when the energy levels are about to be degenerate

Second order perturbation: To calculate correction in second order, we will make use of λ² equation (11),

( ˆH₀−E_n⁽⁰⁾)ψ⁽²⁾_n = −( ˆH⁰−E_n⁽¹⁾)ψ⁽¹⁾_n +E_n⁽²⁾ψ⁽⁰⁾_n (19) and go straight for expansion of ψ⁽²⁾n andψn⁽¹⁾ in terms of unperturbed wavefunction ψn⁽⁰⁾,

ψ⁽¹⁾_n = X

m

c⁽¹⁾_m ψ_m⁽⁰⁾ and ψ_n⁽²⁾ = X

m

c⁽²⁾_m ψ⁽⁰⁾_m . (20) Substituting the above expression forψ⁽²⁾_n and ψ_n⁽¹⁾ in (19), we obtain,

( ˆH0−E_n⁽⁰⁾)X

m

c⁽²⁾_m ψ_m⁽⁰⁾ = −( ˆH⁰−E_n⁽¹⁾)X

m

c⁽¹⁾_m ψ_m⁽⁰⁾+E_n⁽²⁾ψ_n⁽⁰⁾ X

m

(E_m⁽⁰⁾−E_n⁽⁰⁾)c⁽²⁾_m ψ_m⁽⁰⁾ = −X

m

( ˆH⁰−E_n⁽¹⁾)c⁽¹⁾_m ψ_m⁽⁰⁾+E_n⁽²⁾ψ_n⁽⁰⁾. As before, we take inner product with the unperturbed wavefunctionψ_k⁽⁰⁾,

X

m

(E_m⁽⁰⁾−E_n⁽⁰⁾)c⁽²⁾_m

ψ⁽⁰⁾_k , ψ_m⁽⁰⁾

=

−X

m

ψ_k⁽⁰⁾,Hˆ⁰ψ_m⁽⁰⁾

c⁽¹⁾_m +X

m

E_n⁽¹⁾c⁽¹⁾_m

ψ⁽⁰⁾_k , ψ⁽⁰⁾_m

+E_n⁽²⁾

ψ_k⁽⁰⁾, ψ⁽⁰⁾_n

or, X

m

(E⁽⁰⁾_m −E⁽⁰⁾_n )c⁽²⁾_m δ_km=−X

m

c⁽¹⁾_m Hˆ_km⁰ +X

m

E_n⁽¹⁾c⁽¹⁾_m δ_km+E_n⁽²⁾δ_kn or, (E_k⁽⁰⁾−E_n⁽⁰⁾)c⁽²⁾_k =−X

m

c⁽¹⁾_m Hˆ_km⁰ +E_n⁽¹⁾c⁽¹⁾_k +E_n⁽²⁾δ_kn. (21) Fork=nwe get the second order correction to energy, keeping in mind that c⁽¹⁾_n = 0,

E_n⁽²⁾ = X

m6=n

c⁽¹⁾_m Hˆ_nm⁰ = X

m6=n

Hˆ_mn⁰ Hˆ_nm⁰ En⁽⁰⁾−Em⁽⁰⁾

= X

m6=n

|Hˆ_mn⁰ |² E⁽⁰⁾n −E⁽⁰⁾m

. (22)

Fork6=n, we will getc⁽²⁾_k knowingEn⁽¹⁾ (13) andc⁽¹⁾m (17), c⁽²⁾_k (E_k⁽⁰⁾−E_n⁽⁰⁾) = −X

m

c⁽¹⁾_m Hˆ_km⁰ +c⁽¹⁾_k Hˆ_nn⁰ or, c⁽²⁾_k (E_k⁽⁰⁾−E_n⁽⁰⁾) = −X

m6=n

Hˆ_mn⁰ Hˆ_km⁰ En⁽⁰⁾−Em⁽⁰⁾

+

Hˆ_kn⁰ Hˆ_nn⁰ En⁽⁰⁾−E_k⁽⁰⁾ or, c⁽²⁾_k = X

m6=n

Hˆ_km⁰ Hˆ_mn⁰

(En⁽⁰⁾−Em⁽⁰⁾)(En⁽⁰⁾−E_k⁽⁰⁾)

− Hˆ_kn⁰ Hˆ_nn⁰ (En⁽⁰⁾−E_k⁽⁰⁾)²

. (23)

Therefore, the energy E_n and wavefunctionψ_n of the full Hamiltonian (1) to second order in perturbation theory are,

E_n = E_n⁽⁰⁾+ ˆH_nn⁰ + X

m6=n

|Hˆ_mn⁰ |² En⁽⁰⁾−Em⁽⁰⁾

+· · · (24)

ψn = ψ_n⁽⁰⁾+ X

m6=n

"

Hˆ_mn⁰ E_n⁽⁰⁾−E_m⁽⁰⁾

# ψ_m⁽⁰⁾

+X

m6=n



 X

k6=n

Hˆ_mk⁰ Hˆ_kn⁰

(En⁽⁰⁾−E_k⁽⁰⁾)(En⁽⁰⁾−Em⁽⁰⁾)

− Hˆ_mn⁰ Hˆ_nn⁰ (En⁽⁰⁾−Em⁽⁰⁾)²



ψ_m⁽⁰⁾+· · · (25) 1

For perturbation theory to work, the corrections it produces must be small (not wildly different fromE_n⁽⁰⁾). But onward second order corrections in energy (24) and first order in wavefunction (25) contain the term that must be small,

Hˆ_mn⁰ En⁽⁰⁾−Em⁽⁰⁾

1, n6=m

otherwise it has potential to grow large if E_n⁽⁰⁾ ≈ E⁽⁰⁾_m , i.e. when the energy levels are about to be degenerate. Therefore, degenerate energy levels have to be treated differently in perturbation theory.

Examples

1. Using first order perturbation theory, calculate the energy of then-th state for a particle of mass m moving in an infinite potential well of length 2L with wall at x = 0 and x = 2L, which is modified at the bottom by the perturbations: (i) λV₀sin(πx/2L) and (ii) λV0δ(x−L), where λ1.

2. Calculate the energy of the n-th excited state to first order perturbation for a 1-dim infinite potential well of length 2L, with walls at x=−L and x =L, which is modified at the bottom by the following perturbations with V₀ 1,

Hˆ⁰ =

−V₀ −L≤x≤L

0 elsewhere Hˆ⁰ =

−V₀ −L/2≤x≤L/2 0 elsewhere

Hˆ⁰ =

−V₀ −L/2≤x≤0

0 elsewhere Hˆ⁰ =

V₀ 0≤x≤L/2 0 elsewhere

3. For a 1-dim harmonic oscillator, the spring constant changes from k tok(1 +), where is small. Calculate the first order perturbation in the energy.

4. Calculate the first order perturbation in the energy for n-th state of a 1-dim harmonic oscillator subjected to perturbationβ x⁴,β is a constant.

5. Consider a quantum charged 1-dim harmonic oscillator, of chargeq, placed in an electric field E~ = Ex. Find the exact expression for the energy and then use perturbation theoryˆ to calculate the same.

6. For the following set of Hamiltonians, withλ1,

(i) E₀

1 λ λ 3

(ii) E₀









1 +λ 0 0 0

0 8 0 0

0 0 3 −2λ

0 0 −2λ 7









(iii) E₀









−5 3λ 0 0

3λ 5 0 0

0 0 8 −λ

0 0 −λ −8









(iv) E₀









3 2λ 0 0

2λ −3 0 0

0 0 −7 √

2λ

0 0 √

2λ 7







 ,

(a) find the eigenvalues and eigenvectors of the unperturbed Hamiltonian, (b) diagonalize the full Hamiltonian to find the exact eigenvalues and expand each eigenvalue to the second power ofλand (c) using the first and second order perturbation theory find the approximate energy eigenvalues and eigenstates of the full Hamiltonian.

2