to some extet the usual difficulty of dealig with a ifiite dimesioal space. Suppose for example that we were to take as a trial ket a state φi = φ i ii (5.3) i expaded i terms of some orthoormal set of vectors { ii}, ad take the expasio coefficiets φ i (or their real ad imagiary parts) as our variatioal parameters. If the set { ii} is complete, the the family { φi} of trial kets icludes all physical vectors i the state space, ad the resultig variatioal procedure will just geerate the exact eigevectors of H. Suppose, o the other had, that the states { ii} are ot complete, but spa some N dimesioal subspace S N. The variatioal procedure would the search through this fiite dimesioal subspace to fid those states that are closest to beig actual eigestates of the full system. The resultig vectors would the extremize the mea value hhi = hφ H φi = i,j φ i H ij φ j (5.4) take with respect to the states φi i this subspace. I this last expessio, H ij = hi H ji deotes the matrix elemets of H with respect to the orthoormal states { ii} spaig the subspace S N. Suppose, however, we itroduce a ew Hermitia operator H (S) defied oly o the subspace S N ad havig the same matrix elemets H (S) ij = hi H ji = H ij (5.5) as H withi that subspace (but which vaishes outside of S N );then eigevectors of this restricted operator H (S) will be those states φi i S N which extremize the mea value hh (S) i = i,j φ i H (S) ij φ j = i,j φ i H ij φ j, (5.6)

5 Perturbatio Theory for Nodegeerate Levels 137 i.e., they will be precisely the variatioal eigestates of the full Hamiltoia H that we are lookig for. Thus, i this case, applicatio of the variatioal method simply amouts to diagoalizig the matrix represetig H restricted to some fiite subspace of iterest. Moreover, as implied by our previous commets o the variatioal method, the Rayleigh- Ritz method described above will exactly fid ay actual eigevectors of H that lie etirely withi the chose subspace S N. 5. Perturbatio Theory for Nodegeerate Levels We ow tur to a more geeral ad systematic method for determiig the eigevector ad eigevalues for observables with a discrete spectrum. As i the last sectio we use the laguage of eergy eigestates ad Hamiltoia eve though the method itself is perfectly applicable to other observables. For the purposes of statig the iitial problem of iterest, however, we cosider the eigevalue problem for a time-idepedet Hamiltoia H = H (0) + H (1) = H (0) + λv (5.7) havig a discrete odegeerate spectum. IwritigtheHamiltoiaithisform,the eigevalue problem for the operator H (0), which will be referred to as the uperturbed part of the Hamiltoia, is assumed to have bee solved, ad the perturbatio H (1) = λv is presumed to be, i some sese, small compared to H (0). Our goal is to obtai expressios for the eigestates i = (λ)i ad eigevalues ε = ε (λ) of H as a expasio i powers of the small, real parameter λ. These eigestates of the full Hamiltoia H are to be expressed as liear combiatios ad simple fuctios of the kow eigestates (0) i ad eigevalues ε (0) of the uperturbed Hamiltoia H (0). Thus, the exact ad uperturbed states of the system are assumed to satisfy the equatios (H ε ) i =0 h mi = δ,m ih =1 (5.8) (H (0) ε (0) ) (0) i =0 h (0) m (0) i = δ,m (0) ih (0) =1, (5.9) the two relatios o the right of the last two lies idicatig that both sets of states form a ONB for the space of iterest. We wish to idetify, i particular, the uperturbed eigestates (0) i as those to which the exact states i ted as λ 0. This still leaves the relative phase of the two sets of basis vector udetermied, as we could multiply the basis vectors of oe set by a arbitrary set of phases e iφ without affectig the validity of the equatios above. For ozero values of λ, therefore, we further fix the relative phase betwee these two sets of states by requirig that the ier product h (0) i betwee correspodig elemets of these two basis sets be real ad positive. Now, by assumptio, there exist expasios of the full eigestate i ad the correspodig eigeeergy ε of the form i = (0) i + λ (1) i + λ () i + (5.30) ε = ε (0) + λε (1) + λ ε () + (5.31) We will refer to the terms λ k ε (k) ad λ k (k) i as the kth order correctio to the th eigeeergy ad eigestate, respectively. The correspodig correctio to the eergy is also geerally referred to as the kth order eergy shift, for obvious reasos. To determie these correctios, we will simply require that the exact eigestate i satisfy the appropriate eigevalue equatio (H ε ) i =0 (5.3)

6 138 Approximatio Methods for Statioary States to all orders i λ. Upo substitutio of the expasios for i ad ε ito the eigevalue equatio we obtai or (H (0) + λv ) λ k (k) i = λ k ε (k) k=0 k=0 k=0 λ j (j) i = j=0 j=0 k,j=0 λ j+k ε (k) (j) i (5.33) λ k H (0) (k) i + V λ k+1 (k) i λ j+k ε (k) (k) i =0. (5.34) For this equatio to hold for small but arbitrary values of λ, the coefficiets of each power of that parameter must vaish separately. The reaso for this is essetially that the polyomials f k (λ) =λ k form a liearly idepedet set of fuctios o R, soayrelatio of the form P k=0 c kλ k =0ca oly be satisfied for all λ i R if c k =0for all k. Applyig this requiremet to the last equatio geerates a ifiite heirarchy of coupled equatios, oe for each power of λ. The equatio geerated by settig the coefficiet of λ k equal to zero is referred to as the kth order equatio. Collectig coefficiets of the first few powers of λ we obtai after a little rearragemet the zeroth order equatio the first order equatio the secod order equatio the third order equatio (H (0) ε (0) ) (0) i =0, (5.35) (H (0) ε (0) ) (1) i +(V ε (1) ) (0) i =0, (5.36) (H (0) ε (0) ) () i +(V ε (1) ) (1) i ε () (0) i =0, (5.37) (H (0) ε (0) ) (3) i +(V ε (1) ) () i ε () (1) i ε (3) (0) i =0, (5.38) ad fially, after ispectig those which precede it, we deduce for k theformofthe geeral kth order equatio (H (0) ε (0) ) (k) i +(V ε (1) ) (k 1) i k j= ε (j) (k j) i =0. (5.39) As we will demostrate, the structure of these equatios allows for the geeral kth order solutios to be obtaied from those solutios of lower order, allowig for the developmet of a systematic expasio of the eigestates ad eigeeergies i powers of λ. Tobegi the demostratio we ote that the zeroth order equatio (5.35) is already satisfied, by assumptio. From kowledge of the uperturbed states ad eigeeergies, Eq. (5.36) ca be solved to give the first order correctio ε (1) to the eergy. This is most easily doe by simply multiplyig (5.36) o the left by the uperturbed eigebra h (0), i.e., h (0) (H (0) ε (0) ) (1) i + h (0) (V ε (1) ) (0) i =0. (5.40) Sice H (0) is Hermitia (ad ε (0) therefore real) it follows that h (0) (H (0) ε (0) )=0, ad so the first order equatio reduces to the relatio ε (1) = h (0) V (0) i λε (1) = h (0) H (1) (0) i. (5.41)

7 Perturbatio Theory for Nodegeerate Levels 139 Thus, the first order correctio to the eergy eigevalue for the th level is simply the mea value of the perturbig Hamiltoia take with respect to the uperturbed eigefuctios. Note that the first order correctio to the eergy comes from a mea value take with respect to the zeroth order approximatio to the state, cosistet with the remarks made earlier i the cotext of the variatioal method. As we will see, a similar structure persists to all orders of perturbatio theory, amely, a approximatio of the state to kth order geerates a approximatio to the eergy that is correct to order k +1. Now that we have ε (1), we ca put it back i to the first order equatio (5.36) to fid a expasio for the first order correctio (1) i totheeigestate.sicewewatto express this correctio as a expasio (1) i = m m (0) ihm (0) (1) i (5.4) i uperturbed eigestates m (0) i of H (0), we obviously eed to evaluate the expasio coeffiiciets hm (0) (1) i. Thus, we ow take ier products of the first order equatio (5.36) with the other members of this complete set of states, i..e., for the states with m 6=. Multiplyig (5.36) o the left by hm (0) we obtai hm (0) (H (0) ε (0) ) (1) i + hm (0) (V ε (1) ) (0) i =0 (5.43) ad observe that for the first term o the left of this expressio hm (0) (H (0) ε (0) ) (1) i =(ε (0) )hm (0) (1) i, (5.44) while orthogoality of the uperturbed states implies that, for the secod term, hm (0) ε (1) (0) i = ε (1) hm(0) (0) i =0 for m 6=. (5.45) Thus, we obtai after a little rearragemet the followig result hm (0) (1) i = hm(0) V (0) i ε (0) m 6=. (5.46) for the expasio coefficiets of iterest. This procedure for obtaiig the expasio coefficiets for the state (1) i does ot work for the term with m =, sice it just leads, agai, to the expressio ε (1) = h (0) V (0) i for the first order eergy correctio. As it turs out, however, the oe remaiig expasio coefficiet h (0) (1) i ca be evaluated from the ormalizatio coditio h i =1ad our already chose phase covetio. The ormalizatio coditio implies the expasio h i 1 = h (0) + λh (1) + λ h () + [ (0) i + λ (1) i + λ () i + ] h i = h (0) (0) i + λ h (1) (0) i + h (0) (1) i + O(λ ). (5.47) Sice, by assumptio, h (0) (0) i =1, all of the remaiig terms o the right-had side of this expasio must vaish, term-by-term. Thus, to first order ormalizatio of the eigestates requires that ³ 0=h (1) (0) i + h (0) (1) i =Re h (0) (1) i. (5.48) O the other had, we have chose our phase covetio so that the ier product h i h (0) i = h (0) + λh (1) + λ h () + (0) i = h (0) (0) i + λh (1) (0) i + λ h () (0) i + (5.49)

8 140 Approximatio Methods for Statioary States is real ad positive. Settig the imagiary part equal to zero gives the coditio h i λ k Im h (k) (0) i =0, (5.50) k=1 which agai requires, for arbitrary λ, that each ier product i the sum be separately real. Combiig this with (5.48) we deduce, therefore, that h (0) (1) i =0. (5.51) By our simple choice of phase, the, the first order correctio (1) i is forced to be orthogoal to the uperturbed eigestate (0) i. Usig this fact we the ed up with the followig expasio (1) i = m (0) ihm (0) V (0) i (5.5) m6= ε (0) for the first order correctio, ad similar expasios i = (0) i hm (0) λv (0) i m (0) i + O(λ ) (5.53) m6= ε (0) i = (0) i hm (0) H (1) (0) i m (0) i + O(λ ) (5.54) m6= ε (0) for the full eigestate, correct to first order i the perturbatio. This expressio shows that the pertubatio H (1) mixes the eigestates of H (0), bywhichisreferredtothe fact that the eigestates of H are liear combiatios of the uperturbed eigestates. Note also that the presece of the eergy deomiators ε (0) appearig i the expasio coefficiets i this expressio ted to mix together states close together i eergy more strogly tha states that are eergetically disparate. This makes it clear why we assumed from the outset that the uperturbed spectrum was o-degeerate, sice the method we have developed clearly must fail whe applied to perturbatios that coect degeerate states. It is also clear that a implicit coditio for the perturbatio expasio to coverge, i.e., that the correctio terms be sufficietly small is that hm (0) H (1) (0) i ε (0) (5.55) for all states m (0) i coected to the state (0) i by the perturbig Hamiltoia. Whe the first order correctio to the eergy vaishes, or higher accuracy is required, it is ecessary to go to higher order i the perturbatio expasio. To obtai the secod order eergy correctio we proceed as follows: multiply the secod order equatio (5.37) o the left by the uperturbed eigebra h (0) to obtai h (0) (H (0) ε (0) ) () i + h (0) (V ε (1) ) (1) i ε () h (0) (0) i =0. (5.56) Agai usig the fact that h (0) is a eigebra of H 0, alog with the orthogoality relatio h (0) (1) i =0deduced above, we fid that ε () = h (0) V (1) i. (5.57) Isertig the expasio deduced above for (1) i, we the obtai the secod order eergy shift λ ε () = h (0) λv m (0) ihm (0) λv (0) i = m6= ε (0) m6= λv m ε (0) (5.58)

9 Perturbatio Theory for Nodegeerate Levels 141 where the quatities λv m = hm (0) λv (0) i = H (1) m are just the matrix elemets of the perturbatio betwee the uperturbed eigestates. Thus, the full eigeeergies, correct to secod order, are give by the expressio ε = ε (0) + H (1) m6= H (1) m ε (0) + O(λ 3 ). (5.59) I problems ivolvig weak perturbatios it usually suffiices to determie correctios ad eergy shifts to lowest o-vaishig order i the perturbatio, ad so it is uusal that oe eeds to go beyod secod order for simple problems i perturbatio theory. Exceptios to this geeral observatio arise quite ofte whe dealig with may-body problems, where diagramatic methods have bee developed that take the ideas of perturbatio theory to a extrememly high level, ad where it is ot ucommo to fid examples where effects of the perturbatio are calculated to all orders. It is worth poitig out, however, that the first order correctio to the eergy cotais o iformatio i it about ay chages that occur i the eigestates of the system as a result of the perturbatio. This iformatio appears for the first time i the secod order eergy shift, as the derivatio above makes clear. I may cases it is ot possible to perform the sum i (5.59) exactly, ad so it is useful to develop simple meas for estimatig the magitude of the secod order eergy shift. As it turs out, it is ofte straightforward to develop upper ad lower bouds for the magitude of the chage i eergy that occurs i ay give eigestate. For example, a geeral upper boud for the secod order shift ca be obtaied for ay odegeerate level by observig that ε () = m6= H (1) m ε (0) ε (0) m m6= ε (0) H (1) m ε (0) m (5.60) where i the right had side we have a sum of positive defiite terms that will always be larger i magitude tha a similar sum i which some of the correspodig terms are positive ad some egative, depedig upo where the eergy of each level lies relative to the oe of iterest. Moreover, each term i the sum o the right ca itself be bouded, sice the eergy deomiators are bouded from below by that associated with the level closest i eergy to the state (0) i. If we deote by ε the eergy spacig betwee level ad the state closest i eergy to it, the ε (0) ε (0) m ε ad so ε () 1 ε m6= H (1) m = λ ε m6= V m We ca perform the ifiite sum by removig the restrictio o the summatio idex. We do this latter trick by addig ad subtractig the quatity V = hv i, where hv i = V = h (0) V (0) i is the mea value of the perturbatio take with respect to the uperturbed eigestate. (It is, essetially, just the first order eergy correctio ε (1) ). Performig this operatio allows us to write the upper boud above i the form " # ε () λ hm (0) V (0) i hv i. (5.61) ε m where the sum is ow urestricted. But sice the uperturbed states form a complete set

10 14 Approximatio Methods for Statioary States of states we ca ow rewrite the sum as hm (0) V (0) i = h (0) V m (0) ihm (0) V (0) i m m = h (0) V (0) i = hv i (5.6) which is just the mea value of the square of the perturbatio take with respect to the uperturbed eigestate. Makig this substitio above ad recogizig the root-measquare statistical ucertaity V = hv i hv i (5.63) associated with the perturbig operator take with respect to the uperturbed state of iterest, we obtai our fial result for the upper boud ε () λ V ε = H (1) ε (5.64) i the secod order eergy shift. We ote i passig that this gives aother ituitively reasoable measure for determiig the validity of the perturbatio expasio, which requires for the smalless of ε () that the ucertaity i H (1) be small relative to the eergy level spacig associated with the uperturbed states, i.e., that H (1) / ε << 1. For the groud state eergy (or more geerally the extremal eigevalue) it is also possible to determie a lower boud o the magitude of the secod order eergy shift. For the groud state, such a boud follows from the fact that i this case the secod order eergy shift ε () 0 = m6= H (1) m ε (0) 0 (5.65) is always egative (the chage i state always leads to a lower eergy, cosistet with the variatioal priciple), because the eergy deomiators are always positive. We ca thus write ε () 0 = m6= H (1) m ε (0) 0 H(1) m ε (0) 0 (5.66) where i the right-had side we have used the fact that ay sigle term i the sum is less tha or equal to the total sum of all the positive defitite terms therei. The maximal term i the sum (which is usually oe of the low-lyig excited states closest i eergy to the groud state) ca thus be used to provide a reasoable lower boud for the the secod order shift i the groud state eergy. As a applicatio of the techiques of odegeerate perturbatio theory we cosider the example of a harmoically boud electro to which a uiform electric field is applied. Thus, we take for our Hamiltoia H = H 0 + ˆV (5.67) where H 0 = P m + 1 mω (5.68) is a simple oe-dimesioal harmoic oscillator describig the boud electro, ad the perturbig field is described by the potetial ˆV = ee = f. (5.69)

11 Perturbatio Theory for Nodegeerate Levels 143 Our goal is to treat ˆV as a small perturbatio ad calculate relevat correctios to the eergy levels ad eigestates i the presece of the applied electric field. To this ed we recall the stadard trasformatios r mω q = ~ p = P (5.70) m~ω a = 1 (q + ip) a + = 1 (q ip) N = a + a (5.71) that allow us to put the harmoic oscillator part of the problem i a simpler, dimesioless form H 0 = N + 1 r ~ ~ω ˆV = f mω q = λq = λ (a + + a) (5.7) where r ~ λ = f (5.73) mω is a (presumed small) measure of the stregth of the applied field. With these defiitios, the uperturbed states of H 0 are the usual oscillator states i which obey N i = i H 0 i =( + 1 )~ω i = ε(0) i h 0 i = δ, 0. (5.74) We also have the relatios a + i = +1 +1i a i = 1i (5.75) i terms of which we readily determie that the first order eergy shift due to the applied field ε (1) = h ˆV i = λ h (a + + a) i = λ h a + i + h a i ª = λ +1h +1i + h 1i ª = 0 (5.76) vaishes due to the orthogoality of the uperturbed states. So the first order eergy shift vaishes ad we must go to secod order to calculate the eergy shift. Physically this vaishig of the first order eergy shift occurs for the uperturbed states because they have equal weight o each side of the origi, ad so the et chage i eergy due to the liear applied potetial vaishes. We ca aticipate that the secod order correctio will cause a lowerig of the eergy as the electro displaces i the presece of the field, lowerig its potetial eergy i the process. To see this we first calculate the first order correctio to the eigestates. I the preset problem we will deote by ˆi the exact eigestates of the system that are presumed to have a expasio ˆi = i + λ i (1) + λ i () + (5.77) i powers of the small parameter λ. The first order correctio is give, accordig to the results of the last sectio, by the expressio λ i (1) = m6= hm ˆV i mi = λ ε (0) ε (0) m 0 6= hm (a + + a) i mi. (5.78) ( m)~ω

12 144 Approximatio Methods for Statioary States Lettig a ad a + act to the right we fid after a little calculatio that, except for the groud state, this reduces to a sum of just two terms λ i (1) = λ +1 +1i + 1i ~ω ~ω λ = +1 +1i 1i. (5.79) ~ω Thus, to first order the exact eigestates of H ca be writte ˆi = i + λ ~ω +1 +1i 1i + O(λ ). (5.80) Thus, the perturbatio mixes oly the states immediately above ad below the uperturbed level. Note that the correspodig expressio for the groud state does ot cotai the secod term i the above expressio, i.e., ˆ0i = 0i + λ ~ω 1i + O(λ ). (5.81) We ow cosider the secod order eergy shift ε () = hm ˆV i m6= ε (0) ε (0) m (5.8) which will also (except for the groud state) cotai just two terms: ( ) ε () = λ h +1 a + i h 1 a i + ~ω ~ω = λ Thus, to secod order, we fid ½ +1 ~ω + ~ω ε = ε (0) ¾ = λ ~ω = f mω. (5.83) λ ~ω = f mω. (5.84) We ote that for this particular problem the eergy shift is the same for all states, that is, all of the eergies of the system are lowered by the same amout i the presece of the field. It turs out that the secod order eergy correctio for this problem gives the exact eigeeergies (eve though the first order correctio to the state does ot give the exact eigestates). This result is physically iuitive, sice it correspods to the fact that a classical mass-sprig system whe hug i a gravitatiol field simply stretches, or displaces, to a ew equilibrium positio, thereby lowerig its potetial eergy, but cotiues to oscillate with the same frequecy as it would if it were left uperturbed. To establish this result i the preset cotext we perform a caoical trasform to a ew set of variables ˆq = q λ ˆp = p (5.85) ~ω which has the positio coordiate ow cetered at the ew force ceter at q = λ/~ω. This trasformatio preserves the commutatio relatios [ˆq, ˆp] =[q, p] =i (5.86)

13 Perturbatio Theory for Nodegeerate Levels 145 ad allows us to write the Hamiltoia i terms of the ew coordiate ad mometa i the form H = ~ω (q + p ) λq = ~ω ˆq + λ λ ˆq + ~ω ~ ω +ˆp µ λ ˆq + λ ~ω = ~ω (ˆq +ˆp ) λ ~ω. (5.87) We ca ow itroduce operators â = ˆq + iˆp â + = ˆq iˆp ˆN =â + â i the usual way, so that the Hamiltoia takes the form H =(ˆN + 1 λ )~ω ~ω of a oscillator of frequecy ω lowered uiformly i eergy by a amout λ /~ω. Of course this oscillator has its equilibrium positio at ˆq = q λ/~ω =0, i.e., shifted with respect to the uperturbed oscillator, ad its eergy levels are i exact agreemet with those foud to secod order i the applied field usig perturbatio theory. I fact, this displacemet of the oscillator uder the actio of the field makes it clear that the exact eigestates of the system satisfy the equatio hˆq ˆi = φ (ˆq) =φ (q λ/~ω) =φ (q ε), where ε = λ/~ω, i.e., they are just the uperturbed oscillator states shifted alog the x-axis, ad cetered at the ew equilibrium positio q = ε. The uitary operator which effects this trasformatio is the correspodig traslatio operator T (ε) =e ipε which, i the positio represetatio has the effect of displacig the wave fuctio, i.e., T (ε)ψ(q) =ψ(q ε). Thus, we expect that the uperturbed ad perturbed eigestates are related through the relatio ˆi = T (ε) i = e ipε i. For small ε (or small λ) we ca expad the expoetial as T (ε) ' 1 ipε so that ˆi '(1 ipε) i = i ipε i. Usig the fact that ipε = ε(a + a)/ ad substitutig back i the defiitio of ε = λ/~ω we recover the result ˆi ' i + λ +1i λ 1i ~ω ~ω that we obtaied usig the techiques of first order perturbatio theory.

14 146 Approximatio Methods for Statioary States 5.3 Perturbatio Theory for Degeerate States The expressios that we derived above for the first order correctio to the groud state ad the secod order correctio to the eergy are clearly iappropriate to situatios i which degeerate or early-degeerate eigestates are coected by the perturbatio, sice the correspodig correctios all diverge as the spacig betwee the eergy levels goes to zero. This divergece is a idicatio of the stregth with which the perturbatio tries to mix together states that are very-early degeerate (or exactly so), ad suggest that we might wish to treat differyl those states that are kow i advace to be very closely related i eergy. I this sectio, therefore, we discuss the geeral approach take to deal with problems of this sort. We assume, as before, that the Hamiltoia of iterest ca be separated ito two parts, which we ow write i the simplied form H = H 0 + V (5.88) where, sice we will ot be developig a systematic expasio i powers of the perturbatio we have o eed for the more complex otatio used previously. We agai seek the exact eigestates ad eigeeergies of H, expressed as a expasio i eigestates of H 0, the latter of which are assumed to be at least partially degeerate. We will deote by { φ, τi} a arbitrary othoormal basis of eigestates of H 0, where the idex τ is icluded to distiguish betwee the differet liearly idepedet basis states of H 0 havig the same uperturbed eergy ε (0).Thebasisstates { φ, τi τ =1,,N } (5.89) with fixed eergy ε (0) form a basis for a eigesubspace S(ε (0) ) of H 0 correspodig to that particular degeerate eergy. The dimesio N of this subspace is just the degeeracy of the correspodig eigevalue ε (0) of H 0. It is importat to poit out that, due to the degeeracy, our choice of the basis set { φ, τi} is ot uique; ay uitary trasformatio carried out withi ay oe of the eigespaces S(ε (0) ) geerates a ew basis { χ, τi} that ca be used as readily as ay other for expadig the exact eigestates of H. Ay such basis set will satisfy the obvious eigevalue, orthogoality, ad completeess relatios H 0 φ, τi = ε (0) φ, τi hφ 0, τ 0 φ, τi = δ 0,δ τ 0,τ φ, τihφ, τ =1 H 0 χ, τi = ε (0) χ, τi hχ 0, τ 0 χ, τi = δ 0,δ τ 0,τ,τ,τ χ, τihχ, τ =1. (5.90) We ow observe that the divergeces that reder the formulae of odegeerate perturbatio theory iapplicable really oly arise if the perturbatio actually coects states withi each eigesubspace, i.e., if there exists o-zero matrix elemets V τ,0 τ 0 = hφ, τ V φ, τ 0 i of the perturbatio coectig basis states of the same eergy. Thus, our previous formulae ca, i fact be applied (at least to the level that we have developed them), uder two coceivable circumstaces, oe ivolvig the diagoal matrix elemets of V ad oe ivolvig the off-diagoal elemets: 1. If the first order correctio ε (1),τ = hφ, τ V φ, τi is distict for all the basis states φ, τi i each eigespace S(ε (0) ), the the degeeracy is lifted i the first order of the perturbatio. Provided the magitude of this splittig of the eergy levels by the perturbatio is large compared to the matrix elemets of V that coect these states

15 Perturbatio Theory for Degeerate States 147 wecathesimply redefie what we call the uperturbed ad the perturbig part of the Hamiltoia. I other words, although we orgially decomposed the Hamiltoia i the form H = H 0 + V, where, i a represetatio of eigestates of H 0, H 0 = φ, τiε (0) hφ, τ,τ V = φ, τiv τ, 0 τ 0hφ 0, τ 0,,τ; 0,τ 0 we ca ow iclude the diagoal part of V i a redefied Ĥ0 such that H = Ĥ0 + ˆV, but ow, Ĥ 0 = φ, τi[ε (0) + ε (1),τ ]hφ, τ,τ ˆV = φ, τiv τ, 0 τ 0hφ 0, τ 0,τ6= 0,τ 0 where the perturbatio ˆV ow has o diagoal matrix elemets i this represetatio. I this situatio, states withi S(ε (0) ) are ow o loger degeerate, so we ca proceed as before to apply the formulae of o-degeerate perturbatio theory, with the eergy deomiators ow icludig the first order shifts, so o divergeces occur.. If, o the other had, the off-diagoal part of the perturbatio ˆV just happes to vaish betwee all the basis states withi a give eigesubspace S(ε (0) ), the (at least to secod order) the problematic terms i the perturbatio expasio ever actually arise; thus if the submatrix [V ] represetig the perturbatio withi the degeerate subspace is diagoal, we ca actually proceed as though there were o degeeracy. I passig we might commet regardig the first of these circumstaces that the act of icludig the diagoal part of the perturbatio V i a redefied Ĥ0 ca always be performed, eve whe it does ot etirely lift the degeeracy. We may therefore assume without loss of geerality i what follows that such a operatio has already bee carried out, ad hece that the perturbatio has o diagoal compoets i the basis of iterest. Regardig the secod circumstace metioed above, it might be thought that, i actual practice, the vaishig of the off-diagoal matrix elemets of V withi S(ε (0) ) would occur i so few circumstaces that it hardly merits attetio. To the cotrary, thereisaseseiwhichitalways ca be made to occur. To uderstad this commet, ad i a the process reveal the basic techique that is geerally employed for dealig with degeerate states, we ote that the off-diagoal matrix elemets of the perturbatio V take betwee basis states i a give degeerate subspace of H 0 deped upo which set of basis states of H 0 we choose to begi with. If, e.g., we choose a set { φ, τi} we get oe set of matrix elemets V τ;τ 0 = hφ, τ V φ, τi, defiig a certai submatrix [V ],whileifwechoose,istead,ayotherset{ χ, τi} we obtai a completely differet set of matrix elemets Ṽ τ;τ 0 = hχ, τ V χ, τ 0 i

16 148 Approximatio Methods for Statioary States defiig a differet submatrix [Ṽ ] represetig the perturbatio V withi this degeerate eigesubspace. Thus, as we would expect, the submatrix [V ] or [Ṽ ] represetig the perturbatio withi the eigespace S(ε (0) ) depeds upo the particular basis set ({ φ, τi} or { χ, τi}) that we chose to work i. I light of circumstace, above, the questio that arises is the followig: uder what circumstaces ca wei fid a represetatio of basis states { χ, τi} withi S(ε (0) ) for which the submatrix hṽ represetig the perturbatio i that subspace is strictly diagoal? Isofar as the perturbatio V itself is presumed to be a observable (ad thus Hermitia), ay submatrix [V ] or [Ṽ ] represetig V withi such a subspace must itself be a Hermitia, ad related to ay of the other matrices represetig V withi this subspace by a uitary (sub)traformatio. For a fiite N dimesioal subspace, however, we kow that it is always possible to fid a represetatio that diagoalizes ay Hermitia matrix. For each subspace we just have to go through the usual procedure of fidig the roots ε,τ to the characteristic equatio det ([V ] ε) =0 for the N dimesioal submatrix [V ] that represets V withi a give eigespace S(ε (0) ), ad the solve the resultig liear equatios to fid those combiatios { χ, τi} of the origial basis vectors { φ, τi} that are also eigevectors of the submatrix [V ]. I this ew represetatio, by costructio, o elemets of the ew basis set { χ, τi} havig the same eergy uperturbed eergy are coected to oe aother by ozero matrix elemets of the perturbatio. Moreover, the diagoalizatio of V withi each eigespace S(ε (0) ) provides a ew set of eigevalues ε,τ (the roots of the characteristic equatio det [V ε] computed withi the subspace) which will form the diagoal elemets of the matrix represetig V i this represetatio. These diagoal elemets ca the be combied with those of H 0 to obtai ew uperturbed eigeeergies (correct to first order) ε,τ = ε (0) + ε,τ that will themselves ofte at least partially lift the degeeracy. The particular states foud durig the diagoalizatio ca the be chose as a ew set of zeroth order states with which to pursue higher order correctios, accordig to the techiques of odegeerate perturbatio theory. Thus, the basic result of degeerate perturbatio theory is ot a explicit formula, as it is i the odegeerate case. Rather it is a simple prescriptio: diagoalize the perturbatio V withi the degeerate subspaces of H 0 to determie a ew basis for proceedig, if ecessary, with the determiatio of higher order correctios usig stadard techiques Applicatio: Stark Effect of the =Level of Hydroge We cosider as a applicatio of the ideas developed above the splittig of the spectral lies observed i the absorbtio ad emissio spectra of the hydroge atom whe it is placed i a uiform DC electric field, the so-called Stark effect. The relevat Hamiltoia ca be writte i the expected form H = H 0 + V, where H = p m e r is the usual oe describig a sigle electro boud to the proto of a hydroge atom, ad the perturbatio V = Fz = Frcos θ

17 Perturbatio Theory for Degeerate States 149 describes the costat force F = ee exerted o the electro by a uiform field orieted alog the egative z-axis. We iitially take as our uperturbed states the stadard boud eigestates, l, mi of the hydroge atom H 0, l, mi = ε (0), l, mi ε (0) = ε 0 which have both a rotatioal ad a accidetal degeeracy of the eergy levels. The degeeracy g of the th eigeeergy (or the dimesio N of the associated eigespace S ) is give by the expressio 1 g = = (` +1). `=0 The first order correctio to the eergy of the state, l, mi due to the applied electric field vaishes, i.e., ε (1) lm = F hlm Z lmi =0 reflectig the fact that the mea positio of the electro i ay of the stadard hydroge atom eigestates is the origi. To use perturbatio theory to fid o-vaishig correctios to the eergy due to the applied field we must hadle the degeeracies. Cosider, e.g., the four-fold degeerate =level, which is spaed by the four lmi states, 0, 0i, 1, 0i, 1, 1i, 1, 1i. Withi the subspace S the submatrix represetig H 0 is, of course, diagoal ε (0) ε (0) [H 0 ]= ε (0) ε (0) To proceed we eed to costruct the matrix [V ] represetig the perturbatio withi this subspace. Thus we eed to evaluate the matrix elemets Z h,l,m z,l 0,m 0 i = d 3 r ψ,`,mzψ,`0,m 0. But the perturbig operator is clearly just the z compoet of the vector operator R. ~ Accordig to the Wiger-Eckart theorem such a operator ca oly coect states havig the same z-compoet of agular mometum, i.e., those for which m = m 0. Thus, of the states i the =maifold, the oly ozero matrix elemets for this perturbatio occur betwee the state, 0, 0i ad the state, 1, 0i. Hece, withi this subspace the matrix of iterest has the form [V ]= 0 η 0 0 η where η = h, 0, 0 Fz, 1, 0i. This latter itegral is readily evaluated i the positio represetatio, usig the kow form, h~r, 0, 0i = ψ,0,0 (~r) = 1 8a 3 ( r a )e r/a Y 0 0 (θ, φ)

18 150 Approximatio Methods for Statioary States h~r, 1, 0i = ψ,1,0 (~r) = 1 1 r 8a 3 3 a e r/a Y1 0 (θ, φ) of the hydroge =wave fuctios. Substitutig ito the itegral of iterest we fid after a short calculatio that η = F Z 16a 4 r 4 ( r Z π a )e r/a dr Thus, 0 [V ]= 0 3Fa 0 0 3Fa si θ cos θdθ = 3Fa. Diagoalizig [V ] we set det(v ε) = ε [ε (3Fa) ]=0ad fid the eigevalues ˆε,1,1 = ˆε,1, 1 =0 ˆε,+ = +3Fa ˆε, = 3Fa which we ca add to the =hydrogeic eergies to provide the first order correctios. Thus, to first order, the =eigeeergies i the presece of the field take the form ε,+ = ε (0) +3Fa ε, = ε (0) 3Fa ε,1,1 = ε,1, 1 = ε (0) which correspod, respectively, to ew zeroth order states., +i =, i =, 0, 0i +, 0, 1i, 0, 0i +, 0, 1i, 1, 1i, 1, 1i. Qualitatively we see that the four-fold degeerate =hydrogeic level is split by the field ito three separate levels, with the odegeerate lower ad higher eergy states splittig off liearlyt i the applied field from the remaiig two-fold degeerate subspace correspodig to the uperturbed eergies. With these ew basis states (i which ` is o loger a ecessarily good quatum) oe ca, i priciple, ivestigate higher order correctios to the eergy.