(p.277) Appendix C Introduction to Perturbation Theory
(p.277) Appendix C Introduction to Perturbation Theory
(p.277) Appendix C
Introduction to Perturbation Theory
C.1 Non-degenerate perturbation theory
We give here a summary of the principles of perturbation theory, for reference and to define the notation.
We are faced with a Hamiltonian ℋ that is too complicated to handle directly. We suppose that it differs by a small ‘perturbation’ ℋ′ from a closely related unperturbed or ‘zeroth-order’ Hamiltonian ℋ0 describing a problem that we can solve:
Here λ may be a physical quantity describing the strength of the perturbation, such as the magnitude of an electric or magnetic field, but often it is just a parameter that we can vary hypothetically from 0 for the unperturbed problem to 1 for the problem that we want to solve.
Suppose that the eigenfunctions of the unperturbed problem are |n 0〉, with eigenvalues :
We want to find |n〉 and Wn satisfying ℋ|n〉 = Wn|n〉. We assume at this stage that |n〉 is a non-degenerate state, well separated in energy from other states. Expand |n〉 and Wn as power series in λ:
Without loss of generality, we can require that |n 0〉 is normalized and that all the corrections to the zeroth-order wavefunction are orthogonal to it:
so that 〈n|n 0〉 = 〈n 0|n 0〉 = 1. This is intermediate normalization. Substitute the expressions (C.1.3) and (C.1.4) into (ℋ − Wn)|n〉 = 0, and we get
For sufficiently small λ, we expect the power series to converge, and in that case we can equate coefficients of powers of λ:
(p.278) and so on. The first of these equations is the zeroth-order problem, an eigenvalue equation which we suppose solved. The rest are inhomogeneous differential equations. Multiply the first-order equation (C.1.8) by 〈n 0| and integrate:
Because is hermitian, it follows that for any wavefunction ψ,
and since |n 0〉 is normalized we obtain
Thus the first-order energy is the expectation value of the perturbation operator for the unperturbed wavefunction |n 0〉. Indeed the total energy to first order is
An alternative way to write this result is:
Thus the expectation value of the complete Hamiltonian over the unperturbed wavefunction gives the total energy correct to first order in λ.
Now multiply the second-order equation (C.1.9) by 〈n 0|, to give
and we have the second-order energy. To evaluate this, we need the first-order wavefunction |n′〉, and to obtain it, we have to solve the inhomogeneous differential equation (C.1.8).
C.1.1 Rayleigh–Schrödinger perturbation theory
The standard way to do this, used by Rayleigh for classical oscillators and adapted for quantum mechanics by Schrödinger, is to expand |n′〉 in terms of the unperturbed eigenfunctions:
where the prime on the summation sign conventionally indicates that we are to omit the term with k = n. This is to ensure that 〈n 0|n′〉 = 0, in accordance with (C.1.5). Substitute this expression into (C.1.8) and remember that :
(p.279) Multiply by 〈p 0|, and remember that the eigenfunctions |k 0 i of the unperturbed Hamiltonian are orthonormal. We obtain
so that the first-order wavefunction is
Here the second form introduces an abbreviated notation for the ‘energy denominator’ and for the matrix element . We substitute this result into (C.1.17) to obtain the second-order energy:
Here we have a sum over the excited states of the system. It is important to realize that this extends over all excited states, including continuum states.
C.1.2 Unsöld’s average-energy approximation
A crude but useful approximation to the second-order energy can be obtained by replacing all the energy denominators in the Rayleigh–Schrödinger expression (C.1.22) for the second-order energy by some ‘average’ value ∆:
If |n〉 is the ground state and ∆ the first excitation energy, then ∆pn ≥ ∆ and the approximate expression must be larger in magnitude than the exact. Often ∆ is taken to be the first ionization energy, but this does not give a lower bound to because the sum over states includes an integral over the continuum states.
(p.280) C.2 The resolvent
We study here a formal treatment of the perturbation problem which is sometimes useful. It is used in §6.3.1 in the development of iterative symmetry-forcing perturbation methods. First consider the operator Pk ≡ |k 0〉〈k 0|, where |k 0〉 is one of the normalized eigenstates of ℋ0. If we apply this to a wavefunction ∑j cj|j 0〉, we obtain ∑j cj|k 0〉〈k 0|j 0〉 = ∑j cj|k 0〉δ1 = ck|k 0〉. That is, Pk projects from a general wavefunction the |k 0〉 component. It is a projection operator. All projection operators satisfy , easily verified in this case. We can also define a complementary projection operator Qk = 1 − Pk, which removes the |k 0〉 component from any wavefunction, leaving the rest of the wavefunction unchanged. It is easy to verify that also.
Now consider the operator
Operating on state |n 0〉 this gives zero, because Qn|n 0〉 = 0. (This can be made rigorous by replacing by z in the denominator, and taking the limit after applying the operator (McWeeny 1989).) Otherwise it behaves like an inverse of the operator . It is called the reduced resolvent.
and in general
By expressing Rn in the form (with the prime on the sum denoting k as usual that the term k = n is omitted from the sum) we can recover the formulae of Rayleigh– Schrödinger perturbation theory.
C.3 Degenerate perturbation theory
If the state |n 0〉 is degenerate (a set of components |nα 0〉 all with energy ) then we cannot apply the Rayleigh–Schrödinger formulae (C.1.21) and (C.1.17) because some of the energy denominators vanish. Also, if we take a particular perturbed state and allow λ to tend to zero, there is no reason to expect that the resulting zeroth-order state |nm 0〉 will coincide with one of the |nα 0〉. Rather we expect to get a linear combination of them:
If we multiply by 〈nβ 0|, the first term disappears as before, because , and we get a set of secular equations:
Consequently we need to solve these secular equations to find the first-order energies and the corresponding eigenvectors c αm. Alternatively, since , we can replace (C.3.3) by
where Hnβ,nα is the matrix element of the complete Hamiltonian and W is the energy correct to first order. Note that (C.3.5) is the result we would get from a variational treatment using the trial function (C.3.1), and that consequently the same result applies in the ‘nearly degenerate’ case, where the energy separations among a set of unperturbed states, although non-zero, are not large compared with the matrix elements of the perturbation. In such a case the formulae don’t blow up, but the perturbation series fails to converge and the degenerate version must be used.
Now we can follow the Rayleigh–Schrodinger procedure to find the first-order wavefunction as before:
where the sum is taken over the components |pq 0〉 i of the other unperturbed states, and as before. Similarly
C.4 Time-dependent perturbation theory
Here we consider a Hamiltonian ℋ = ℋ0 + ℋ′ consisting of a time-independent part ℋ0 and a time-dependent perturbation ℋ′. The unperturbed states are stationary:
where the |n〉 are the eigenstates of the unperturbed Hamiltonian: ℋ0|n〉 = Wn|n〉. The perturbed problem does not have stationary states; the solution Ψ must satisfy Schrodinger’s time-dependent equation:
(p.282) To solve this, we use Dirac’s method of variation of constants: let
If the Hamiltonian were time-independent the ak would be constant (and their values would be arbitrary). In the presence of the small time-dependent perturbation they evolve slowly with time. Substituting (C.4.3) into (C.4.2) gives
Multiply by and integrate over variables other than time:
where ωpk = (Wp − Wk)/ℏ. We can integrate this formally to obtain
If ℋ′ takes the form ℋ′ = Vˆ f(t), where Vˆ is an operator independent of time and f(t) is a function only of the time, then
All of this is exact, but is rarely useful as it stands since the expression for ap(t) involves all the ak (including ap) at times from 0 to t. We must therefore approximate: we assume that Vˆ is sufficiently small that the ak will not change much over the interval from 0 to t, so that we can put ak(τ) = ak(0) in (C.4.8), giving
If necessary we could substitute this result back into the r.h.s. of (C.4.8) to obtain a second-order formula, but in practice (C.4.9) is usually adequate.
For simplicity we now consider the case where the system is definitely in the stationary state n at time 0, so that ak(0) = δkn. Then ap(t) is the probability amplitude for a transition to state p having occurred by time t:
The development of the theory from this point depends on the form of the function f(t) that describes the time-dependence of the perturbation. See, for example, the discussion in §2.5 of the response of a molecule to an oscillating electric field.