UNIVERSITY OF CALIFORNIA, SANTA BARBARA, DEPT. OF PHYSICS

HANDOUT PHYSICS 215B Winter 2014

The “Brillouin-Wigner” formulation of (time-independent) perturbation theory displays thestructure of perturbation theory in more systematic form than the more familiar “Rayleigh-Schrodinger” formulation. The latter is obtained straightforwardly from the former.

(A): Brillouin-Wigner Perturbation Theory: non-degenerate case

• Setup:H = H0 + λV ,

H0|m〉 = εm|m〉, (where 〈m|k〉 = δm,k)(1)

Pick a fixed eigenket |n〉 of the unperturbed Hamiltonian H0, and suppose that the corre-sponding non-interacting eigenvalue εn is non-degenerate. Let us denote the correspondingeigenstate |N〉 of the perturbed Hamiltonian H (with energy En), which evolves with λfrom the unperturbed ket |n >, by

|N〉 = |n〉+ λ|N (1)〉+ λ2|N (2)〉+ ... (2)

so thatH|N〉 =

(H0 + λV

)|N〉 = En|N〉 (3)

and

En = εn + λE(1)n + λ2E

(2)n + .... (4)

It will prove convenient to choose the following normalization of |N〉:

〈n|N〉 = 1 (for all λ), implying [using(1, 2)] : 〈n|N (k)〉 = 0, (k = 1, 2, 3, ...).(5)

The ket |N〉 will then not be normalized.

Let us rewrite (3) as (En − H0

)|N〉 = λV |N〉. (6)

Using the projection operators

DEF : Qn := 1− |n〉〈n|, Pn := |n〉〈n|and the ”resolvent”

DEF : G := Qn

(En − H0

)−1=∑m

′(En − εm)−1 |m〉〈m|,

(7)

all of which commute with H0,

[H0, Qn] = [H0, Pn] = [H0, G] = 0,

Eq. (6) implies:

Qn|N〉 = λGV |N〉, (8)

Adding Pn|N〉 = |n〉〈n|N〉 = |n〉 [which follows from Eq.s(2,5)] to both sides of Eq.(8)yields

|N〉 = Pn|N〉+ Qn|N〉 = Pn|N〉+ λGV |N〉 = |n〉+ λGV |N〉which can be rewritten as

[1− λGV ]|N〉 = |n〉or equivalently

•RESULT(STATES) : |N〉 ={1− [GλV ]

}−1|n〉 ={1 + [GλV ] + [GλV ]2 + ...

}|n〉,

(9)

When written out explicitly in the basis of unperturbed states {|m〉}, this reads

|N〉 = |n〉+

(∑m1

′(En − εm1)

−1 |m1〉〈m1|λV

)|n〉+

+

(∑m1

′(En − εm1)

−1 |m1〉〈m1|λV

)(∑m2

′(En − εm2)

−1 |m2〉〈m2|λV

)|n〉+ ...

where the superscript ‘prime’ on the sum means m 6= n. Here the ”resolvent” G definedin Eq.(7) still contains the “full” (exact, and unknown) energies En [see Eq. (4)], which,in turn, can be be found order by order in perturbation theory as follows:

Taking the inner product of both sides of Eq.(6) with |n〉, we obtain

•RESULT(ENERGIES) : En = εn + 〈n|λV |N〉,

(En − εn) = 〈n|λV |N〉 = 〈n|λV{1 + [GλV ] + [GλV ]2 + ...

}|n〉

(10)

where the last equation follows from (9). Expanding out the intrinsic dependence of Gon λ (appearing in the denominators via En) the first few terms turn out to be [in thenotation of (4)]

E(1)n = 〈n|V |n〉,

E(2)n = 〈n|V gV |n〉,

E(3)n = 〈n|V gV gV |n〉 − 〈n|V |n〉〈n|V g2V |n〉,

E(4)n = 〈n|V gV gV gV |n〉 − 〈n|V gV |n〉〈n|V g2V |n〉+

+ 〈n|V |n〉〈n|V |n〉〈n|V g3V |n〉 − 〈n|V |n〉〈n|(V gV g2V + V g2V gV

)|n〉

(11)

2

etc. ..., where

g ≡ Qn(εn − H0

)−1(12)

with Qn defined in (7) above is the expansion of G from Eq.(7) to lowest (zeroth) order inλ.

• RAYLEIGH-SCHRODINGER PERTURBATION THEORY:

The more familiar form of perturbation theory, called “Rayleigh-Schrodinger” perturbationtheory, is obtained from the above “Brillouin-Wigner” perturbation theory by expandingthe intrinsic dependence of G on λ in Eq.(9) upon making use of Eq.s (4,11).

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(B): Brillouin-Wigner Perturbation Theory: degenerate case

• Setup:H = H0 + λV ,

H0|m〉 = εm|m〉, (where 〈m|k〉 = δm,k)(13)

Pick a fixed eigenket |n〉 of the unperturbed Hamiltonian H0, and suppose that the cor-responding non-interacting eigenvalue εn is g-fold degenerate. There is thus a set of gorthonormal eigenkets |n; a〉, corresponding to the same eigenvalue εn,

H0|n; a〉 = εn|n; a〉, (a = 1, ...g). (14)

In the following, we will describe a procedure [see Eq. (24), below] which specifies a (ingeneral different) set of g orthonormal kets |n;α〉,

|n;α〉 =

g∑a=1

|n; a〉 〈n; a|n;α〉, (α = 1, ...g) (15)

where Ua,α := 〈n; a|n;α〉 is a unitary g × g-matrix.

In analogy to the non-degenerate case, let us denote the eigenstate of the perturbed Hamil-tonian H (with energy En,α), which evolves with λ from the unperturbed ket |n;α〉, by

|N ;α〉 = |n;α〉+ λ|N (1);α〉+ λ2|N (2);α〉+ ... (16)

so thatH|N ;α〉 =

(H0 + λV

)|N ;α〉 = En,α|N ;α〉 (17)

and

En,α = εn + λE(1)n,α + λ2E

(2)n,α + .... (18)

It will again prove convenient to choose the following normalization of |N ;α〉:

〈n;α|N ;α〉 = 1 (for all λ), implying [using(15, 16)] : 〈n;α|N (k);α〉 = 0, (k = 1, 2, 3, ...)(19)

The ket |N ;α〉 will then again not be normalized.

Let us rewrite (17) as (En,α − H0

)|N ;α〉 = λV |N ;α〉 (20)

Using the projection operators

DEF : Qn := 1−g∑a=1

|n; a〉〈n; a|, Pn :=

g∑a=1

|n; a〉〈n; a|

and the ”resolvent”

DEF : G := Qn

(En,α − H0

)−1=∑m

′ (En,α − εm

)−1 |m〉〈m|

(21)

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(the ’prime’ on the sum indicates omission of all the kets |n; a〉 with a = 1, 2, ..., g from the

sum), all of which commute with H0,

[H0, Qn] = [H0, Pn] = [H0, G] = 0,

Eq. (20) implies:

Qn|N ;α〉 = λGV |N ;α〉. (22)

Adding Pn|N ;α〉 = |n;α〉〈n;α|N ;α〉 = |n;α〉 [which follows from Eq.s (16,19)] to bothsides of Eq.(22) yields

|N ;α〉 = Pn|N ;α〉+ Qn|N ;α〉 = Pn|N ;α〉+ λGV |N ;α〉 = |n;α〉+ λGV |N ;α〉

which can be rewritten as[1− λGV ]|N ;α〉 = |n;α〉

or equivalently

•RESULT(STATES) :

|N ;α〉 ={1− [GλV ]

}−1|n;α〉 ={1 + [GλV ] + [GλV ]2 + ...

}|n;α〉. (23)

In order to obtain an expression for the energies En,α we start from Eq. (20), act on both

sides from the left with Pn, and make use of Eq.(23)

(En,α − H0)Pn|N ;α〉 = λPnV |N ;α〉 = λPnV{1− [GλV ]

}−1|n;α〉.

Using again [see below Eq.(22)]

Pn|N ;α〉 = |n;α〉〈n;α|N ;α〉 = |n;α〉

we obtain upon taking the inner product with |n;β〉

(En,α − εn)〈n;β|n;α〉 = λ〈n;β|V{1− [GλV ]

}−1|n;α〉.

In other words

•RESULT(ENERGIES) : (En,α−εn)δα,β = λ〈n;β|V{1−[GλV ]

}−1|n;α〉,

i.e. the kets |n;α〉 are chosen [see Eq. (15)] so as to diagonalize the g × g-matrix

〈n, b|V{1− [GλV ]

}−1|n; a〉 = 〈n, b|[λV + λV (GλV ) + ...]|n, a〉, (a, b = 1, ..., g). (24)

The resulting eigenvalues of this g × g-matrix are (En,α − εn).

To be specific: To first order in λ we need to diagonalize the g × g- matrix

λ〈n; b|V |n; a〉, (a, b = 1, 2, ..., g)

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while to second order in λ we have to diagonalize the g × g-matrix

λ〈n; b|V |n; a〉+ λ2〈n; b|V GV |n; a〉+ ... =

= λ〈n; b|V |n; a〉+ λ2〈n; b|V gV |n; a〉+O(λ3), (a, b = 1, 2, ..., g),

in order to obtain the energies En,α and the corresponding eigenvectors |n;α〉 = Pn|N ;α〉to the respective order. - Here

g ≡ Qn(εn − H0

)−1

is the expansion of G, defined in Eq.(21), to lowest (zeroth) order in λ [compare alsoEq.s(7,12)].

The full eigenkets |N ;α〉 (as opposed to their projections Pn|N ;α〉) are then given by Eq.(23).

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Ref.: Lectures on Quantum Mechanics, by: Gordon Baym (Benjamin/Cummings Publish-ing Comp., Reading (1969))

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