Quantum Mechanics Perturbation

First Order Non-Degenerate Perturbation The-
ory
En1 = ⟨ψn0 |W |ψn0 ⟩
- A term must vanish
Formula: Finding the first order correction to the wave function
X
ψn1 = 1
Cmn 0
ψm + ...
m̸=n

Knowing that:
ψn = ψn0 + λψn1 + λ2 ψn2 + . . .
And that:
H0 ψn0 + W ψn0 = En0 ψn0 + En1 ψn0
Collecting similar terms:
(H0 − En0 )ψn1 = En1 ψn0 + W ψn0


(H0 − En0 )ψn1 = −(W − En1 )ψn0
Substituting:
X
(H0 − En0 ) 1
Cmn 0
ψm = −(W − En1 )ψn0
m̸=n

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We are solving for the coefficient Cmn .
⇒ We can get rid of the sums by using the Kronecker delta δ:
⇒ We can if we use ⟨ψn0 |ψm 0
⟩ = δnm .
Now multiplying both sides by ⟨ψl0 |:
X
⟨ψl0 |(H0 − En0 ) 1
Cmn ψm 0
⟩ = −⟨ψl0 |(W − En1 )ψn0 ⟩
m̸=n

Simplifying:
X
⟨ψl0 |(El0 − En0 ) 1
Cmn 0
ψm ⟩ = −⟨ψl0 |(W − En1 )ψn0 ⟩
m̸=n

Using orthogonality:
X
(El0 − En0 ) 1
Cmn ⟨ψl0 |ψm
0
⟩ = −⟨ψl0 |(W − En1 )ψn0 ⟩
m̸=n

Since ⟨ψl0 |ψm
0
⟩ = δlm , we get:
(El0 − En0 )Cmn
1
δlm = −⟨ψl0 |(W − En1 )ψn0 ⟩
P
When l = m, we have δlm = 1.
Therefore:
0
(Em − En0 )Cmn
1
= −⟨ψm 0
|W |ψn0 ⟩


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