QUANTUM MECHANICS A (SPA-5319)
Time-Independent Perturbation Theory
It is often the case that nature will provide us with systems where an exact analytical solution
is impossible to obtain, or where an exact analytical solution is difficult to obtain. The
physicist’s answer to such a situation is often to attempt some sort of approximate method
in order to reach some approximate solutions. A very common method for obtaining the
allowed energies for some new situations (potentials) where these resemble an already
known and solved system is Time-Independent Perturbation Theory.
Suppose we have solved the time-independent Schrödinger equation for a given potential:
̂ 𝑜 𝛹𝑛𝑜 = 𝐸𝑛𝑜 𝛹𝑛𝑜
𝐻
And have obtained the set of orthogonal eigenfunctions 𝛹𝑛𝑜 . Consider, for example, the
𝐿
𝐿
2
2
infinite square well with 𝑉 = 0 for − ≤ 𝑥 ≤ , and 𝑉 = ∞ elsewhere. This gives solutions:
2
𝛹𝑛𝑜 = √𝐿 cos(
2
= √𝐿 sin(
and allowed energies 𝐸𝑛 =
𝑛𝜋𝑥
𝑛𝜋𝑥
𝐿
ℏ2 𝜋 2 𝑛 2
2𝑚𝐿2
)
for even parity, n odd
)
for odd parity, n even
𝐿
. If we now perturb (or change) the potential slightly we
would like to solve for our new TISE:
̂ Ψ𝑛 = 𝐸𝑛 Ψ𝑛
𝐻
An example of this can be a well with a “step” at the bottom. In general, we expect this to be
mathematically complicated and we may not be able to solve exactly for this new potential.
Perturbation theory allows us to obtain approximate solutions starting with the unperturbed
case (see figure 1).
V0
-L/2
0
=
+
-L/2
L/2
V0
L/2
0
L/2
Figure 1: a perturbed well with a bump can be viewed as the original well plus a perturbation.
The first step is to write the perturbed Hamiltonian as the unperturbed Hamiltonian plus a
perturbation:
̂=𝐻
̂ 𝑜 + 𝜆𝐻
̂1 ,
𝐻
where 𝜆 is some constant.
We can use a similar approach to write the energies and wavefunctions as the original
unperturbed terms plus a first order correction (), followed by a second order correction ()
and so on:
𝛹𝑛 = 𝛹𝑛𝑜 + 𝜆𝛹𝑛1 +𝜆2 𝛹𝑛2 + ⋯
𝐸𝑛 = 𝐸𝑛𝑜 + 𝜆𝐸𝑛1 +𝜆2 𝐸𝑛2 + ⋯
Substituting this into the time-independent Schrödinger equation, collecting like-powers of
 and letting 𝜆 = 1 yields a very important result1:
𝐸𝑛1
∞
̂1 𝛹𝑛𝑜 𝑑𝑥
= ∫ 𝛹𝑛𝑜∗ 𝐻
−∞
That is, the first order energy correction is simply the expectation value of the perturbation
on the unperturbed wavefunction.
1
We can only do this because the Hamiltonian is Hermitian: ∫ 𝑥𝐴̂𝑦 = ∫ 𝐴̂𝑥𝑦
Consider now the perturbed infinite potential well shown in figure 1. Let’s calculate the firstorder energy correction using the even parity (n odd) solutions.
2
Ψ𝑛 = √𝐿 cos(
𝑛𝜋𝑥
𝐿
)
𝐿
𝐿
for − 2 ≤ 𝑥 ≤ 2 ,
= 0 elsewhere
The perturbation is given by:
̂
𝐻1 = 𝑉0
𝐿
0≤𝑥≤2 ,
for
= 0 elsewhere
Using the result for the correction, derived above, we find
𝐸𝑛1
𝐿
∞
=
̂1 𝛹𝑛𝑜 𝑑𝑥
∫ 𝛹𝑛𝑜∗ 𝐻
−∞
This yields the result 𝐸𝑛1 =
𝑉0
2
2 2
𝑛𝜋𝑥
𝑛𝜋𝑥
= ∫ cos(
) 𝑉0 cos(
) 𝑑𝑥
𝐿 0
𝐿
𝐿
. So 𝐸𝑛 = 𝐸𝑛𝑜 +
𝑉0
2
is the approximate result we expect. The
bump raises all the allowed energy levels by a constant value (= V0/2).
The question immediately arises: Can we do the same for the wavefunction? i.e. just as we
wrote 𝐸𝑛 = 𝐸𝑛𝑜 + 𝐸𝑛1 , can we do the same for the wavefunction and write: 𝛹𝑛 = 𝛹𝑛𝑜 + 𝛹𝑛1 ?
If so, what is 𝛹𝑛1 (the first-order correction to the unperturbed wavefunction)?
In principle we can, but generally speaking perturbation theory provides accurate energy
results but very poor approximations for wavefunctions.