Approximate Methods. Time-independent Perturbation Theory .
Approximate methods. Time-independent perturbation theoryVariational principles. Semiclassical approximation.There exist only a handful of problems in quantum mechanics which can be solved exactly.More often one is faced with a potential or a Hamiltonian for which exact methods are unavailableand approximate solutions must be found. Here we review three approximate methods each ofwhich has its own strengths and weaknesses.(i) Perturbation theory: useful when the Hamiltonian can be written as an exactly solvable pieceplus ’small’ correction;(ii) Variational method: in the standard form helps to determine ground state properties in acomplex system. The major drawback is lack of control over the accuracy (which often is remarkably good!). Can be re-formulated to deal with excited states, but quickly becomes inaccurate.(iii) Semiclassical approximation: deals with slowly-varying potentials.Time-independent perturbation theoryIn perturbation theory one considers corrections to energy eigenstates and eigenvalues whichoriginate from ’small’ terms, let’s denote them V̂ , added to the exactly solvable Hamiltonian Ĥ0 :Ĥ Ĥ0 V̂ .(0)(0)The conditions for naming V̂ ’small’ will be quantified below. Let En and ψn be eigenenergiesand the corresponding eigenstates of Ĥ0 :Ĥ0 ψn(0) i En ψn(0) i ,(0)n 0, 1, 2, 3, . . . .(0)The goal is to determine corrections to ψn and En which take us closer and closer to the exacteigenstates ψn with exact eigenenergies ψn under the assumption that V̂ is small. Since V̂ is amatrix and has a dimension of energy the notion of small can not be attached to it directly withoutcomparing matrix elements of V̂ to some energy scale. To avoid this complication we introducean auxiliary quantity λ, replace V̂ with λV̂ , and pretend that λ 0 is a small parameter. At theend of the calculation we set λ 1. In essence, we simplify the bookkeeping of collecting termswhich are proportional to the same power of V̂ terms. So, we start solving the eigenvalue equation[Ĥ0 λV̂ ] ψn i En ψn i ,by assuming the existence of expansionsEn En(0) λEn(1) λ2 En(2) · · · λk En(k) . . . , ψn Zn ψn(0) λψn(1) λ2 ψn(2) · · · λk ψn(k) . . . ,(1)where the global normalization factor has to be established at the end of the calculation (thisallows one to demand that the wavefunction correction is orthogonal to the un-perturbed state).Plugging these expansions back to the eigenvalue equation we get hi [Ĥ0 λV̂ ] ψn(0) λψn(1) λ2 ψn(2) . . . En(0) λEn(1) λ2 En(2) . . .ψn(0) λψn(1) λ2 ψn(2) . . . .The rest of the theory requires that we express both the l.h.s and the r.h.s. of this equation aspower series expansions in λ and then equate expressions in front of each λk , i.e. we match Taylor1
series expansions about the λ 0 point. This is why this procedure is called the ’perturbationtheory’. Now the l.h.s is (0)(1)(0)2(2)(1)k(k)(k 1)Ĥ0 ψn λ Ĥ0 ψn V̂ ψn λ Ĥ0 ψn V̂ ψn · · · λ Ĥ0 ψn V̂ ψn . ,and the r.h.s. isEn(0) ψn(0) λ En(0) ψn(1) En(1) ψn(0) λ2 En(0) ψn(2) En(1) ψn(1) En(2) ψn(0) · · · λkkXEn(m) ψn(k m) . . . .m 0Matching power series we get a set of coupled equations which admit an easy step-by-step solution.The first equation isĤ0 ψn(1) V̂ ψn(0) En(0) ψn(1) En(1) ψn(0) .(2)(0)We project it on the ψlwe obtainstate to replace operators with numbers and matrix elements. For l nV̂nn ψn(0) En(1) ψn(0) En(1) V̂nn .(3)which immediately solves for the lowest, or first, order correction to the energy eigenvalue. It isnothing but the average of the perturbation term over the unperturbed state. For l 6 n we obtain(0)(0)(0)El hψl ψn(1) i V̂ln En(0) hψl ψn(1) i V̂ln(0)hψl ψn(1) i (0)El(0).(4) Enwhich solves for the lowest-order correction to the wavefunctionX (0)V̂ln.ψn(1) ψl(0)(0)El Enl6 n(5)(0)(0)We immediately see that an admixture of states with l 6 n is small as long as V̂ln /(El En ) 1, and the sum over l converges fast enough. Thus is a major condition which justifies the use ofthe perturbation theory.(k 1)This wonderful conspiracy that one only needs to know wavefunction corrections up to ψn(k)(k)to find En which is then used to determine ψn persists in higher orders, making the wholescheme slightly increasing in complexity for larger k, but still easily manageable. Consider nowsecond-order corrections. ProjectingĤ0 ψn(2) V̂ ψn(1) En(0) ψn(2) En(1) ψn(1) En(2) ψn(0) ,(0)onto the state ψlwith l n we get the second order correction to energy ashψn(0) V̂ ψn(1) i En(2) ,(0)(6)(1)(0)En(2) V̂nl V̂ln(0)(0)l6 n El EnX.(7)(k)Note that hψn ψn i 0 as well as hψn ψn i 0 for any k by construction, see Eq. (1). Sincefor Hermitian operators V̂nl V̂ln we can write the answer in the form which makes it clear thatthe second-order correction to the ground state is always negativeEn(2) V̂ln 2X(0)l6 nEl2(0) En,(8)
because for n 0 all denominators are positive. This result has numerous important implicationsin physics. For example, if the ground state of Ĥ0 is degenerate and matrix elements of V̂ are zerobetween all ground states (say by symmetry) then the the best state is the one which admits largermatrix elements between the ground state manifold and excited states. Second-order perturbationtheory for energy is also behind many effective interactions such as the VdW force between neutralatoms without dipole moments, or any force mediated by an exchange of an intermediate particlefor that matter. Below I also discuss how it leads to the notion of effective Hamiltonian underappropriate conditions; after all nearly everything in physics is an effective description at therelevant energy scale.(0)By projecting (6) on ψl with l 6 n we find the second-order correction to the wavefunction(0)(0)hψl ψn(2) i (1)(1)(0)(1)hψl V̂ ψn i En hψl ψn i(0)El(0).(9) En(1)(1)What is left is to plug in here expressions for ψn and En : X (0) XV̂lk V̂knV̂ln V̂nn .ψn(2) ψl (0)(0)(0)(0)(0)(0) 2(El En )l6 nk6 n (El En )(Ek En )(10)SinceEn(k) hψn(0) V̂ ψn(k 1) i ,holds for any k, it does not cost us anything to write the next-order correction to energyEn(3) V̂nl V̂lk V̂knX(0)l,k6 n(El(0)(0)(0) En )(Ek En )XV̂nn V̂nl 2(0)l6 n(El(0) En )2.(11)We will stop here since the protocol is clear. Let us now turn to examples of how it works.Take a fake problem when the potential is represented by two harmonic oscillator terms11U (x) U0 (x) V (x) mω 2 x2 mν 2 x2 ,22with ν ω. The exact solution is, of course, readily obtained by introducing Ω (ω 2 ν 2 )1/2which also can be Taylor expanded as ν2ν4Ω ω 1 2 4 . .2ω8ωThe exact energies are 1En Ω n 2 .In perturbation theory we haveEn(1) 2mν 2 hn x2 nimν 2 (2n 1)11ν2 2 hn mν x ni ω n ,224mω2 2ω 2exactly as expected from exact E0 with Ω expanded in a series.3
Doing the check for the wavefunction is also straightforward conceptually but more messy:ψn(1) X(0) hl mνψl2 x2 /2 niω(l n)l6 n ν 2 X (0) hl [ˆ(a) ˆ(a)† ]2 niψl.4ω 2l nl6 nSubstituting herehl [ˆ(a) ˆ(a)† ]2 ni (2n 1)δl,n pp(n 1)(n 2)δl,n 2 (n)(n 1)δl,n 2we get pν2 p(0)(0) (n 1)(n 2)ψ n(n 1)ψn 2n 2 ,8ω 2(the second term is present only if n 2 0). To compare this expression with exact solution we(0) have to take ψn ( mΩx) and differentiate it with respect to Ωψn(1) ψn(1) δΩIt is convenient to use y ψn(1)d (0) ψ ( mΩx)dΩ n Ω ων 2 d (0) ψ ( mωx) .2ω dω n mωx at this point ν 2 mω 1/4 11yy220 H(y) HH(y)e y /2 .(y) nnnn2ω 2 π4222 n!The rest is lengthy algebra of manipulating with Hermite polynomials usingHn 0 (y) 2yHn (y) Hn 1 (y)to write square brackets as 1 1Hn yHn 1 y 2 Hn ,. 2 2and then applying1yHn nHn 1 Hn 1 ,2couple of times to get rid of the powers of y in front of polynomials 1 111. Hn yHn 1 y(nHn 1 Hn 1 ) Hn yHn 1 2nyHn 1 2 224 111 1n(n 1)Hn (n 1)Hn Hn 2 2n (n 1)Hn 2 Hn Hn 2 Hn 2 .42282(0)The final step is to attach the proper normalization factors to Hermite polynomials to obtain ψn 2(0)and ψn 21 2n n! n(n 1)1Hn 2 Hn 2 28ppn(n 1)(n 1)(n 2)11ppHn 2 Hn 2 .n 2n 2442(n 2)!2(n 2)!4
This leads to pν 2 p(0)(0)(n)(n 1)ψ (n 1)(n 2)ψ,n 2n 28ω 2which matches the result of the perturbation theory precisely.ψn(1) Problem 35. Shifted harmonic oscillator by perturbation theoryConsider a harmonic oscillator accompanied by a constant force f which is considered to be small1V (x) mω 2 x2 f x .2a). Show that this system can be solved exactly by using a shifted coordinatey x f,mω 2and write exact expressions for energy eigenvalues and eigenfunctions.(1)b). Use perturbation theory (by considering the force term as a perturbation) to calculate En ,(2)(1)En , and ψn . Compare you answers with the exact expression and demonstrate match whenexact expressions are expanded in powers of f .Problem 36. Anharmonic oscillator1 .Evaluate the effect of a small anharmonic term of the formVanh x3 ,on the spectrum of harmonic oscillator with mass m and frequency ω.a). Start by showing that3hk x ni 12mω 3/2 p(n 1)(n 2)(n 3)δk,n 3 3(n 1)3/2 δk,n 1 3n3/2 pδk,n 1 n(n 1)(n 2)δk,n 3 .(1)b). Calculate the first-order energy shift En in perturbation theory in Vanh .(2)c). Calculate the second-order energy shift En .Degenerate perturbation theory. Effective Hamiltonians.(k)We observe that the perturbation theory fails when denominators in expressions for En and(k)(0)(0)ψn are either zero, or comparable to the mixing matrix element, i.e. when El En . V̂ln .Often this is a sign that the perturbation theory has to be abandoned altogether. However, thereare important cases when the condition of validity of the perturbation theory is violated onlyfor two, of several, say N , isolated levels which happen to be near degenerate. Lets denote thecorresponding N -dimensional Hilbert space by νA while the rest of the Hilbert space is νB5
In this case we proceed as follows:(0)(1) Account for the diagonal matrix elements of V̂nn in the renormalized spectrum n En En .Formally, they are exact energy eigenstates of the Hamiltonian Ĥ00 Ĥ0 D̂, where D̂nm V̂nn δnm . I thus suggest that we always transfer diagonal matrix elements of V̂ to Ĥ0 and consider0 0. At this point we check againthe rest as perturbation V̂ 0 V̂ D̂. By construction V̂nnwhether conditions for the validity of the perturbation theory are violated for V̂ 0 and the spectrum{ n } to identify the troubled group of states νA , if any. Suppose that we do have a group of N states for which the perturbation theory cannot beapplied directly.0We split the matrix V̂kk0 into two matrixes Âkk 0 andB̂kk0 such that B̂ does not have non-zero matrix elements between states which both belong to νA , while has non-zero matrix elements only between the states in the νA subspace. This is illustrated in the figurewhich explains which matrix elements are assigned to and B̂. By definition, the perturbation theory canbe applied for B̂. We thus proceed as usual and calculate corrections to energy levels and wavefunctionsfor all levels. The final result must be regarded as anapproximate solution for the energy spectrum and eigenstates of Ĥ00 B̂. Let us denote them as { n } and{ψ̃n }. 0 *aVˆ ' * b * c 0 0 0 0Aˆ 0 e* 0 0ab0e*e0f*g*0 0 e 0 0 0 0 0 c f ˆ ˆ A Bg 0 0 *aBˆ * b * cab0000f*g* A subspace We now account for the effects of matrix  and new matrix elements generated in the neardegenerate subspace νA . In some cases we have non-zero  from the very outset and in theleading approximation it is possible to skip the previous step completely and proceed with thediagonalization of the Hamiltonian matrix(A)Ĥnm n δnm Ânm ,ψn , ψm νA ,in the νA subspace. For the two-level example shown in the figure we readily findr 2 3( 2 3 )2E A23 2 ,24i.e. we deal with this problem exactly the same way as with any other finite-dimensional space.One immediate improvement would be to use corrected spectrum n instead of n . However,to be consistent we must also account for corrections to the matrix  introduced by the changeof basis states from ψn to ψ̃n and new matrix elements generated from higher orders in B̂. Thisis especially important in cases when  is small, or even zero! This brings us to the notion ofeffective Hamiltonian in the selected energy subspace νA . Needless to say that effective Hamiltonians and theories are found in physics all over the place. More precisely, all we do in physicsis effective Hamiltonians and models which emerge after high-energy modes are ’absorbed’ intoeffective constants and potentials. Here I will discuss how things are done at the level of thelowest-order perturbation theory (first for the wavefunction).6c f g 0
We have Xψ̃n Zn ψn(0) (0)ψll6 νA B̂ln Zn ψn(0) l nX(0)ψl Cln ,(12)l6 νAwith Z-factors close to unity because 1/2 X2Zn 1 Cln Zn 1 l6 νANow1 X Cln 2 .2l6 νAÈmn hψ̃m  ψ̃n i Zm Zn Âmn Âmn .I will not keep terms proportional to AB 2 here, though it is possible at a little cost (see below fornormalized and orthogonal functions).Given new wavefunctions we also find that the B̂ matrix in the νA subspace now has non-zeromatrix elements. They may be small, i.e. proportional to B̂ 2 , but given near degeneracy of energylevels in νA their effect on these levels may be strong and even non-perturbative. These matrixelements are (keeping only terms B̂ 2 ): *iX (0)X (0)Xh(0) B̂ln B̂ml Cln ψl Clm B̂ ψn(0) ψl Cln ClmÂ0m6 n ψml6 νAl6 νA 2XB̂ml B̂lnl6 νAl6 νA l ( n m )/2( l m )( l n )h iFinally, we have to account for contributions coming from Ĥ00n6 m. If we naively use wa-vefunctions (12) we will get a meaningless expression because it will not be invariant under thechoice of where the energy is counted from. This is because the overlapX hψ̃m ψ̃n i Omn Cln 6 0 ,Clml6 νAis non-zero. By small rotation of states (again, keeping only the leading small terms)(0)em ψm X1X(0)Onm ψn(0) Clm ψl ,2 nl6 νAwe fix orthogonality (and normalization!) to leading order. Now* h iX (0)XX (0)X11(0)(0)(0)Â00m6 n Ĥ00 ψm ψl Clm Okm ψk Ĥ00 ψn(0) ψl Cln Oin ψi 22n6 ml6 νAXl6 νA ClmCln kl6 νA X l ( m n )/211 l ( m n ) Bml Bln Â0m6 n .2( l n )( l m )2l6 νA7i
Combining all pieces together we finally getih(ef f )Ĥmn n δnm  Â0 Â00m6 n n δnm Âmn XB̂ml B̂lnl6 νA l ( n m )/2(1 δnm ) .( l m )( l n )Establishing an effective Hamiltonian for the N -dimensional subspace is important when l n n m , for all l 6 νA and n, m νA , i.e. when near degenerate states are separated bylarge energy gaps from the rest of the spectrum. In this case we can use a simplified expression(ef f )Ĥnm n δnm Âmn X B̂ml B̂ln(1 δnm ) , l Al6 νAPwhere A N 1 n νA n is the ’central’ energy of the νA subspace. This completes our derivation. Notice that for exact degeneracy of n levels and zero Â-matrix, an arbitrary small mixingof levels generated by Â0 -matrix is a non-perturbative effect.Problem 37. Degenerate perturbation theory.Go back to the Three-Level Problem (# 26), i.e. 0 0 Ĥ 0 0 , U(it is exactly the same Problem; I simply re-ordered the matrix for transparency) and addressit now in the limit of large U using degenerate perturbation theory and the effective Hamiltonian approach. At the end compare your final result to the result of exact calculation fromProblem 34.Problem 38. Perturbation theory for the two level system.Well, we already know the exact solution for the two level system. Now try to reproduce knownresults by considering the case of large bias, or small mixing, /ξ 1, and applying the perturbation theory in .a). Calculate the second order corrections to energies E1,2 ξ. Compare with the exact resultat the same level of accuracy.b). Calculate the first order corrections to the energy wavefunctions. Compare with the exactresult at the same level of accuracy.Variational principleConsider the quantityJ[ψ] hψ Ĥ ψi,hψ ψiwhich can be constructed for any state. The fancy name for it is ’functional’ because it convertsan input in the form of a function into an output which is a number. Next we expand ψ in the8
energy basisψ(x) Xcn ψn ,Ĥψn En ψn ,nand write J[ψ] asPPEn cn 2E0 n cn 2nJ P P E0 .22n cn n cn The inequality follows from the fact that En E0 for all n. Thus J is minimized when ψ ψ0in which case J E0 . However any other choice will yield something larger. A good choiceof wavefunction will result in an answer which is closer to the exact value. The protocol thenis to make a variational guess of ψ containing one or more parameters {αi }, calculate J({αi }),and minimize J with respect to this parameter set. For example, consider the standard harmonicoscillator Hamiltonian for which the ground state energy is E0 ω/2. Of course, the actualground state wavefunction is a Gaussian, but suppose we don’t know this and choose insteadψ(x) N (a2 x2 )2 for x aand zero otherwise.First, choose N to normalize the wavefunctionZa1 2dx ψ(x) N2 aZa22 9 2562 4dx(a x ) N aThen2hψ x ψi N2Zadxx2 (a2 x2 )4 N 2 a11 aAlso315 ad2hψ 2 ψi hψ 0 ψ 0 i N 2dxZr N 315.256a9256315 2a2 a .3465346511adx[4x(a2 x2 )]2 N 2 a7 a2563 2 .105aHence3mω 2 a2J 2ma222 dJ3mω 2 a 0dama311 2a 33.mωSubstituting this back to J we finally getEmin3ω 2 33 33ωω 222r2311! 1.044ω,2which is only 4% above the true ground state despite the crude guess for the wave function.Today, the whole machinery is developed for variational calculations of many-body groundstates with the trial states containing hundreds and even thousands of variational parameters. Atthis level of complexity, the evaluation and minimization of J is done by computers (in the multidimensional Hilbert space of particle coordinates, J is often simulated by Monte Carlo). The mostrecent advance came from quantum information science when Vidal and Verstraete introducedmatrix product and tensor network states. In quantum chemistry calculations people can do 200electrons and calculate energies with relative accuracy of about 0.1%, a remarkable achievement.But even this is not enough because 0.1% on 1 keV energy is about 10000 K—not sufficient tomake reliable predictions for chemical reaction rates at room temperature. Still .9
We have discussed the variational principle for the ground state. Can it be generalized toexcited states? The answer is yes. First, the trivial way. Sometimes the symmetry of the groundand excited states is different, say one is even and the other is odd. By taking variational stateswhich respect different symmetries we obtain a standard variational principle for the ground statesin each of the different symmetry sectors. This works because expansion coefficients cn are nonzero only in the same-symmetry sector; the rest of the proof is exactly as before with E0 beingreplaced with the lowest energy in a given symmetry sector. This observation can be carried outalso at the level of conserved quantum numbers: for example, if the total momentum is conserved,then taking trial states with different total momentum we have a variational principle in thismomentum sector. Same for the conserved total angular momentum, or magnetization componenton the symmetry axis, etc. etc.An idea of using trial states orthogonal to the variational ground state (in the same symmetry,or conserved quantum number sector) in attempt to determine the first excited level might workin practice for the lowest excited state, though, strictly speaking, it is not based on a solid mathematical foundation because of an unknown contribution from an admixture of the exact groundstate.There is, however, an extension of the variational principle which works for any energy level,in principle. Consider quantity, called ’local energy’, defined byE(x) ψ (x)Ĥψ(x). ψ(x) 2For an arbitrary state it is a function of x, but whenever ψ(x) coincides with one of the energyeigenstates it becomes a constant (!). This immediately suggests that a functional looking at localfluctuations of E(x) about its average (averaging can be done using ψ 2 ):DEJloc (E(x) hE(x)i)2 hE 2 (x)i hE(x)i2 ,which is non-negative by construction, reaches a minimum value of Jloc 0 when the trial stateis hitting one of the eigenstates. Not only we have a general variational principle, but also somemeasure of estimating the accuracy by monitoring how close is the minimum value of Jloc to zero.Once the best state is determined (by minimization of Jloc with respect to variational parameters)one proceeds with the calculation of energy using the standard expression for J.Problem 39. Anharmonic oscillator2 .Consider a harmonic oscillator with additional x4 term1V (x) mω 2 x2 x4 .2a). Use the variational principle with a Gaussian trial wavefunction ψ(x) 1πa2 1/42 /2a2e x,in order to estimate the ground state energy E0 .b). Evaluate your variational answer for E0 analytically in the limit of vanishing .c). Use first-order perturbation theory in order to calculate E0 , and compare with the result found10
in b).Problem 40. Quartic potential.Use Gaussian trial wavefunction to estimate the ground state energy for the quartic potentialV (x) gx4 (save time by paying attention to the previous problem :) Show that your answer isEmin 4/3 g 1/33.4m2Compare this to the ’exact’ answer 0.668(g/m2 )1/3 .The semiclassical approximationThis method paves the road connecting QM with the classical mechanics. It is also a usefultool for obtaining accurate results under circumstances when the potential is relatively smooth.It was developed by Wentzel, Kramers, and Brillouin, giving it the ’WKB’ name. Well, someactually call it LG Liouville–Green method by arguing that borrowing math. tools for solvingmath. equation when it happens to be relevant for physics does not qualify for name giving, someadd another latter to the end, WKBJ with J Jeffreys, some call it ’eikonal approximation’, etc. Iwill generically call it the semiclassical approximation. The idea here is to write the wavefunctionasψ(x) A(x) eiB(x)/ ,where A(x), B(x) are real functions. Now demand that the form satisfy the time independentSchrödinger Equation . In preparation for that, computedψi A0 eiB(x)/ AB 0 eiB(x)/ ,dx d2 ψ2i 0 0 iB(x)/ ii200 iB(x)/ 00 iB(x)/ Ae ABe ABe A(B 0 )2 eiB(x)/ .dx2 2We then have 2i 0ii2 20000002A (x) A (x)B (x) A(x)B (x) 2 A(x)(B (x)) (V (x) E)A(x) 0 , 2m let us consider the limit of 0, which is formally the same as assuming that higher-orderderivative of the wavefunction amplitude is small, and start solving the Schrödinger Equationapproximately by dealing separately with different powers of in the above expression. In theleading approximation we findA00 0 ,neglect this derivative ,2A0 B 0 AB 00 0 ,pA(B 0 )2 2m(V (x) E)A 0 B 0 2m(E V (x))Z B xp2m(E V (x))dx .x011
The usual way of writing the last expression isZ xpp(x)dx ,where p(x) 2m(E V (x)) ,B(x) x0where p(x) is called the semiclassical momentum; correspondingly, v(x) p(x)/m is called thesemiclassical velocity. If E V (x) is negative, i.e. we are dealing with the classicallyforbiddenpregion, the value of p(x) becomes purely imaginary, p(x) iκ(x), where κ(x) 2m(V (x) E).I will also call κ(x) a semiclassical momentum in the forbidden region.Knowing B(x) we solve for A(x) using the second equationd ln A1 d ln B 0 dx2 dx CA(x) p.p(x)This completes the derivation of the semiclassical approximation for the wavefunction Z xCiψ(x) pp(x)dx .exp x0p(x)(13)There is also good physical reasoning for this functional form. Wepknow that a particle movingipxin a constant potential is described by a plane wave Ce with p 2m(E V ). If potential issmooth, i.e. does not change much on the scale of the particle wavelength then locally the planewave description should be also valid, but with locally defined momentum p(x) adjusted to thecurrent value of the potential. The only problem with the replacementZ xppx 2m(E V (x))dx ,is that if we do not do something with the wave amplitude then the current density j C 2 p(x)/mwill decrease while for smooth potentials we do not expect any reflections (with exponentially goodaccuracy), i.e. the current density should remain constant. This leads toC C(x) p(x) 1/2 .We can now explicitly check under what conditions it is possible to neglect the second derivativeof A. To this end require thatA(x)(B 0 (x))2,A00 (x) 2If L is the length scale over which the A(x) function changes substantially then A00 A/L2 leadingto λ(x)L(x) ,(14)p(x)2πi.e. the amplitude has to change a little (in relative terms) over the particle wavelength.We can not consider the theory of the the semiclassical approximation complete until we discussthe matching conditions. The problem is that in the vicinity of points where E V (x) calledthe classical ’turning points’, the semiclassical momentum goes to zero. This invalidates condition(14) leaving us with an incomplete description because even if we can use (13) in different regionsin space, to the left and to the right of the turning point, we do not know yet how to match themto establish a common state and find the energy quantization condition. The only exception from12
this is the case of a stepwise change in V (x) at point x0 — then the semiclassical approximationmay be valid all the way to point x0 from both sides and the matching conditions are standard,i.e. continuity of ψ(x) and its derivative.One of the many equivalent solutions for the generic turning point is as follows. In the vicinityof the turning point a we can write V (x) E V (x) V (a) f (x a) and try to solve theSchrödinger Equation equation (let’s assume that f is positive) 1 d2 f (x a) ψ(x) 0 .2m dx2we have to do it only once (!) because in terms of the dimensionless variable z (2mf )1/3 (x a)we have a universal (system independent) equation 2 d z ψ 0.dz 2Its solutions are special (Airy) functions Ai(z) and Bi(z). What is crucial for our semiclassicaltheory is their asymptotic form for large positive and negative z:Ai(z) limz 13/2 1/4 e (2/3)z ,2 πz13/2e(2/3)z ,1/4πz 12 3/2 πz ,Ai( z) lim 1/4 cosz 34πz 12 3/2 πBi( z) lim 1/4 sinz ,z 34πzBi(z) limz These are exact relations across the turning point.Now, if we look at the semiclassical solutions in the same asymptotic regions we find thatZ aZ app 22p(x)dx 2mfa x dx 2mf (a x)3/2 ( z)3/2 , for z 0 ,33xxandZaxZpp(x)dx 2mfxp(x a)dx pa222mf (x a)3/2 z 3/2 , for z 0 .33This allows us to write the general solution to the left of the turning point as 12 3/2 π2 3/2 πψ( z) lim 1/4 α cosz β sinz ,z 3434πzwith arbitrary α and β, and to the right of the turning point asψ(z) limz hi1(2/3)z 3/2 (2/3)z 3/2 δe,γeπz 1/413
with arbitrary γ and δ. All what is left is to notice that our expression to the left can be re-writtenidentically asψ( z) αAi( z) βBi( z) ,and to the right asψ( z) 2γAi(z) δBi(z) .To agree with the exact solution we must then have γ α/2 and δ β. These are our matchingconditions across the turning point. To summarize: Z a Z a απβπψx a pcosp(x)dx psinp(x)dx 44p(x)p(x)xx Z x Z x α/2βψa x pexp κ(x)dx pexpκ(x)dx .κ(x)κ(x)aaNote, that in this matching rule the integrals are always increasing as we move away from theturning point. We do not need to do anything to obtain matching conditions when the classicallyforbidden region is to the left of the turning point x b, i.e. when f 0. Simply notice that ifwe change z z we will get the same problem with f f . Thus Z x Z x απβπψb x pcosp(x)dx psinp(x)dx 44p(x)p(x)bbψx b Z b Z b α/2β pexp κ(x)dx pexpκ(x)dx .κ(x)κ(x)xxAgain, the integrals are all increasing as we move away from the turning point. With this, thesemiclassical approximation is complete and can be used as a tool to establish interesting resultsunder rather generic conditions when exact solutions are not available.Consider particle transmission through the large smoothbarrier shown to the left. For the reflection/transmissionsetup in region III we require that there is only an outgoingwaveRxδψIII pei b p(x)dx iπ/4 p(x) Z x Z x δππpcosp(x)dx i sinp(x)dx 44p(x)bb II I IIIEabIn the second region, the wavefunction is a superposition of increasing and decreasing exponentials Z b Z b δ/2 iδexp κ(x)dx pexpκ(x)dx .(15)ψx b pκ(x)κ(x)xxWe immediately fix the amplitudes in this expression using matching conditions; in a way it evenlooks easier than ’continuity of ψ and its derivative’ because we can work our way from right to14x
left in a unique way and build the solution, i.e. no need for solving a set of linear equations forfree coefficients. We rewrite (15) identically as Z x Z x ieC δe C δ/2κ(x)dx .(16)expκ(x)dx pexp ψx b pκ(x)κ(x)aabut now we count all integrals from point x a to prepare for applying the matching conditionsfor the second point. HereZ bC κ(x)dx ,ais the semiclassical action for the underbarrier motion. According to the conditions of validity, thelength of the underbarrier passage has to be much longer than the typical wavelength 2π /κ(x),i.e. the value of C is large. Applying the second matching condition we get Z a Z a e C δ/2π 2ieC δπψx a psinp(x)dx pcosp(x)dx .(17)44p(x)p(x)xxWe can write it asψx a pαp(x) Zexp ixaπp(x)dx i4 β pp(x) Zexp ixaπp(x)dx i4 .(18)with α iδ eC e C /4 , β iδ eC e C /4 .Assuming that the potential goes to zero on both sides of the barrier as x we have thecurrent density of incoming/reflected/transmited particles being proportional to α 2 , β 2 , and δ 2 ,
Variational principles. Semiclassical approximation. There exist only a handful of problems in quantum mechanics which can be solved exactly. More often one is faced with a potential or a Hamiltonian for which exact methods are unavailable and approximate solutions must be found. Here we review three approximate methods each of
Quantum mechanics 2 - Lecture 2 Igor Luka cevi c UJJS, Dept. of Physics, Osijek 17. listopada 2012. Igor Luka cevi c Perturbation theory. Contents Time-independent nondegenerate perturbation theory Time-independent degenerate perturbation theory Time-
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
Fundamentals and Applications of Perturbation Methods in Fluid Dynamics Theory and Exercises - JMBC Course - 2018 Sjoerd Rienstra Singularity is almost invariably a clue (Sherlock Holmes, The Boscombe Valley Mystery) 1 07-03-2018. FUNDAMENTALS AND APPLICATIONS OF PERTURBATION METHODS IN FLUID DYNAMICS
of the correlation energy that is missing from Hartree-Fock calculations. 2 Time-independent perturbation theory 2.1 Non-degenerate systems The approach that we describe in this section is also known as "Rayleigh-Schr odinger perturbation theory". We wish to find approximate solutions of the time-independent
similarities with perturbation theory of quantum mechanics (Rae 1992). They need useful approaches for formulating nonstationary or stochastic system dynamics. Of course, the system behaviors are represented differently. Perturbation theory assumes well-defined system behavior, e.g. the wave equation (
4 Classical perturbation theory 4 . Celestial Mechanics greatly motivated the advances of perturbation . different mathematical settings by Arnold ([16]) and Moser ([37]). The overall theory is known with the acronym of KAM theory
4.1 Perturbation theory, Feynman diagrams As as been presented for QED, a natural scheme is to assume that g is small and perform aseriesexpansioninpowersofg. This amounts to consider that the interaction terms are small, and represents a small perturbation of the free theory. Thus we expand the interaction term in the functional integral exp 4 .
E. Kreyszig, “Advanced Engineering Mathematics”, 8th edition, John Wiley and Sons (1999). 3. M. R. Spiegel, “Advanced Mathematics for Engineers and Scientists”, Schaum Outline Series, McGraw Hill, (1971). 4. Chandrika Prasad, Reena Garg, "Advanced Engineering Mathematics", Khanna Publishing house. RCH-054: Statistical Design of Experiments (3:1:0) UNIT 1 Introduction: Strategy of .
