CHAPTER 5: STATIONARY PERTURBATION THEORY

(From Cohen-Tannoudji, Chapter XI)

A. DESCRIPTION OF THE METHOD

Approximation methods to obtain analytical solution of eigenvalue problems.

1. Statement of the problem

We consider a time-independent perturbation

Ĥ = Ĥ0 + W (5.1)
of the time-independent Hamiltonian Ĥ0, whose eigenvalues and eigenvectors are
known and which captures the essential physics, by an additional term

W = λŴ (5.2)
λ  1 (5.3)
We assume that the eigenvalues and eigenstates of Ĥ0 are known and that the un-
perturbed energies form a discrete spectrum E 0p with eigenvectors |ϕipi (where the
index i refers to degeneracy):

Ĥ0|ϕipi = E 0p|ϕipi (5.4)

where
0
hϕip|ϕip0 i = δ pp0 δii0 (5.5)
XX
|ϕipihϕip| = 1̂ (5.6)
p i

i.e. the states |ϕipi form a basis.

We seek an approximative solution of the full Hamiltonian Ĥ(λ) = Ĥ0 + λŴ

Ĥ(λ)|ψ(λ)i = E(λ)|ψ(λ)i (5.7)

where the eigenvalue and eigenvector can be expanded in terms of λ

E(λ) = ε0 + λε1 + . . . + λqεq + . . . (5.8)

|ψ(λ)i = |0i + λ|1i + . . . + λq|qi + . . . (5.9)

Inserting these into the eigenvalue equation yields

    
 X∞  ∞
 X  X∞ 
 0
Ĥ0 + λŴ  λq|qi =  λq εq0   λq|qi (5.10)
    
q =0
   0  
q=0 q=0
As this equation must hold for any (small) value of λ, it must hold for each power of
λ separately, giving the equations for various orders of the perturbation:
0th-order: is just the eigenvalue equation of the unperturbed Hamiltonian, ε0 = En0

Ĥ0|0i = ε0|0i (5.11)

1st order:
   
Ĥ0 − ε0 |1i + Ŵ − ε1 |0i = 0 (5.12)

2nd order
   
Ĥ0 − ε0 |2i + Ŵ − ε1 |1i − ε2|0i = 0 (5.13)

q-th order
   
Ĥ0 − ε0 |qi + Ŵ − ε1 |q − 1i − ε2|q − 2i . . . − εq|0i = 0 (5.14)
We shall write |ψ(λ)i to be normalized and its phase will be chosen s.t. h0|ψ(λ)i ∈ R.
For 0th order we have

h0|0i = 1 (5.15)

and to the 1st order we get

 
hψ(λ)|ψ(λ)i = [h0| + λh1|] [|0i + λ|1i] + O λ2 (5.16)
 
= h0|0i + λ [h1|0i + h0|1i] + O λ2 (5.17)

Since both h0|0i = 1 and hψ(λ)|ψ(λ)i = 1 we get to the 1st order

λ [h1|0i + h0|1i] = 0
⇒ h0|1i = h1|0i = 0 (5.18)
For the 2nd order we get

hψ(λ)|ψ(λ)i = h0|0i + λ [h1|0i + h0|1i]

 
+λ2 [h2|0i + h0|2i + h1|1i] + O λ3 (5.19)

1
⇒ h0|2i = h2|0i = − h1|1i (5.20)
2

and eventually for q-th order we have

1
h0|qi = hq|0i = − hq − 1|1i + hq − 2|2i + . . . + h2|q − 2i + h1|q − 1i (5.21)

2
B. PERTURBATION OF A NON-DEGENERATE LEVEL

We will try to answer how a nondegenerate eigenvalue and eigenvector of the unper-
turbed Hamiltonian Ĥ0

ε0 = En0
|0i = |φni
are modified by introducing the perturbation W .

We will be seeking the eigenvalue En(λ) of the full Hamiltonian Ĥ(λ) which when
λ → 0 approaches En0 of Ĥ0.

We will assume that λ is small enough for this eigenvalue to remain non-degenerate.
1. First-order corrections

a. ENERGY CORRECTION
Taking the 1st order equation we found above, and projecting onto |ϕni gives
   
hϕn| Ĥ0 − ε0 |1i + hϕn| Ŵ − ε1 |0i = 0 (5.22)
and since |ϕni = |0i is the eigenvector of Ĥ0 with the eigenvalue ε0 = En0, we obtain
the first order correction to the energy
ε1 = hϕn|Ŵ|0i = hϕn|Ŵ|ϕni (5.23)
and the 1st order perturbative expression for the energy eigenvalue of the perturbed
system in the form (recall, W = λŴ )

En(λ) = En0 + hϕn|W|ϕni + O(λ2)

b. EIGENVECTOR CORRECTION

To find the first-order correction to the eigenvector, we must project the first-order
n o
i
equation above onto all the vectors of the |ϕ pi basis other than |ϕni
   
hϕip| Ĥ0 − En0 |1i + hϕip| Ŵ − ε1 |ϕni = 0 (p , n) (5.24)

Since the eigenvectors of Ĥ0 associated with different eigenvalues are orthogonal
ε1hϕip|ϕni = 0 and hϕip|Ĥ0 = hϕip|E 0p, we get
 
E 0p − En0 hϕip|1i + hϕip|Ŵ|ϕni = 0 (5.25)

and
1
hϕip|1i = hϕ
0 p
i |Ŵ|ϕ i
n (p , n) (5.26)
En0 − E p
and, since hϕn|1i = h0|1i = 0, the first order correction to the eigenvector can be
written as
X X hϕip|Ŵ|ϕni
|1i = |ϕ ii (5.27)
p
p,n i En0 − E 0p

The expression for the eigenvector of the perturbed Hamiltonian to the first-order is
thus
X X hϕip|W|ϕni  
|ψn(λ)i = |ϕni + |ϕ i i + O λ2 (5.28)
p
p,n i En0 − E 0p

The perturbation W mixes the state |φni with the other eigenstates of Ĥ0.
2. Second-order corrections

a. ENERGY CORRECTION

We proceed in a way similar to the previous case. We project the 2nd order equation
obtained above onto |ϕni
   
hϕn| Ĥ0 − En0 |2i + hϕn| Ŵ − ε1 |1i − ε2hϕn|ϕni = 0 (5.29)

Since |ϕni = |0i is the eigenvector of Ĥ0 with the eigenvalue ε0 = En0, the first term is
zero and the second order correction becomes

ε2 = hϕn|Ŵ|1i (5.30)
With the expression for |1i obtained above we can write the second order correction
to the energy eigenvalue as

X X hϕip|Ŵ|ϕni 2
ε2 = 0 − E0
(5.31)
p,n i E n p
The 2nd order expression for the energy eigenvalue of the perturbed system be-
comes
X X hϕip|W|ϕni 2
En(λ) = En0 + hϕn|W|ϕni + + O(λ3) (5.32)
p,n i En0 − E 0p
b. EIGENVECTOR CORRECTION

The eigenvector corrections |2i can be obtained by projecting the equation

   
Ĥ0 − ε0 |2i + Ŵ − ε1 |1i − ε2|0i = 0 (5.33)

onto the set of basis vectors |φipi different from |φni and by using the condition

1
h0|2i = h2|0i = − h1|1i (5.34)
2
c. UPPER LIMIT OF ε2

What is the error involved in the 1st order perturbation theory?

Consider
X X hϕip|Ŵ|ϕni 2
ε2 = (5.35)
p,n i En0 − E 0p
and let the absolute value of difference of En0 being studied and that of the nearest
level E 0p be

En0 − E 0p ≥ ∆E (5.36)
then an upper limit for the absolute value of 2 is
1 XX i 2
|ε2| ≤ hϕ p |Ŵ|ϕ n i (5.37)
∆E p,n
i
 1 XX i 2
|ε2| ≤ hϕ p|Ŵ|ϕni (5.38)
∆E p,n
i
1 X X
= hϕn|Ŵ|ϕipihϕip|Ŵ|ϕni (5.39)
∆E p,n
i
 
1  X X
i i

= hϕn|Ŵ   |ϕ pihϕ p| Ŵ|ϕni (5.40)
∆E p,n i

Taking into account the completeness relation

XX
|ϕnihϕn| + |ϕipihϕip| = 1̂ (5.41)
p,n i
allows us to rewrite the inequality as
1 h i
|ε2| ≤ hϕn|Ŵ 1̂ − |ϕnihϕn| Ŵ|ϕni (5.42)
∆E 
1 2
 2  1  2
≤ hϕn|Ŵ |ϕni − hϕn|Ŵ|ϕni = ∆Ŵ (5.43)
∆E ∆E

An upper limit for the 2nd order term in En(λ) = 0 + λ1 + λ22 + . . . is then
1
λ2ε2 ≤ (∆W)2 (5.44)
∆E

This indicates the order of magnitude of the error committed by taking only the 1st
order correction into account.
C. PERTURBATION OF A DEGENERATE STATE

Assume that the level En0 to be gn-fold degenerate, and E0n be the corresponding
gn-fold dimensional eigenspace of Ĥ0.

Now the choice

ε0 = En0 (5.45)

is not sufficient to determine |0i since the equation Ĥ0|0i = 0|0ican be satisfied by
any linear combination of vectors in E0n.
To determine |0i and 1 we project the 1st order equation
   
Ĥ0 − ε0 |1i + Ŵ − ε1 |0i = 0 (5.46)

onto the gn basis vectors |φini:

since Ĥ0|φini = 0|φini for all |φini we obtain the gn relations:

hϕin|Ŵ|0i = ε1hϕin|0i (5.47)

Now using the completeness relation

XX 0 0
hϕn|Ŵ|ϕ p ihϕ p |0i = ε1hϕin|0i
i i i (5.48)
p i0
0
where hϕ p |0i = 0 for all p , n.
i
Taking into account only the terms where p = n we get
gn
X 0 0
hϕin|Ŵ|ϕin ihϕin |0i = ε1hϕin|0i (5.49)
i0 =1
where the gn × gn matrix
0
Ŵ (n) = hϕin|Ŵ|ϕin i (5.50)

is the restriction of Ŵ to the eigenspace E0n.

We can rewrite the equation
gn
X 0 0
hϕin|Ŵ|ϕin ihϕin |0i = ε1hϕin|0i (5.51)
i0 =1
into a vector equation

Ŵ (n)|0i = ε1|0i (5.52)

To calculate the eigenvalues (to the 1st order) and the eigenstates (to the 0th or-
der) of the Hamiltonian corresponding to a degenerate unperturbed state En0, we
have to diagonalize the matrix W (n), which represents the perturbation Ŵ inside the
eigenspace E0n associated with the eigenvalue En0.
The first order effect of the perturbation is given by the various roots of the charac-
j (1)
teristic equation of W (n), 1( j = 1, 2, . . . , fn ).

Since W (n) is self-adjoint, its eigenvalues are real numbers and their degrees of
degeneracy sum to gn.

Each eigenvalue introduces a different energy correction, i.e. under W = λŴ , the
(1)
degenerate levels split to the 1st order into fn distinct sublevels
j (1)
En, j(λ) = En0 + λε1 j = 1, 2, . . . , fn ≤ gn (5.53)
 j
We shall now choose an eigenvalue 1 of W (n):

- if it is non-degenerate, the corresponding vector |0i is completely determined, i.e.

there exists a single eigenvalue E(λ) of Ĥ(λ) which is
j
En0 + λε1 (5.54)

and is non-degenerate;

- if it is q-fold degenerate, the equation

W (n)|0i = 1|0i (5.55)

(1)
indicates only that |0i belongs to the corresponding q-dimensional subspace F j .