1. Perturbation Theory
Lecture 3

2. Interaction picture: Used when Hamiltonian is
function of t.
Here state vectors and operators both evolve with t.
We write
-----(1)
State vector in
Interaction picture
state vector in
Schr. picture

3. Differentiating (i) w.r.t t
Using and
in (2), we get...
-----(2)
-----(3)

4. -----(4)
which is S.E. In interaction picture.

5. Equation of motion for operator :
Operator in interaction picture is related to operator
In Schrödinger picture through
Eq. Of motion will be
-----(5)
-----(6)

6. Time dependent perturbation theory
Hamiltonian is split into two parts: time
independent part and time dependent part
----(1)
Exactly solvable part satisfy
Sol is:
----(2)
----(3)

7. System is subjected to perturbation
And Schrodinger eq will be
When system interact with ,system emit or
absorb the energy and make transition from
one unperturbed state to other.
----(4)
----(5)

8. ----(5) Time evolution eq In I.P. ----(6) ----(7) ----(8)
We solve (5) in interaction picture
----(5)
Time evolution eq
In I.P.
----(6)
----(7)
----(8)

9. Using (7) in (5), we get
Solution of above eq. will be
Above eq.is solved iteratively.
First order approx. Using in (10),
------(11)
----(9)
Initial condition
----(10)

10. Using in (11), we get
------(12)
which is 2nd app.
Repeating above process, we will get
Above series is known as Dyson series used
to calculate state vectors in perturbation theory.