G Quantum Mechanical Perturbation Theory
GQuantum Mechanical Perturbation TheoryQuantum mechanical perturbation theory is a widely used method in solidstate physics. Without the details of derivation, we shall list a number of basicformulas of time-independent (stationary) and time-dependent perturbationtheory below. For simplicity, we shall use the Dirac notation for wavefunctionsand matrix elements.G.1 Time-Independent Perturbation TheoryAssume that the complete solution (eigenfunctions and eigenvalues) of theSchrödinger equation (0) "(0) (0) " Ei ψ i(G.1.1)H0 ψiis known for a system described by a simple Hamiltonian H0 . If the systemis subject to a time-independent (stationary) perturbation described by theHamiltonian H1 – which can be an external perturbation or the interactionbetween the components of the system –, the eigenvalues and eigenfunctionschange. The method for determining the new ones depends on whether theunperturbed energy level in question is degenerate or not.G.1.1 Nondegenerate Perturbation TheoryWe now introduce a fictitious coupling constant λ, whose value will be treatedas a parameter in the calculations and set equal to unity in the final result,and write the full Hamiltonian H H0 H1 asH H0 λH1 .(G.1.2)The parameter λ is purely a bookkeeping device to keep track of the relativeorder of magnitude of the various terms, since the energy eigenvalues andeigenfunctions will be sought in the form of an expansion in powers of λ:
580G Quantum Mechanical Perturbation Theory " (0) " ψi ψ iEi (0)Ei (n) "λ n ψi ,n 1 λ(G.1.3)n(n)Ei .n 1The series is convergent if the perturbation is weak, that is, in addition to theformally introduced parameter λ, the interaction Hamiltonian itself containsa small parameter, the physical coupling constant.By substituting this expansion into the Schrödinger equation and collecting the same powers of λ from both sides, we obtain (0) "(0) (0) "H0 ψi Ei ψ i , (1) " (0) "(0) (1) "(1) (0) "(G.1.4)H0 ψi H1 ψi Ei ψ i Ei ψ i , (2) " (1) " (2) " (1) " (0) "(0)(1)(2)H0 ψi H1 ψi Ei ψ i Ei ψ i Ei ψ iand similar equations for higher-order corrections. The corrections to the energy and wavefunction of any order are related to the lower-order ones by therecursion formula(0) (n) "(1) (n 1) "(H0 Ei ) ψi (H1 Ei ) ψi(G.1.5)(2) (n 2) "(n) (0) " . . . Ei ψ i 0. Ei ψ iMultiplying the second equation in (G.1.4) (which comes from the terms! (0) that are linear in λ) by ψi from the left, the first-order correction to theenergy is! (0) (0) "(1)Ei ψi H1 ψi .(G.1.6)To determine the correction to the wavefunction, the same equation is multi! (0)plied by ψj (j i):(0) ! (0) (1) "ψj ψiEj! (0) (0)(0) ! (0) (1) " ψj H1 ψi Ei ψj ψi .(G.1.7) (n) "Since the eigenfunctions of H0 make up a complete set, the functions ψican be expanded in terms of them: (n) "(n) (0) " ψ (G.1.8)Cij ψj .ij(n)The coefficients Cii are not determined by the previous equations: their values depend on the normalization of the perturbed wavefunction. Substitutingthe previous formula into (G.1.7), we have! (0) (0)(0) (n)(0) (n)Ej Cij ψj H1 ψi Ei Cij ,(G.1.9)
G.1 Time-Independent Perturbation Theoryand hence! (1) " ψ ij i (0) "(0) ψj H1 ψi (0) " ψ.j(0)(0)Ei E jThe second-order correction to the energy is then ! (0) (0) " 2ψj H1 ψi (2),Ei (0)(0)Ei E jj iand the second-order correction to the wavefunction is! (0) (0) "! (0) (0) " (2) "ψj H1 ψkψk H1 ψi(0) ψψj (0) i(0)(0)(0) Ei E jEi Ekj i k i! (0) (0) "! (0) (0) "ψj H1 ψiψi H1 ψi(0) ψj . (0)(0) 2E Ej iij581(G.1.10)(G.1.11)(G.1.12)However, this wavefunction is not normalized to unity. Proper normalizationis ensured by the choice! (0) (0) "! (0) (0) " (2) "ψj H1 ψkψk H1 ψi (0) " ψ ψ (0) ij(0)(0)(0) EE E Ej i k iijik! (0) (0) "! (0) (0) "ψj H1 ψiψi H1 ψi (0) " ψ (G.1.13) (0)j(0) 2E Ej iij! (0) (0) "! (0) (0) "ψi H1 ψjψj H1 ψi (0) "1 ψ. 2 (0)i(0) 2E Ej iijFinally, the third-order correction to the energy is! (0) (0) "! (0) (0) "! (0) (0) "ψi H1 ψjψj H1 ψkψk H1 ψi(3)Ei (0) (0)(0)(0) Ei E jEi Ekj i k i! (0) (0) "! (0) (0) "! (0) (0) "ψi H1 ψjψj H1 ψiψi H1 ψi . (0)(0) 2Ei E jj i(G.1.14)In this Rayleigh–Schrödinger perturbation theory the explicit form ofhigher-order corrections becomes increasingly complicated. A relatively simplerecursion formula can be obtained by introducing the projection operator (0) "! (0) (0) "! (0) ψ(G.1.15)ψi ψj Pi 1 ψ ijj i
582G Quantum Mechanical Perturbation Theory (0) "which projects onto the subspace that is orthogonal to the state ψi . Thenth-order energy correction can then be written as! (0) (n 1) "(n)Ei ψi H1 ψi,(G.1.16)where the matrix element is to be taken with the wavefunction# (n) "1(1) (n 1) " ψ (0)Pi H1 Ei ψiiEi H0 (2) (n 2) "(n 1) (1) "ψi Ei ψi . . . Ei,(G.1.17)which is in the subspace mentioned above.Formally simpler expressions can be obtained when the Brillouin–Wignerperturbation theory is used. The perturbed wavefunction in the Schrödingerequation " " (G.1.18)H0 H1 ψi Ei ψiis then chosen in the form " " " ψi C0 ψ (0) Δψi ,(G.1.19)i (0) ""where Δψi is orthogonal to ψi , and C0 takes care of the appropriatenormalization. After some algebra, the eigenvalue equation reads " " (0) (0) "H0 Ei Δψi H1 ψi C0 Ei Ei ψi .(G.1.20)By applying the projection operator Pi , and exploiting the relations (0) " " "Pi ψ i 0,Pi ψi Δψi(G.1.21)as well as the commutation of Pi and H0 , " " Ei H0 Δψi Pi H1 ψi(G.1.22)is obtained. Its formal solution is " " ψi C0 ψ (0) i "PiH1 ψi .Ei H0(G.1.23)Iteration then yields " ψi C0 n 0and ΔEi n 0 PiH1Ei H0! (0) ψi H1 n (0) " ψ,PiH1Ei H0i n (0) " ψi(G.1.24)(G.1.25)
G.1 Time-Independent Perturbation Theory583for the energy correction. In this method the energy denominator contains the(0)perturbed energy Ei rather than the unperturbed one Ei . To first order inthe interaction,! (0) (0) "(0)(G.1.26)Ei Ei ψi H1 ψi ,while to second order,! (0) (0) " ψi H1 ψi! (0) (0) "! (0) (0) "ψi H1 ψjψj H1 ψi(0)Ei Ei Ei j i(0)Ej(G.1.27) . .It is easy to show that by rearranging the energy denominator and expandingit as111 (0) (0)Ei H0Ei H0 ΔEiEi H0 ΔEi(0)n 0Ei H0 n,(G.1.28)the results of the Rayleigh–Schrödinger perturbation theory are recovered.The formulas of time-dependent perturbation theory can also be used todetermine the ground-state energy and wavefunction of the perturbed system, provided the interaction is assumed to be turned on adiabatically. Theappropriate formulas are given in Section G.2.G.1.2 Degenerate Perturbation TheoryIn the previous subsection we studied the shift of nondegenerate energy levelsdue to the perturbation. For degenerate levels a slightly different method hasto be used because the formal application of the previous formulas would yieldvanishing energy denominators.Assuming that the ith energy level of the unperturbed system is p-fold(0)(0) "degenerate – that is, the same energy Ei belongs to each of the states ψi1 , (0) " (0) " ψ–, any linear combination of these degenerate eigenstatesi2 , . . . , ψipis also an eigenstate of H0 with the same energy. We shall use such linearcombinations to determine the perturbed states. We write the wavefunctionsof the states of the perturbed system that arise from the degenerate states as (0) " "" ψ (G.1.29)cik ψik cn ψn(0) ,kn iwhere the cik are of order unity, whereas the other coefficients cn that specifythe mixing with the unperturbed eigenstates whose energy is different from(0)Ei are small, proportional to the perturbation. By substituting this form! (0) into the Schrödinger equation, and multiplying both sides by ψij from theleft,
584G Quantum Mechanical Perturbation Theory!ΔEcij (0) "(0) ψij H1 ψik cik ! "(0) ψij H1 ψn(0) cn(G.1.30)n ikis obtained. Since the coefficients cn are small, the second term on the righthand side can be neglected in calculating the leading-order energy correction,which is given by#! (0) "(0) ψij H1 ψik δjk ΔE cik 0 .(G.1.31)kThis homogeneous system of equations has nontrivial solutions if the determinant of the coefficient matrix vanishes: ! (0) "(0) (G.1.32)det ψij H1 ψik δjk ΔE 0 .The solutions of this pth-order equation– that is, the eigenvalues of the matrix! (0) (0) "made up of the matrix elements ψij H1 ψik – specify the eventual splittingof the initially p-fold degenerate level, i.e., the shift of the perturbed levelswith respect to the unperturbed one. Thus the interaction Hamiltonian needsto be diagonalized on the subspace of the degenerate states of H0 . In general,the degeneracy is lifted at least partially by the perturbation. As discussed inAppendix D on group theory, the symmetry properties of the full Hamiltoniandetermine which irreducible representations appear, and what the degree ofdegeneracy is for each new level.G.2 Time-Dependent Perturbation TheoryIf the perturbation depends explicitly on time, no stationary states can arise.We may then be interested in the evolutionof the system: What states can (0)"if the perturbation is turnedbe reached at time t from an initial state ψion suddenly at time t0 ? The answer lies in the solution of the time-dependentSchrödinger equation "" ψi (t) .H0 λH1 (t) ψi (t) i t " The wavefunction ψi (t) is sought in the form (0) "(0)cij (t) ψj e iEj t/ , " ψi (t) (G.2.1)(G.2.2)jsubject to the initial conditioncij (t0 ) δij .(G.2.3)Since the time dependence of the unperturbed state has been written outexplicitly, the functions cij (t) are expected to vary slowly in time. Expandingthe coefficients once again into powers of λ,
G.2 Time-Dependent Perturbation Theory (0)(r)λr cij (t) ,cij (t) cij (t) 585(G.2.4)r 1where, naturally, the zeroth-order term is a constant:(0)cij (t) δij .(G.2.5)Substituting this series expansion into the Schrödinger equation, we find (r)c (t) i t ij(0)ei(Ej(0) Ek )t/ ! (0) " (r 1)(0) ψj H1 (t) ψk cik (t) .(G.2.6)kThe explicit formulas for the first two terms obtained through iteration are(1)cij (t)i t! (0) "(0)(0)(0) ψj H1 (t1 ) ψi ei(Ej Ei )t1 / dt1 ,(G.2.7)t0and (2)cij (t) i 2 t t1dt1t0!dt2 (0) "(0)(0)(0) ψj H1 (t1 ) ψk ei(Ej Ek )t1 / kt0 (0) "! (0) (0)(0) ψk H1 (t2 ) ψi ei(Ek Ei )t2 / (G.2.8)In the interaction picture the time dependence of an arbitrary operator Ois given byÔ(t) eiH0 t/ Oe iH0 t/ .(G.2.9)Using this form for the Hamiltonian, which may have an intrinsic time de(n)pendence as well, the first two coefficients cij can be written in terms of theoperatorsĤ1 (t) eiH0 t/ H1 (t)e iH0 t/ (G.2.10)as(1)cij (t) i t (0) "! (0) ψj Ĥ1 (t1 ) ψi dt1(G.2.11)t0and (2)cij (t) i 2 t t1dt1t0t0 (0) "(0) ψk Ĥ1 (t2 ) ψi .! (0) "! (0) ψj Ĥ1 (t1 ) ψkdt2k(G.2.12) (0) "constitute a complete set, the previousSince the intermediate states ψkformula simplifies to
586G Quantum Mechanical Perturbation Theory (2)cij (t) i 2 t t1dt1t0 (0) "! (0) dt2 ψj Ĥ1 (t1 )Ĥ1 (t2 ) ψi .(G.2.13)t0The same result is obtained when the double integral on the t1 , t2 plane isevaluated in reverse order: (2)cij (t) i 2 t tdt2t0 (0) "! (0) dt1 ψj Ĥ1 (t1 )Ĥ1 (t2 ) ψi ,(G.2.14)t2or by swapping the notation of the two time variables: (2)cij (t) i 2 t tdt1t0 (0) "! (0) dt2 ψj Ĥ1 (t2 )Ĥ1 (t1 ) ψi .(G.2.15)t1Using these two formulas, the coefficient can also be written as (2)cij (t) i 212 t tdt1t0! (0) (0) "dt2 ψj T Ĥ1 (t1 )Ĥ1 (t2 ) ψi ,(G.2.16)t0where T is the time-ordering operator, which orders the operators in a productin descending order of their time argument. Its action can be written in termsof the Heaviside step function as T Ĥ1 (t1 )Ĥ1 (t2 ) θ(t1 t2 )Ĥ1 (t1 )Ĥ1 (t2 ) θ(t2 t1 )Ĥ1 (t2 )Ĥ1 (t1 ) .(G.2.17)Generalizing this to arbitrary orders, and setting λ 1,t n 1 n t t1icij (t) δji dt1 dt2 . . .dtn n 1 t0t0t0(G.2.18) (0) "! (0) ψj Ĥ1 (t1 )Ĥ1 (t2 ) . . . Ĥ1 (tn ) ψi ,or, in time-ordered form, 1cij (t) δji n!n 1 i n t tdt1t0 tdt2 . . .t0dtnt0(G.2.19)! (0) (0) " ψj T Ĥ1 (t1 )Ĥ1 (t2 ) . . . Ĥ1 (tn ) ψi .The time evolution of the wavefunction between times t0 and t is thereforegoverned by the operator S(t, t0 ): "" ψ(t) S(t, t0 ) ψ(t0 ) ,(G.2.20)
G.2 Time-Dependent Perturbation Theory587where S(t, t0 ) n 0 i n t t1dt1t0t n 1dtn Ĥ1 (t1 )Ĥ1 (t2 ) . . . Ĥ1 (tn ) ,dt2 . . .t0t0(G.2.21)or 1S(t, t0 ) n!n 0 i n t tdt1t0 tdt2 . . .t0 dtn T Ĥ1 (t1 )Ĥ1 (t2 ) . . . Ĥ1 (tn ) .t0(G.2.22) (0) "Now consider a system whose initial wavefunction at t0 0 is ψi , anda constant perturbation that acts for a finite period of time. According to(0) "that becomes admixed to the initial(G.2.7), the amplitude of the state ψjstate at time t is given by(0)(0)! (0) (0) " ei(Ej Ei )t/ 1(1)cij ψj H1 ψi(0)(0)Ej E i(G.2.23)in the lowest order (0) "of perturbation theory. The transition probability from(0) " state ψito ψjis thenWi j (0) (0) ! (1) 2 (0) (0) " 2 1 cos (Ej Ei )t/ cij 2 ψj H1 ψi (0)(0) 2Ej E i(G.2.24)in the same order. For large values of the time, the formula on the right-handside gives significant probabilities only for states whose energy difference is atmost of order 2π /t. Sincelimt 1 cos(x x0 )t πtδ(x x0 ) ,(x x0 )2the transition rate in the t limit is2π ! (0) (0) " 2dWi j(0)(0) wi j ψj H1 ψi δ(Ej Ei ) .dt (G.2.25)(G.2.26)As mentioned in the previous section, the formulas of time-dependent perturbation theory can also be used to specify the energy shifts due to a stationary perturbation, provided the interaction is assumed to be turned onadiabatically at t0 . Inserting a factor exp( α t ) in the interactionHamiltonian, which specifies the adiabatic switch-on by means of an infinitesimally small α, we haveĤ1 (t) eiH0 t/ H1 e iH0 t/ e α t .(G.2.27)
588G Quantum Mechanical Perturbation Theory "in the interaction picture. Using the ground state Ψ0 of energy E0 of theunperturbed system, the energy correction due to the perturbation is "! Ψ0 H1 S(0, ) Ψ0 " ,ΔE ! (G.2.28)Ψ0 S(0, ) Ψ0and the wavefunction is " Ψ0 "S(0, ) Ψ ! ". Ψ0 S(0, ) Ψ0(G.2.29)As J. Goldstone (1957) pointed out, the same result may be formulated in a slightly different way. Considering a many-particle system with anondegenerate ground state, the contribution of each term in the perturbation expansion can be represented by time-ordered diagrams that show theintermediate states through which the system gets back to the ground state.This representation contains terms in which some of the particles participating in the intermediate processes are in no way connected to the incomingand outgoing particles. It can be demonstrated that the contributions of thedisconnected parts are exactly canceled by the denominator in (G.2.28) and(G.2.29), so ΔE " Ψ 6 Ψ0 H1n 0 n 0 n 71 H1 Ψ0,E0 H0con n "H1 Ψ0 con ,1E0 H0(G.2.30)where the label “con” indicates that only the contribution of connected diagrams need to be taken into account. It should be noted that instead ofGoldstone’s time-ordered diagrams, the perturbation series for the groundstate energy can also be represented in terms of Feynman diagrams, whichare more commonly used in the many-body problem. Only connected diagrams need to be considered in that representation, too.Reference1. C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics, JohnWiley & Sons, New York (1977).
HSecond QuantizationThe quantum mechanical wavefunction is most often considered as the function of the space and time variables when the solutions of the Schrödingerequation are sought. In principle, this approach is applicable even when thesystem is made up of a large number of interacting particles. However, it isthen much more convenient to use the occupation-number representation forthe wavefunction. We shall introduce the creation and annihilation operators,and express the Hamiltonian in terms of them, too.H.1 Occupation-Number RepresentationIt was mentioned in Chapter 12 on the quantum mechanical treatment oflattice vibrations that the eigenstates of the harmonic oscillator can be characterized by the quantum number n that can take nonnegative integer values.Using the linear combinations of the position variable x and its conjugatemomentum, it is possible to construct operators a† and a that increase anddecrease this quantum number. We may say that when these ladder operatorsare applied to an eigenstate, they create an additional quantum or annihilate an existing one. Consequently, these operators are called the creation andannihilation operators of the elementary quantum or excitation. States canbe characterized by the number of quanta they contain – that is, by the occupation number. Using Dirac’s notation, the state ψn of quantum numbern – which can be constructed from the ground state of the oscillator by then-fold application of the creation operator a† , and thus contains n quanta –will henceforth be denoted by n . The requirement that such states shouldalso be normalized to unity leads to a n n n 1 ,a† n n 1 n 1 .(H.1.1)The operators a and a† of the quantum mechanical oscillator satisfy thebosonic commutation relation.
590H Second QuantizationThis occupation-number representation can be equally applied to manyparticle systems made up of fermions (e.g., electrons) or bosons (e.g., phonons,magnons). Any state of an interacting system consisting of N particles canbe expanded in terms of the complete set of states of the noninteractingsystem. The eigenstates of the noninteracting many-particle system can, inturn, be expressed in terms of the one-particle eigenstates. When the oneparticle problem is solved for the noninteracting system, the complete setφ1 (ξ), φ2 (ξ), . . . , φi (ξ), . . . of one-particle states is obtained, where the collective notation ξ is used for the spatial variable r and spin s of the particles:ξ (r, s).Such complete sets are the set of eigenfunctions for the harmonic oscillator,and the system of plane waves, Bloch functions, or Wannier functions forelectrons. The construction of the complete set of many-particle functionsfrom one-particle functions is different for bosons and fermions: for bosons,several particles may be in the same state, whereas this possibility is excludedby the Pauli principle for fermions. The two cases must therefore be treatedseparately.BosonsFor noninteracting bosons the states of the many-particle system are describedby means of those combinations of the one-particle states that are completelysymmetric with respect to the interchange of the space and spin variables.If an N -particle system contains n1 , n2 , . . . , nk , . . . particles in the statesφ1 , φ2 , . . . , φk , . . . , wherenk N,(H.1.2)kthe wavefunction with the required symmetry properties is Φn1 ,n2 ,.,nk ,. n1 !n2 ! . . . nk ! . . .N! 1/2φp1 (ξ1 )φp2 (ξ2 ) . . . φpN (ξN ) ,P(H.1.3)where 1, 2, . . . , k, . . . occurs among the indices pi exactly n1 , n2 , . . . , nk , . . .times, and summation is over all possible permutations of the indices.It turns out practical to introduce a more concise notation. If the wavefunctions of the one-particles states are known, the wavefunction Φ is unambiguously characterized by the numbers n1 , n2 , . . . , nk , . . . that specify theoccupation of each one-particle state, therefore the above-defined state can beconcisely denoted byΦn1 ,n2 ,.,nk ,. n1 , n2 , . . . , nk , . . . .(H.1.4)This is the occupation-number representation, while the vector space spannedby the set of all such basis states with nonnegative integers nk for bosons iscalled the Fock space
Quantum Mechanical Perturbation Theory Quantum mechanical perturbation theory is a widely used method in solid-state physics. Without the details of derivation, we shall list a number of basic formulas of time-independent (stationary) and time-dependent perturbation theory below. For simp
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