PHY 6646 Spring 2004 K. Ingersent

Brillouin-Wigner Perturbation Theory

Brillouin-Wigner (BW) perturbation theory is less widely used than the Rayleigh-Schrodinger(RS) version. At rst order in the perturbation, the two theories are equivalent. However,BW perturbation theory extends more easily to higher orders, and avoids the need for sep-arate treatment of nondegenerate and degenerate levels.

Suppose that the unperturbed Hamiltonian H0 has discrete eigenvalues "n and or-thonormal eigenkets jni. For each unperturbed eigenstate, we can dene a pair ofcomplementary projection operators,

Pn = jnihnj and Qn = I Pn =Xm6=n

Pm: (1)

Using the spectral representation H0 =P

n "nPn, one can see that [H0; Qn] = 0.

We write the perturbed eigenproblem for H = H0 + H1 (where 0 1) in theform

(En H0)j ni = H1j ni: (2)Acting with Qn from the left, and taking advantage of [H0; Qn] = 0,

Qnj ni = RnH1j ni; (3)where

Rn = (En H0)1Qn = Qn(En H0)1; (4)which has a spectral representation

Rn =Xm6=n

PmEn"m :

Note that Rn is not a projection operator because R2n 6= Rn.

Using Eqs. (1) and (3), and adopting the usual convention hnj ni = 1, we can writej ni = (Pn +Qn)j ni = jni+ RnH1j ni: (5)

This equation can be solved iteratively in powers of . Inserting j ni correct throughorder j1 on the right-hand side of Eq. (5) yields the order-j result on the left:

To order 0 : j ni = jniTo order 1 : j ni = jni+ RnH1jni;To order 2 : j ni = jni+ RnH1jni+ (RnH1)2jni;

: : :

Thus, the full solution is

j ni =1Xj=0

(RnH1)jjni (1 RnH1)1jni: (6)

1

Substituting the spectral representation of Rn into Eq. (6), we obtain

j ni = jni+ X

m1 6=njm1ihm1jH1jni

En"m1+ 2

Xm1 6=n

Xm2 6=n

jm1ihm1jH1jm2ihm2jH1jni(En"m1)(En"m2)

+ : : :

+ jX

m1 6=n

Xm2 6=n

Xm3 6=n

Xmj 6=n

jm1ihm1jH1jm2ihm2jH1jm3i hmjjH1jni(En"m1)(En"m2) (En"mj)

+ : : : (7)

We can obtain En by applying hnj to Eq. (2), and recalling that hnj ni = 1:En = "n + hnjH1j ni; (8)

or, making use of Eq. (7),

En = "n + hnjH1jni+ 2X

m1 6=n

hnjH1jm1ihm1jH1jniEn"m1

+ : : :

+ j+1X

m1 6=n

Xm2 6=n

Xmj 6=n

hnjH1jm1ihm1jH1jm2i hmjjH1jni(En"m1)(En"m2) (En"mj)

+ : : : (9)

The BW formulation has the advantages that (i) the terms of order 2 and higher inEq. (7), and the terms of order 3 and higher in Eq. (9) are much simpler than theircounterparts in RS perturbation theory, and (ii) there is no need for special versionsof these equations to handle degeneracies among unperturbed states.

A disadvantage of the BW approach is that the perturbed energy En appears on bothsides of Eq. (9). One way around this is to substitute a trial value of En on theright-hand side, then iterate Eq. (9) to convergence. Alternatively, one can substitutelower-order approximations for En on the right side of Eq. (9). For example, if theunperturbed problem has no degeneracy, so that j"m "nj = O (0) for m 6= n, then

En = "n + hnjH1jni+ 2X

m1 6=n

hnjH1jm1ihm1jH1jni"n+hnjH1jni"m1

+ 3X

m1 6=n

Xm2 6=n

hnjH1jm1ihm1jH1jm2i("n"m1)("n"m2)

+O4

: (10)

A condition for convergence of the BW perturbation series is jhmjH1jm0ij jEn"mjfor every pair of states jm 6= ni and jm0i that can be reached from jni under repeatedapplication of H1. (State jmi can be reached from jni under H1 if there is any nonvan-ishing product of matrix elements hmjH1jm1ihm1jH1jm2i hmj1jH1jmjihmjjH1jni.)This convergence condition is clearly violated if the exact eigenvalue En happens toequal any unperturbed eigenvalue "m, where m 6= n and the state jmi can be reachedfrom jni under repeated application of H1. In such cases, the operator Rn dened inEq. (4) does not exist. One can still get an approximate eigenenergy by solving Eq. (9)to order j, but this value will not converge to En with increasing j.

To avoid this, and other problems arising from degeneracy (or near-degeneracy) ofthe unperturbed eigenvalues, it is often necessary to diagonalize H1 in each nearly-degenerate subspace before applying BW theory.

One has to weigh up the pros and cons of the BW and RS methods in deciding whichone to use for a particular problem.

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