PERTURBATION THEORY
Time Independent Theory
A Necessity:
Exactly solvable problems are very few and do not
really represent a real physical system completely.
In many cases, the primary interaction on the system
maybe exactly solvable and is hence the dominant
contribution.
We develop a theory which takes the exact solution as a
base to build on, for more accurate description of the
system.
Recipe of the Theory
Take the Hamiltonian of an exactly solvable problem.
Add a perturbative term to the original Hamiltonian,
satisfying the condition,
The terms in the perturbation need to be smaller than
the energy differences of the original Hamiltonian.
Introduce a parameter l as the co-efficient of the
perturbative term in the Hamiltonian, to observe the
effect of the Hamiltonian.
Eigen Value changes with perturbation intensity
Observation: Start with a
negligible value of l such
that the perturbation is
very minuscule to start
with. This causes the
change in the Eigen-values
and functions to vary
slightly from the original
Hamiltonian. Smoothly
varying the parameter l
smoothly varies the new
solution as well.
E40
E30
E20
E10
l1
l
Taylor Expansion
The implication is, the existence of a continuous
Eigen-functions and Eigen-values:
 l  and E l  for an
Hamiltonian parameterized by l as
i
i
ˆ
ˆ
ˆ
H  H 0  lW



Hence for small values of perturbation, the solutions
can be expanded in Taylor series around the known
solution with l = 0, as       l     l     l     ...
0
i
1
i
i
0 
1 
2
2
3
3
i
 2
i
 3
E i  E i  lE i  l E i  l E i
for the eigen-value equation: Hˆ 
i

The original equation: Hˆ

0
 E i i
 i  E i  i
2
3
 ...