1. Perturbation Theory
Lecture 4

2. Transition probability:
Transition probability from initial unperturbed to
Final unperturbed state will be
(1)
where, we used Dyson series

3. And
(2)
(3)
Transition frequency

4. Transition probability in terms of expansion
coefficients will be
First order transition probability will be
(4)
(5)
(6)

5. Transition probability for constant perturbation
Here V(t) does not depend upon t and thus
(7)
(8)

6. Height peak goes like
t2 and width like t and
Thus area under peak and
Thus transition probability
Increases in proportion to t
Transition probability is maximum when

7. When , central peak becomes narrow and
high like delta function .
Using
We write
(9)
(10)

8. Using
We have
Transition rate i.e. Transition probability per
unit time will be
(11)
(12)

9. Transition probability into continuum of final states
Energy density of final states
Total transition rate
(13)
which is Fermi-Golden rule.

10. Transition probability for harmonic perturbation
Perturbation depend harmonically on time
(14)
Transition probability
(15)

11. Neglecting crossed terms
(16)
Using , we get
(17)

13. In the limit
For non-zero transition rate

15. For continuum case
For hermitian perturbation
We have
Which is detailed balancing equation.

16. Problem: