Quantum Physics III Chapter 1: Perturbation Theory

Chapter 1 Perturbation theory c B. Zwiebach It is often the case that the Hamiltonian of a system differs slightly from a Hamiltonian that is well studied and completely understood. This is a situation where perturbation theory can be useful. Perturbation theory allows us to make statements about the Hamiltonian of the system using what we know about the well studied Hamiltonian. The well studied Hamiltonian could be the that of the simple harmonic oscillator in one, two, or three dimensions. In a diatomic molecule, for example, the potential that controls the vibrations is not exactly quadratic; it has extra terms that make the vibrations slightly anharmonic. In that situation the extra terms in the potential represent perturbations of the Hamiltonian. The hydrogen atom Hamiltonian is also a well understood system. If we place the atom inside a weak external magnetic field or electric field, the situation is described by adding some small terms to the hydrogen Hamiltonian. Similarly, the interaction between the magnetic moments of the proton and the electron can be incorporated by modifying the original hydrogen atom Hamiltonian. The interaction between two neutral hydrogen atoms at a distance, leading to the van der Waals force can be studied in perturbation theory by thinking of the two atoms as electric dipoles. The Hamiltonian of interest is written as the understood, original Hamiltonian H (0) , plus a perturbation δH: H (0) + δH . (1.0.1) Since H (0) is Hermitian and the sum must be a Hermitian Hamiltonian, the perturbation operator δH must also be Hermitian. It is convenient to introduce a unit-free constant λ ∈ [0, 1] and to consider, instead, a λ-dependent Hamiltonian H(λ) that takes the form H(λ) = H (0) + λ δH . (1.0.2) When λ = 1 we have the Hamiltonian of interest, but λ allows us to consider a family of Hamiltonians that interpolate from H (0) , when λ is equal to zero, to the Hamiltonian of interest for λ equal to one. In many cases perturbations can be turned on and off; think, 1 2 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY for example, of an atom in an external magnetic field that can be varied continuously. In that case we can view λ as the parameter that allows us to turn on the perturbation by letting λ 6= 0. The parameter λ is also useful in organizing the perturbation analysis, as we will see below. We spoke of a Hamiltonian that differs slightly from H (0) . In order to use perturbation theory we need λδH to be a ‘small’ perturbation of the Hamiltonian H (0) . We will have to deal with the meaning of small. At first sight we may imagine that small means that, viewed as matrices, the largest entries in λδH are smaller than the largest entries in H (0) . While this is necessary, more is needed, as we will see in our analysis. An additional advantage of using λ is that by taking it to be sufficiently small we can surely make λδH small. We assume that the Hamiltonian H (0) is understood, namely, we know the eigenstates and eigenvalues of H (0) . We want to know the eigenstates and eigenvalues of H(λ). One may be able to calculate those exactly, but this is seldom a realistic possibility. Diagonalizing δH is seldom useful, since δH and H do not generally commute and therefore δH eigenstates are not eigenstates of H(λ). In perturbation theory the key assumption is that the eigenvalues and eigenvectors of H(λ) can be found as series expansions in powers of λ. We hope, of course, that there are some values of λ for which the series converges, or at least gives useful information. Figure 1.1: The energy eigenvalues of H(λ) as λ varies from zero to one. On the λ = 0 vertical axis the H (0) eigenstates are represented by heavy dots. By the time λ = 1 the dots have shifted. In Figure 1.1 we illustrate some of the phenomena that we may see in the spectrum of a system with Hamiltonian H(λ). We show how the energies of the various states may change as the parameter λ is increased from zero. The two lowest energy eigenstates are non-degenerate and their energies can go up and down as λ varies. Next up in energy we have two degenerate states of H (0) (the Hamiltonian as λ = 0). The perturbation splits the 3 1.1. NONDEGENERATE PERTURBATION THEORY two levels, and that happens generically. In the figure, the perturbation splits the levels to first order in λ, as shown by the different slopes of the two curves that meet at λ = 0. In other words, viewed as power series in λ the energies of the two states have different linear terms in λ. The last level shown corresponds to four degenerate states. The perturbation to first order in λ splits the states into a group of three states and a fourth. To second order in λ the three states split further. A single Hamiltonian can exhibit behavior like this, with many possible variations. To analyze the evolution of states and energies as functions of λ we have two possible cases: (i) we are following a non-degenerate state or, (ii) we are following a collection of degenerate states. The challenges are quite different and therefore we must analyze them separately. Clearly both situations can occur for a single Hamiltonian, depending on the spectrum of H (0) . To follow a non-degenerate state we use non-degenerate perturbation theory. To follow a set of degenerate states we use degenerate perturbation theory. Since Hamiltonians H (0) generally have both non-degenerate and degenerate states we need to consider both types of perturbation theory. We begin with non-degenerate perturbation theory. 1.1 Nondegenerate perturbation theory We begin by describing the original Hamiltonian H (0) . We assume this Hamiltonian has a discrete spectrum with an orthonormal basis |k(0) i of energy eigenstates, where k ∈ Z is a label that ranges over a possibly infinite set of values: (0) H (0) |k(0) i = Ek |k(0) i , hk(0) |l(0) i = δkl . (1.1.1) We will let k = 0 denote the ground state and we order the states so that the energies generally increase as the value of the label increases, so that (0) E0 (0) ≤ E1 (0) ≤ E2 (0) ≤ E3 ≤ ... (1.1.2) The equal signs are needed because some states may be degenerate. In this section we focus on a non-degenerate state |n(0) i with fixed n. This means that |n(0) i is a single state that is separated by some finite energy from all the states with more energy and from all the states with less energy. In other words the following must be part of the sequence of inequalities in (1.1.2) (0) (0) . . . ≤ En−1 < En(0) < En+1 ≤ . . . (0) (1.1.3) (0) If the chosen state is the ground state, we have instead E0 < E1 . As the perturbation is turned on by making λ different from zero, the energy eigenstate |n(0) i of H (0) will evolve into some energy eigenstate |niλ of H(λ) with energy En (λ): H(λ)|niλ = En (λ) |niλ , (1.1.4) 4 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY where |niλ=0 = |n(0) i , and En (λ = 0) = En(0) . (1.1.5) As we said, the solution is assumed to take the form of a regular power series expansion in λ. To make this clear consider a function f (λ) such that its derivatives to all orders exist for λ = 0. In that case we have a Taylor expansion ∞ X 1 (n) f (0) λn = f (0) + f ′ (0) λ + 12 f ′′ (0) λ2 + f (λ) = n! 3 1 ′′′ 3! f (0) λ n=0 + ··· (1.1.6) The expansion is a power series in λ, with coefficients f (0), f ′ (0), etc, that are λ independent and reflect the value of the function and its derivatives at λ = 0. For our problem we note the values of |niλ and En (λ) for λ = 0 (1.1.5) and write: |niλ = |n(0) i + λ|n(1) i + λ2 |n(2) i + λ3 |n(3) i + . . . , En (λ) = En(0) + λ En(1) + λ2 En(2) + λ3 En(3) + . . . (1.1.7) The superscripts on the states and energies denote the power of λ that accompanies them in the above expressions. The above equations are a natural assumption; they state that the perturbed states and energies, being functions of λ, admit a Taylor expansion around λ = 0. Our aim is to calculate the states |n(1) i , |n(2) i , |n(3) i , . . . (1.1.8) En(1) , En(2) , En(3) , . . . (1.1.9) and the energies Note that all these states and energies are, by definition, λ independent. Here |n(1) i is the (1) leading correction to the state |n(0) i as we turn on λ, and En is the leading correction to the energy as we turn on λ. We will not impose the requirement that |niλ is normalized. It suffices that |niλ is normalizable, which it will be for sufficiently small perturbations. For λ = 1 we would find the solution for H(1) = H (0) + δH in the form |ni ≡ |ni1 = |n(0) i + |n(1) i + |n(2) i + |n(3) i + . . . , En ≡ En (1) = En(0) + En(1) + En(2) + En(3) + . . . (1.1.10) Substituting the ansatz (1.1.7) into the Schrödinger equation (1.1.4) we will find the conditions for such solution to exist:
H (0) + λδH − En (λ) |niλ = 0 , (1.1.11) which more explicitly takes the form   (H (0) − En(0) ) − λ(En(1) − δH) − λ2 En(2) − λ3 En(3) − . . . − λk En(k) + . . .   |n(0) i + λ|n(1) i + λ2 |n(2) i + λ3 |n(3) i + . . . + λk |n(k) i + . . . = 0 . (1.1.12) 1.1. NONDEGENERATE PERTURBATION THEORY 5 Multiplying out we get a series in λ with coefficients λ-independent vectors in the state space of the theory. If this is to vanish for all values of λ those coefficients must be zero. Collecting the coefficients for each power of λ we λ0 : (H (0) − En(0) ) |n(0) i = 0 , λ1 : (H (0) − En(0) ) |n(1) i = (En(1) − δH)|n(0) i , λ2 : (H (0) − En(0) ) |n(2) i = (En(1) − δH)|n(1) i + En(2) |n(0) i , λ3 : (H (0) − En(0) ) |n(3) i = (En(1) − δH)|n(2) i + En(2) |n(1) i + En(3) |n(0) i , .. . λk : .. . .. . (H (0) − En(0) ) |n(k) i = (En(1) − δH)|n(k−1) i + En(2) |n(k−2) i + . . . + En(k) |n(0) i . (1.1.13) Each equation is the condition that the coefficient multiplying the power of λ indicated to the left vanishes. That power is reflected as the sum of superscripts on each term, counting δH as having superscript one. This gives a simple consistency check on our equations. These (1) (2) are equations for the kets |n(1) i, |n(2) i, . . . as well as the energy corrections En , En , . . .. Note again that λ does not enter into the equations, and thus the kets and energy corrections are λ independent. The first equation, corresponding to λ0 , is satisfied by construction. The second equation, corresponding to λ1 , should allow us to solve for the first correction |n(1) i to the state (1) and the first correction En to the energy. Once these are known, the equation corre(2) sponding to λ2 involves only the unknowns |n(2) i and En , and should determine them. At each stage each equation has only two unknowns: a state correction |n(k) i and an energy (k) correction En . A useful choice. We now claim that without loss of generality we can assume that all the state corrections |nk i, with k ≥ 1 contain no vector along |n(0) i. Explicitly: 0 = hn(0) |n(1) i = hn(0) |n(2) i = hn(0) |n(3) i = . . . . (1.1.14) To show this we explain how we can manipulate a solution that does not have this property into one that does. Suppose you have solution in which the state corrections |n(k) i have components along |n(0) i: |n(k) i = |n(k) i′ − ak |n(0) i, k ≥ 1, (1.1.15) with some constants ak and with |n(k) i′ orthogonal to |n(0) i. Then the solution for the full 6 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY corrected state is

|niλ = |n(0) i + λ |n(1) i′ − a1 |n(0) i + λ2 |n(2) i′ − a2 |n(0) i + . . .
= 1 − a1 λ − a2 λ2 − . . . |n(0) i + λ|n(1) i′ + λ2 |n(2) i′ + . . . (1.1.16) Since this is an eigenstate of the Hamiltonian H(λ), it will still be an eigenstate if we change its normalization by dividing it by any function of λ. Dividing by the coefficient of |n(0) i we have the physically identical solution |ni′λ given by |ni′λ = |n(0) i +   1
λ|n(1) i′ + λ2 |n(2) i′ + . . . 1 − a1 λ − a2 λ2 − . . . (1.1.17) We can expand the denominator so that we get
|ni′λ = |n(0) i + λ|n(1) i′ + λ2 |n(2) i′ + a1 |n(1) i′ + . . . (1.1.18) The explicit expressions do not matter, the key point, actually visible in (1.1.17), is that we have a physically identical solution of the same equation in which the state corrections are all orthogonal to |n(0) i. This shows that we can impose the conditions (1.1.14) without loss of generality. Solving the equations. Let us finally begin solving equations (1.1.13). For this we note that the Schrodinger equation for the ket |n(0) i implies that for the bra we have hn(0) |(H (0) − En(0) ) = 0 . (1.1.19) This means that acting with hn(0) | on the left-hand side of any of the equations in (1.1.13) will give zero. Consistency requires that acting with hn(0) | on the right-hand side of any of the equations in (1.1.13) also give zero, and presumably some interesting information. For the λ-equation this gives: 0 = hn(0) |(En(1) − δH)|n(0) i . (1.1.20) (1) Since |n(0) i is normalized and En is a number, this means that En(1) = hn(0) |δH|n(0) i . (1.1.21) This is the most famous result in perturbation theory: the first correction to the energy of an energy eigenstate is simply the expectation value of the correction to the Hamiltonian in the uncorrected state. You need not know the correction to the state to determine the first correction to the energy! Note that the hermicity of δH implies the required reality of the energy correction. We can actually find some interesting formulae (but not yet fully explicit!) for the higher energy corrections. For the λ2 equation, acting with hn(0) | on the right-hand side gives   (1.1.22) 0 = hn(0) | (En(1) − δH)|n(1) i + En(2) |n(0) i . 1.1. NONDEGENERATE PERTURBATION THEORY 7 (1) Recalling our orthogonality assumption, we have hn(0) |n(1) i = 0 and the term with En drops out. We get En(2) = hn(0) |δH|n(1) i , (1.1.23) which states that the second correction to the energy is determined if we have the first correction |n(1) i to the state. Note that this expression is not explicit enough to make it (2) manifest that En is real. This and the earlier result for the first correction to the energy have a simple generalization. Acting with hn(0) | on the last equation of (1.1.13) we get   0 = hn(0) | (En(1) − δH)|n(k−1) i + En(2) |n(k−2) i + . . . + En(k) |n(0) i . (1.1.24) Using the orthogonality of |n(0) i and all the state corrections, we have 0 = −hn(0) |δH)|nk−1 i + En(k) , (1.1.25) En(k) = hn(0) | δH |n(k−1) i . (1.1.26) and therefore we have At any stage of the recursive solution, the energy at a fixed order is known if the state correction is known to previous order. So it is time to calculate the corrections to the states! Let us solve for the first correction |n(1) i to the state. This state must be some particular superposition of the original energy eigenstates |k(0) i. For this we look at the equation (H (0) − En(0) ) |n(1) i = (En(1) − δH)|n(0) i . (1.1.27) This is a vector equation: the left-hand side vector set equal to the right-hand side vector. As in any vector equation, we can check it using a basis set of vectors. Forming the inner product of each and every basis vector with both the left-hand side and the right-hand side, we must get equal numbers. We already acted on the above equation with hn(0) | to figure (1) out En . The remaining information in this equation can be obtained by acting with all the states hk(0) | with k 6= n: hk(0) |(H (0) − En(0) ) |n(1) i = hk(0) |(En(1) − δH)|n(0) i . (1.1.28) On the left-hand side we can let H (0) act on the bra. On the right-hand side we note that (1) with k 6= n the term with En vanishes (0) (Ek − En(0) ) hk(0) | n(1) i = −hk(0) |δH|n(0) i . (1.1.29) To simplify notation we define the matrix elements of δH in the original basis δHmn ≡ hm(0) |δH|n(0) i . (1.1.30) 8 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY Note that the Hermiticity of δH implies that δHnm = (δHmn )∗ . (1.1.31) With this notation, equation (1.1.29) gives hk(0) | n(1) i = − δHkn (0) Ek (0) − En , k 6= n . (1.1.32) Since we now know the overlap of |n(1) i with all basis states, this means that the state has been determined. Indeed we can use the completeness of the basis to write X X |n(1) i = |k(0) ihk(0) |n(1) i = |k(0) ihk(0) |n(1) i , (1.1.33) k k6=n since the term with k = n does not contribute because of the orthogonality assumption. Using the overlaps (1.1.32) we now get |n(1) i = − X |k(0) iδHkn (0) k6=n (0) Ek − En . (1.1.34) This shows that the first correction |n(1) i can have components along all basis states, except |n(0) i. The component along a state |k(0) i vanishes if the perturbation δH does not couple |n(0) i to |k(0) i, namely, if δHkn vanishes. Note that the assumption of non-degeneracy is needed here. We are summing over all states |k(0) i that are not |n(0) i and if any such state has the same H (0) energy as |n(0) i the energy denominator will vanish causing trouble! Now that we have the first order correction to the states we can compute the second order correction to the energy. Using (1.1.23) we have X hn(0) |δH|k(0) iδHkn . (1.1.35) En(2) = hn(0) |δH|n(1) i = − (0) (0) Ek − En k6=n In the last numerator we have hn(0) |δH|k(0) i = δHnk = (δHkn )∗ and therefore X |δHkn |2 En(2) = − . (0) (0) E − E n k6=n k (1.1.36) (2) This is the second-order energy correction. This explicit formula makes the reality of En manifest. In summary, going back to (1.1.7), we have that the states and energies for H(λ) = H (0) + λδH are, to this order, |niλ = |n(0) i − λ En (λ) = En(0) X δHkn (0) k6=n Ek (0) − En + λ δHnn − λ 2 X k6=n |k(0) i + O(λ2 ) , |δHkn |2 (0) (0) Ek − En (1.1.37) 3 + O(λ ) , 9 1.1. NONDEGENERATE PERTURBATION THEORY Remarks: 1. The first order corrected energy of the (non-degenerate) ground state overstates the true exact ground state energy. To see this consider the first order corrected ground (0) (1) state energy E0 + λE0 . Writing this in terms of expectation values, with |0(0) i denoting the unperturbed ground state, we have (0) (1) = h0(0) |H (0) |0(0) i + λh0(0) |δH|0(0) i E0 + λE0 = h0(0) |(H (0) + λδH)|0(0) i (1.1.38) = h0(0) |H(λ)|0(0) i . By the variational principle, the expectation value of the Hamiltonian on an arbitrary (normalized) state is larger than the ground state energy E0 (λ), therefore (0) (1) E0 + λE0 = h0(0) |H(λ)|0(0) i ≥ E0 (λ) , (1.1.39) which is what we wanted to prove. Given this overestimate at first order, the second order correction to the ground state energy is always negative. Indeed, − λ2 X k6=0 |δHk0 |2 (0) (0) Ek − E0 , (1.1.40) (0) and each term is negative because the unperturbed excited state energies Ek (k 6= 0) (0) exceed the unperturbed ground state energy E0 . 2. The second order correction to the energy of the |n(0) i eigenstate exhibits level repulsion: the levels with k > n push the state down and the levels with k < n push the state up. Indeed, − λ2 X k6=n |δHkn |2 (0) (0) Ek − En = − λ2 X k>n |δHkn |2 (0) (0) Ek − En + λ2 X k 1. Table 1 shows f (z) evaluated for z = 0.9, 1.2, and 1.5. The various approximations to the full series are shown. For our expansion of (1.1.45), convergence for |z| < 1 implies convergence when |λ||V | (0) 1 2 |E1 (0) (0) − (0) E2 | (0) (0) < 1 =⇒ |λV | < 12 |E1 − E2 | . (1.1.47) For |λV | > 12 |E1 − E2 | the perturbation series does not converge. We learn that for convergence the perturbation must be small compared with energy differences in H (0) . It is not sufficient that the magnitude of the matrix elements of λδH be small compared to those in H (0) , energy differences matter. This result leads us to expect complications when energy differences go to zero and H (0) has degeneracies. 12 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY z 0.9 1.2 1.5 f (z) 1.34536 1.56205 1.80278 f8 (z) 1.33939 1.47946 1.20297 f14 (z) 1.33939 1.67280 4.82288 f20 (z) 1.34490 1.36568 -18.4895 f30 (z) 1.34545 2.23047 641.772 Table 1.1: f (z) ≡ 1.1.2 √ 1 + z2 = P i ci z i and fn (z) = Pn i=0 ci z i. Example: Anharmonic oscillator Consider the simple harmonic oscillator H (0) = p̂2 + 1 mω 2 x̂2 . 2m 2 (1.1.48) We want to explore the effect of a perturbation proportional to ∼ x̂4 . This has the effect of changing the original quadratic potential for a more complicated potential that includes a quartic term. To do analysis in a clear way, we must consider units. Using the constants ~, m, ω of the harmonic oscillator a length scale d can be uniquely build: d2 = ~ . mω (1.1.49) The unit-free coordinate x̂/d then has a simple expression in terms of creation and annihilation operators: r ~ x̂ (1.1.50) x̂ = (â + ↠) = d √12 (â + ↠) → = √12 (â + ↠) . 2mω d It follows that an x̂4 perturbation with units of energy takes the form δH = ~ω x̂4 m2 ω 3 4 = x̂ = d4 ~ 1 4 ~ω(â + ↠)4 . (1.1.51) Using the unit-free parameter λ, the perturbed Hamiltonian will therefore be H(λ) = H (0) + λ m2 ω 3 4 x̂ = H (0) + λ 41 ~ω(â + ↠)4 . ~ (1.1.52) We will identify the states |k(0) i of H (0) with the number eigenstates |ki, k = 0, 1, . . . , of the harmonic oscillator. Recall that (0) Ek = ~ω(k + 21 ) , (↠)k |ki = √ |0i . k! (1.1.53) 13 1.1. NONDEGENERATE PERTURBATION THEORY The Hamiltonian H(λ) defines an anharmonic oscillator. In a classical anharmonic oscillator the frequency of oscillation depends on the amplitude of oscillation. In the quantum harmonic oscillator all levels are equally spaced. The frequencies associated with transitions between various levels are therefore integer multiples (i.e. harmonics) of the basic frequency associated to a transition between the first excited state and the ground state. In the quantum anharmonic oscillator the spacing between the energy levels is not uniform. (1) First the simplest question: What is the first-order correction E0 to the energy of the ground state? For this, following (1.1.21) we simply calculate the expectation value of the perturbation on the ground state: (1) E0 = h0| 41 ~ω(â + ↠)4 |0i = 1 4 ~ωh0|(â + ↠)4 |0i = 3 4 ~ω , (1.1.54) where we used h0|(â + ↠)4 |0i = 3 , (1.1.55) as you should verify. It follows that the corrected energy is (0) (1) E0 (λ) = E0 + λE0 + O(λ2 ) = 21 ~ω + λ 34 ~ω + O(λ2 ) =
1 + 23 λ + O(λ2 ) (1.1.56) 1 2 ~ω We note that the energy of the ground state increases with λ > 0. This is reasonable as the quartic term in the modified potential squeezes the ground state. How about second order correction to the ground state energy? For this we use (1.1.36) taking n = 0: (2) E0 = − X k6=0 |δHk0 |2 (0) (0) Ek − E0 . (1.1.57) The sum is over all k ≥ 1 such that δHk0 is non-vanishing. Here δHk0 = 1 4 ~ω hk|(â + ↠)4 |0i . (1.1.58) We consider (â + ↠)4 |0i which corresponds, up to constants to acting on the ground state wavefunction ϕ0 with x4 . This should give an even wavefunction. So (â + ↠)4 |0i must be a superposition of |0i, |2i, and |4i. We cannot get states with higher number because there are at most four creation operators acting on the vacuum. A short calculation (do it!) confirms that √ √ (1.1.59) (â + ↠)4 |0i = 3|0i + 6 2 |2i + 4! |4i . This immediately gives δH00 = 3 4 ~ω , δH20 = √ 3 2 2 ~ω , δH40 = √ 6 2 ~ω , (1.1.60) the first of which we had already determined and is not needed for the second order computation. Back in (1.1.57) we have (2) E0 = − (~ω)2 9 (~ω)2 3 |δH20 |2 |δH40 |2 − = − − =− 2~ω 4~ω 2~ω 2 4~ω 2 9 4 + 3 8
~ω = − 21 8 ~ω . (1.1.61) 14 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY Therefor the corrected ground state energy to quadratic order is (0) (1) (2) E0 + λE0 + λ2 E0 = 1 2 ~ω 1 + 32 λ − 21 2 4 λ
. (1.1.62) The computation can be carried to higher order, as first done by Bender and Wu (Phys. Rev.184 (1969)1231). They find that1
2 7 333 3 30885 4 916731 5 65518401 6 E0 (λ) = 21 ~ω 1 + 32 λ − 21 4 λ + 8 λ − 64 λ + 128 λ − 512 λ + O(λ ) (1.1.63) As it turns out the coefficients keep growing and the series does not converge for any nonzero λ; the radius of convergence is actually zero! This does not mean the series is not useful. It is an asymptotic expansion. This means that for a given small value of λ the magnitude of successive terms generally decrease until, at some point, they start growing again. A good approximation to the desired answer is obtained by including only the part of the sum where the terms are decreasing. (1) Exercise 3. Calculate the first order correction En to the energy for the state |ni of number n. Exhibit the anharmonicity of the oscillator by using this result to find, to first order in λ, the energy separation ∆En (λ) = En (λ) − En−1 (λ) between levels. Let us now find the first order correction to the ground-state wavefunction. Using (1.1.34) with n = 0 we have X δHk0 |0(1) i = − |ki . (1.1.64) (0) (0) k6=0 Ek − E0 We then find |0(1) i = − = √ δH40 δH20 |2i − |4i = − 34 2|2i − 2~ω 4~ω − 34 † † â â |0i − 1 16 1 16 √ 4!|4i (1.1.65) † † † † â â â â |0i . This means that to first order the ground state of the perturbed oscillator is |0iλ = |0i − λ 1.2  3 4 ↠↠|0i + 1 16  ↠↠↠↠|0i + O(λ2 ) . (1.1.66) Degenerate perturbation theory If the spectrum of H (0) has degenerate states, as shown in Figure 1.1, tracking the evolution of those states as λ becomes nonzero presents new challenges. We first show that naive extrapolation of our results for a non-degenerate state do not work. We will also be able to appreciate the basic difficulty. 1 Bender and Wu’s results, in eqns. (2.12) of their paper must all be multiplied by a factor of 2, as they take A0 = 1/2. 15 1.2. DEGENERATE PERTURBATION THEORY 1.2.1 Degenerate toy model Consider an example with two-by-two matrices. The unperturbed matrix H (0) will be set equal to the identity matrix:   1 0 (0) H = . (1.2.1) 0 1 We have a degeneracy here as the two eigenvalues are identical (and equal to one). The perturbation matrix δH is chosen to be off diagonal:   0 1 δH = . (1.2.2) 1 0 We then have H(λ) = H (0) + λδH =  1 λ λ 1  . Using labels n = 1, 2 the unperturbed eigenstates can be taken to be     0 1 (0) (0) (0) (0) , E1 = E2 = 1 , , |2 i = |1 i = 1 0 (1.2.3) (1.2.4) with the corresponding eigenvalues indicated as well. To first order in λ, the eigenvalues (0) predicted from non-degenerate perturbation theory (1.1.37) are En (λ) = En + λδHnn . This gives (0) E1 (λ) = E1 + λδH11 = 1 + λ · 0 = 1 ? (0) (1.2.5) E1 (λ) = E2 + λδH22 = 1 + λ · 0 = 1 ? The eigenvalues are unperturbed to first order in λ since the matrix δH is off-diagonal. These answers, however, are wrong. We can compute the exact eigenvalues of H(λ) and they are 1 ± λ. There is also a problem with the state corrections. Equation (1.1.37) states that X δHkn |k(0) i . (1.2.6) |niλ = |n(0) i − λ (0) (0) − E E n k6=n k (0) (0) but this time, with E1 = E2 the denominator is zero, and the δH matrix element is also zero, giving us an ambiguous result. So what can we do? A direct calculation shows that   1 1 H(λ) has eigenvectors √ with eigenvalue = 1 + λ , (1.2.7) 2 1   1 1 with eigenvalue = 1 − λ . (1.2.8) and √ 2 −1 You may think the eigenvectors jump from those of H (0) indicated in (1.2.4) to those of H(λ) as soon as λ becomes nonzero. Such discontinuity is totally against the spirit of 16 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY perturbation theory. Happily, this is not really true. The eigenvectors of H (0) are in fact ambiguous, precisely due to the degeneracy. The eigenvectors of H (0) are actually the span of the two vectors listed in (1.2.4). The perturbation selected a particular combination of these eigenvectors. This particular combination is the one that we should use even for λ = 0. The lesson is that to get states that vary continuously as λ is turned on we must choose the basis in the degenerate subspace of H (0) carefully. We will call that carefully selected basis the “good” basis. 1.2.2 Systematic Analysis Again, we are looking at the perturbed Hamiltonian H(λ) = H (0) + λδH , (1.2.9) where H (0) has known eigenvectors and eigenvalues. We will focus this time on a degenerate subspace of eigenvectors of dimension N > 1, that is, a space with N linearly independent eigenstates of the same energy. In the basis of eigenstates, H (0) is a diagonal matrix that contains a string of N > 1 identical entries: (0) (0) H (0) = diag { E1 , E2 , . . . , En(0) , . . . , En(0), . . . } . {z } | (1.2.10) |n(0) ; 1i , |n(0) ; 2i, . . . , |n(0) ; N i . (1.2.11) hn(0) ; p |n(0) ; li = δp,l , (1.2.12) N In the degenerate subspace we choose a collection of N orthonormal eigenstates Accordingly, we have H (0) |n(0) ; ki = En(0) |n(0) ; ki . (1.2.13) This set of vectors span a degenerate subspace of dimension N that we will call VN VN ≡ span{|n(0) ; ki, k = 1, . . . N } . (1.2.14) The total state space of the theory, denoted by H is written as a direct sum: H = VN ⊕ V̂ , (1.2.15) where V̂ is spanned by those eigenstates of H (0) that are not in VN . We denote by |p(0) i with p ∈ Z a basis for V̂ . That basis may include both degenerate and non degenerate states. Together with the states in VN we have an orthonormal basis for the whole state space: hp(0) |q (0) i = δpq , hp(0) |n(0) ; ki = 0 . (1.2.16) 17 1.2. DEGENERATE PERTURBATION THEORY Our notation distinguishes the states in VN from those in V̂ because the former have two labels and the latter have only one. We now consider the evolution of the degenerate eigenstates as we turn on the perturbation. Again we assume that the states vary continuously in λ and thus write: |n(0) ; ki → |n; kiλ = |n(0) ; ki + λ|n(1) ; ki + λ2 |n(2) ; ki + O(λ3 ) , En(0) → En,k (λ) = En(0) + (1) λ En,k +λ 2 (2) En,k (1.2.17) 3 + O(λ ) . These equations hold for k = 1, . . . , N . Note that for each value of k the energy corrections might be different and that’s why the energy corrections carry the label k. Our goal is to (p) find the state corrections |n(p) ; ki and the energy corrections En,k for p ≥ 1 and for each k. As before we demand that |n(p) ; ki for p ≥ 1 has no component along |n(0) ; ki, i.e. hn(0) ; k|n(p) ; ki = 0 for p ≥ 1 (1.2.18) Note, however, that |n(p) ; ki may have components along |n(0) ; ℓi with ℓ 6= k. So |n(0) ; ki may and in fact will have a component in VN . The perturbed eigenstates must satisfy H(λ) |n; kiλ = En,k (λ) |n; kiλ , (1.2.19) and substituting the perturbative expansions above we obtain equations completely analogous to the ones in Eq.(1.1.13) λ0 : (H 0 − En(0) ) |n(0) ; ki = 0 , λ1 : (H 0 − En(0) ) |n(1) ; ki = (En,k − δH)|n(0) ; ki , λ2 : (H 0 − En(0) ) |n(2) ; ki = (En,k − δH)|n(1) ; ki + En,k |n(0) ; ki , .. . (1.2.20) (1) (1) .. . (1.2.21) (2) (1.2.22) .. . In the following we will discuss a solution to first order for the case in which the degeneracy in VN is completely broken to first order in perturbation theory; that is, the first order corrections to the energies split the N states completely. Our solution will proceed in three steps: 1. Hit the O(λ) equation with hn(0) ; ℓ| to learn that δH must be diagonal in the chosen basis for VN and to determine the first-order energy shifts. 2. Use the O(λ) equation to calculate the components of |n(1) ; ki in V̂ . 3. Hit the O(λ2 ) equation with hn(0) ; ℓ| to determine the second order energy correction (2) En,k and the component of |n(1) ; ki in VN . 18 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY (0) Step 1. Recalling that hn(0) ; ℓ|(H (0) − En ) = 0, as we hit the O(λ) equation with hn(0) ; ℓ| the left-hand side vanishes and we find (1) hn(0) ; ℓ|(En,k − δH)|n(0) ; ki = 0 . (1.2.23) Since the basis states in VN are orthonormal, this implies that (1) hn(0) ; ℓ| δH |n(0) ; ki = En,k δℓ,k . (1.2.24) This equation holds for all k, ℓ = 1, . . . , N . Remarkably, this equation is telling us that the basis |n(0) ; ki must be chosen to make the matrix δH diagonal in the subspace VN ! This is required in order to get the perturbation theory going. Setting ℓ equal to k we read the values of the first order energy shifts (1) En,k = hn(0) ; k|δH|n(0) ; ki = δHnk,nk , (1.2.25) where the last equality is a definition. The energies to first order are then En,k (λ) = En(0) + λ δHnk,nk . (1.2.26) A few remarks: 1. The above result for the first order energy shifts is true always, even if the degeneracy is not lifted. The degeneracy is lifted when (1) (1) En,k 6= En,ℓ , whenever k 6= ℓ , (1.2.27) for all values of k, ℓ = 1, . . . , N . This assumption will be used in the later steps. If the degeneracy is lifted, the basis states |n(0) ; ki that make δH diagonal in VN are called “good states” or a “good basis”. This means that they are the basis states in VN that get deformed continuously as λ becomes non-zero. If the degeneracy is not lifted to first order the determination of the good basis has to be attempted to second order. 2. The perturbation δH is diagonalized in the subspace VN . The perturbation δH is not diagonal on the whole space H, only within the block representing VN is δH a diagonal matrix. Alternatively we can see this via the action of δH on the basis states. Introducing a resolution of the identity, we have X X δH|n(0) ; ℓi = |n(0) ; qi hn(0) ; q|δH|n(0) ; ℓi + |p(0) i hp(0) |δH|n(0) ; ℓi q = X q p (1) En,ℓ δℓ,q |n(0) ; qi (1) = En,ℓ |n(0) ; ℓi + X p + X p |p (0) i hp (0) |δH|n(0) ; ℓi (1.2.28) |p(0) i hp(0) |δH|n(0) ; ℓi . This shows that the states |n(0) ; ℓi are almost δH eigenstates with eigenvalues equal to the first order energy corrections. The failure is an extra state along V̂ . 19 1.2. DEGENERATE PERTURBATION THEORY 3. We can sometimes assess without computation that a certain basis in VN makes δH diagonal. Here is a rule: the matrix δH is diagonal for a choice of basis in VN if there is a Hermitian operator K that commutes with δH for which the chosen basis vectors are K eigenstates with different eigenvalues. This is quickly established. Consider two different basis states in VN : |n(0) ; pi and |n(0) ; qi, with p 6= q. Assume these have K eigenvalues λp and λq , respectively. Since [δH, K] = 0: 0 = hn(0) ; p|[δH, K]|n(0) ; qi = (λq − λp )hn(0) ; p| δH |n(0) ; qi . (1.2.29) Since the eigenvalues λp and λq are presumed to be different, the non-diagonal matrix elements of δH vanish. Step 2. The O(λ) equation cannot determine the component of |n(1) ; ki along VN . As we will see later, such piece is required by consistency and gets determined from the O(λ2 ) equation. We now determine the piece of |n(1) ; ki along V̂ . For this we hit the O(λ) equation with hp(0) | to find |p(0) i ⊥ VN (1) hp(0) |(H (0) − En(0) )|n(1) ; ki = hp(0) | ( En,k − δH ) |n(0) ; ki (1.2.30) (Ep(0) − En(0) )hp(0) |n(1) ; ki = −δHp,nk , (1.2.31) and we get where we introduced the matrix element δHp,nk ≡ hp(0) | δH |n(0) ; ki. Our equation above means that the piece of |n(1) ; ki in V̂ is now determined: |n(1) ; ki = − X p δHp,nk (0) Ep − (0) En |p(0) i + |n(1) ; ki VN , (1.2.32) where we included explicitly the still undetermined piece of |n(1) ; ki along VN . Step 3. We now hit we hit the O(λ2 ) equation with hn(0) ; ℓ|. The left-hand side vanishes and using the above expression for |n(1) ; ki we find (1) 0 = − hn(0) ; ℓ|(En,k − δH) + hn (0) (1) ; ℓ|(En,k X p − δH)|n (1) |p(0) i ; ki δHp,nk (0) Ep VN (0) − En (1.2.33) (2) +En,k δk,ℓ . (1) In the first term on the right-hand side, the part proportional to En,k vanishes by orthonormality. On the second line, the term including δH can be simplified because δH is diagonal within VN . Recalling (1.2.28) we have X (1) hn(0) ; ℓ|δH|p(0) i hp(0) | . (1.2.34) hn(0) ; ℓ|δH = En,ℓ hn(0) ; ℓ| + p 20 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY The piece in V̂ vector drops out for our case of interest: hn(0) ; ℓ|δH|n(1) ; ki (1) VN = En,ℓ hn(0) ; ℓ|n(1) ; ki Back into equation (1.2.33) we now get  X δHnℓ,p δHp,nk  (1) (1) (0) (1) − E + E n,ℓ hn ; ℓ|n ; ki n,k (0) (0) Ep − En p VN . (1.2.35) (2) VN + En,k δk,ℓ = 0 . (1.2.36) Setting ℓ = k we can determine the second correction to the energies: X |δHp,nk |2 (2) En,k = − . (0) (0) E − E p n p For k 6= ℓ we get X δHnℓ,p δHp,nk p (0) (0) Ep − En   (1) (1) + En,k − En,ℓ hn(0) ; ℓ|n(1) ; ki VN (1.2.37) = 0. (1.2.38) Had we not included the piece of |n(1) ; ki along the degenerate subspace we would have had an inconsistency, since there is no reason why the first term on the left-hand side must be zero. Now the above equation just fixes the components of |n(1) ; ki in the degenerate (1) (1) subspace as long as En,k 6= En,ℓ : hn(0) ; ℓ|n(1) ; ki VN = − 1 (1) En,k − (1) En,ℓ X δHnℓ,p δHp,nk (0) (0) Ep − En p , k 6= ℓ. (1.2.39) We thus have |n(1) ; ki VN = − X ℓ6=k |n(0) ; ℓi 1 (1) (1) En,k − En,ℓ X δHnℓ,p δHp,nk (0) (0) Ep − En p . (1.2.40) It may seem that this extra piece, found by using the O(λ2 ) equation, is higher order than it should in the perturbation: its numerator contains two powers of δH. But this expression (1) (1) also has a curious energy denominator, En,k − En,ℓ , in which each term has a power of δH. All in all, the correction to the state is properly first order in δH. Summarizing our result we have Degenerate perturbation theory with degeneracies lifted at O(λ): |n; kiλ = |n(0) ; ki − λ X p δHp,nk (0) Ep |p(0) i + (0) − En En,k (λ) = En(0) + λ δHnk,nk − λ2 X (1) ℓ6=k En,k X δHnk,p δHp,nk p (0) Ep − |n(0) ; ℓi (0) En − (1) En,ℓ + O(λ3 ) , X δHnℓ,p δHp,nk  p (0) Ep − (0) En + O(λ2 ) (1) En,k = δHnk,nk . (1.2.41) 21 1.2. DEGENERATE PERTURBATION THEORY 1.2.3 Degeneracy lifted at second order We now investigate the case when the degeneracy is completely unbroken to first order. The situation and the setup is similar to the one we just considered: we have a degenerate subspace VN of dimension N and the rest of the space is called V̂ . This time, however, we will assume that the degeneracy of H (0) is not broken to first order in the perturbation δH. Concretely, this means that on the VN basis |n(0) ; ki with k = 1, . . . , N , we have hn(0) ; ℓ| δH |n(0) ; ki = En(1) δℓ,k . (1.2.42) (1) The first order energy correction is the same, and equal to En , for all basis states in VN . You should compare with (1.2.24), where the energy had an extra subscript to distinguish its various possible values. Because the degeneracy is not broken to first order we do not know at this point what is the good basis in VN . We will consider here the case when the degeneracy is completely lifted to second order. We express our ignorance about good basis vectors by stating that we are searching for the right linear combinations: |ψ (0) i = N X k=1 (0) |n(0) ; ki ak . (1.2.43) (0) For some values of the constants ak with k = 1, . . . , N the state |ψ (0) i will be good. We can think of a(0) as the column vector representation of |ψ (0) i in VN . We have written just one state, |ψ (0) i, even though we are expecting to find N good states to span the degenerate subspace. We therefore adjust the notation to reflect this. We introduce a new index I = 1, . . . , N and write (0) |ψI i = N X k=1 (0) |n(0) ; ki aIk , I = 1, . . . , N . (1.2.44) The index I now labels the different good states and their different vector representations (0) (0) aI . Our most immediate goal is to find those vectors aI and thus the good basis. To (0) do so we will have to consider second order energy corrections. The states |ψI i form an (0) orthonormal basis in VN if the coefficients aI satisfy (0) (0) hψJ |ψI i = δIJ → X (0) (0) (aJk )∗ aIk = δIJ . (1.2.45) k We set up the perturbation theory as usual (2) (1) (0) |ψI iλ = |ψI i + λ|ψI i + λ2 |ψI i + . . . , EnI (λ) = En(0) + λ En(1) +λ 2 (2) EnI +λ 3 (3) EnI (1.2.46) + ... . 22 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY Note that in the energy expansion we have accounted for the degeneracy to zeroth and first order: the index I first appears in the second-order corrections to the energy. Using the Schrödinger equation H(λ)|ψI iλ = EnI (λ)|ψI iλ , (1.2.47) gives the by now familiar equations, of which we list the first four: (0) λ0 : (H (0) − En(0) ) |ψI i = 0 , λ1 : (H (0) − En(0) ) |ψI i = (En(1) − δH)|ψI i , λ2 : (H (0) − En(0) ) |ψI i = (En(1) − δH)|ψI i + EnI |ψI i , λ3 : (H (0) − En(0) ) |ψI i = (En(1) − δH)|ψI i + EnI |ψI i + EnI |ψI i . (1) (0) (2) (1) (2) (0) (3) (2) (2) (1) (1.2.48) (3) (0) The zero-th order equation is trivially satisfied. For the order λ equation the overlap with hn(0) ; ℓ| works out automatically, without giving any new information. Indeed, the left-hand side vanishes and we thus get (0) 0 = hn(0) ; ℓ|(En(1) − δH)|ψI i . (1.2.49) (1) Since hn(0) ; ℓ| is a δH eigenstate with eigenvalue En , up to a vector in V̂ (see (1.2.34)), the above right-hand side vanishes. Acting on the order λ equation with hp(0) | gives useful information: (0) (0) (1) (Ep(0) − En(0) )hp(0) |ψI i = hp(0) |(En(1) − δH)|ψI i = −hp(0) |δH|ψI i , (1.2.50) using the orthogonality of V̂ and VN . Letting (0) δHpI ≡ hp(0) |δH|ψI i , we then have (1) hp(0) |ψI i = − (0) δH pI (0) Ep (0) − En (1.2.51) . (1.2.52) Since the ket |ψI i is still undetermined, it makes sense to write this information about (0) (1) |ψI i in terms of the unknown aI coefficients. We have δHpI ≡ N N X X (0) (0) hp(0) |δH|n(0) ; kiaIk = δHp,nk aIk , k=1 (1.2.53) k=1 Back into (1.2.52) we get 1 (1) hp(0) |ψI i = − (0) Ep − N X (0) En k=1 (0) δHp,nk aIk . (1.2.54) 23 1.2. DEGENERATE PERTURBATION THEORY (1) This gives the piece of |ψI i in V̂ in terms of the unknown zeroth order eigenstates. We have now extracted all the information from the order λ equation. We look now at the order λ2 equation, which contains the second order corrections to the energy and therefore should help us determine the zeroth order good states. We hit that equation with hn(0) ; ℓ| and we get (2) (0) (1) (1) 0 = hn(0) ; ℓ|(En(1) − δH)|ψI i +hn(0) ; ℓ|(En(1) − δH)|ψI i V̂ VN + EnI aIℓ . (1.2.55) (1) Happily, the second term, involving the components of |ψI i along VN , vanishes because (1) of the by now familiar property (1.2.28) adapted to this case. The piece with En on the first term also vanishes. We are thus left with (2) (0) (1) 0 = −hn(0) ; ℓ|δH|ψI i V̂ + EnI aIℓ . (1.2.56) Introducing a resolution of the identity to the immediate right of δH, only the basis states in V̂ contribute and we get X (2) (0) (1) (1.2.57) 0 = − hn(0) ; ℓ|δH|p(0) ihp(0) |ψI i + EnI aIℓ , p where there is no need to copy the |V̂ anymore. Using the result in (1.2.54) we now get 0 = X δHnl,p p N X 1 (0) Ep (0) − En k=1 (0) δHp,nk aIk N X + (2) (0) EnI δℓk aIk (1.2.58) k=1 Reordering sums and multiplying by minus one we get N X k=1 − X δHnl,p δHp,nk p (0) Ep − (0) En − (2) EnI δℓk ! (0) aIk = 0 . (1.2.59) To understand better this equation we define the N × N Hermitian matrix M (2) (2) Mℓ,k ≡ − X δHnℓ,p δHp,nk p (0) (0) Ep − En . (1.2.60) The equation then becomes N   X (0) (2) (2) Mℓ,k − EnI δℓk aIk = 0 . (1.2.61) k=1 Recalling that the Kronecker delta is the matrix representation of the identity, we have   (2) (0) M (2) − EnI 1 aI = 0 . (1.2.62) 24 CHAPTER 1. TIME INDEPENDENT PERTURBATION THEORY (2) This is an eigenvalue equation that tells us that the energy corrections EnI are the eigen(0) values of M (2) and the vectors aI are the associated normalized eigenvectors. These determine, via (1.2.44), our orthonormal basis of good zeroth order states. If δH is known, the matrix M (2) is computable and Hermitian and can therefore be diagonalized. (1) We will leave the computation of the component of |ψI i on the degenerate subspace. That can be done if the degeneracy is completely broken to quadratic order (the eigenvalues of M (2) are all different). Still, it takes some effort and one must use the order λ3 equation. Our results so far are X X (0) (1)  δHpI (0) |p(0) i (0) + |ψI iλ = |ψI i + λ |ψJ i aI,J + O(λ2 ) , (0) E − E n p p J6=I (1.2.63) EIn (λ) = (3) (2) En(0) + λ En(1) + λ2 EIn + λ3 EIn + . . . O(λ3 ) . (1) Here the aI,J are still unknown coefficients that determine the component of the first correction to the states along the degenerate subspace. If you followed the discussion, all other symbols in the above equations have been defined and are computable given δH. (1) The answer for the coefficients aI,J turns out to be 1 (1) aI,J = (2) EnI − (2) EnJ " X δHJp δHpq δHqI (0) p,q (Ep − (0) (0) En )(Eq − (0) En ) − En(1) X p δHJp δHpI (0) The third order corrections to the energy are (3) EIn = X p,q δHIp δHpq δHqI (0) (0) (0) (0) (Ep − En )(Eq − En ) − En(1) X p |δHpI |2 (0) (0)
2 Ep − En (0)
2 Ep − En . # . (1.2.64) (1.2.65)