08perturbation - Perturbation Theory Only a few QM systems...

P460 - perturbation 1
Perturbation Theory
• Only a few QM systems that can be solved exactly:
Hydrogen Atom(simplified), harmonic oscillator,
infinite+finite well
• solve using perturbation theory which starts from a
known solution and makes successive approx-
imations
• start with time independent. V’(x)=V(x)+v(x)
• V(x) has solutions to the S.E. and so known
eigenvalues and eigenfunctions
• let perturbation v(x) be small compared to V(x)
'
) (
/
V of solutions a
x V of ions eigenfunct
l
l
nl n
l
⇒ =
⇒
∑
ψ ψ
ψ
As ψ
l
form complete set of states (linear algebra)
Sometimes Einstein convention used. Implied sum
if 2 of same index

P460 - perturbation 2
Plug into Schrod. Eq.
• know solutions for V
• use orthogonality
• multiply each side by wave function* and integrate
• matrix element of potential v is defined:
∑ ∑ ∑
= + +
= + +
−
−
l nl n l nl l nl
mdx
d
n n n
mdx
d
a E v a a V
E v V
n
ψ ψ ψ
ψ ψ
ψ
/
2
/ / /
2
] [
) (
2
2 2
2
/ 2 2
h
h
∑ ∑ ∑
= +
l nl n l nl l l nl
a E v a E a ψ ψ ψ
/
ml l m
dx δ ψ ψ =
∫
*
) (
/ *
m n nm l m nl
E E a dx v a − =
∑
∫
ψ ψ
dx v v
l m ml
ψ ψ
∫
=
*

P460 - perturbation 3