Perturbation Theory
Lecture 4
Transition probability:
Transition probability from initial unperturbed to
Final unperturbed state will be
---------(1)
where, we used Dyson series
And
---------(2)
---------(3)
Transition frequency
Transition probability in terms of expansion
coefficients will be
---------(4)
---------(5)
First order transition probability will be
---------(6)
Transition probability for constant perturbation
Here V(t) does not depend upon t and thus
---------(7)
---------(8)
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
When
, central peak becomes narrow and
high like delta function .
Using
---------(9)
We write
---------(10)
Using
We have
---------(11)
Transition rate i.e. Transition probability per
unit time will be
---------(12)
Transition probability into continuum of final states
Energy density of final states
Total transition rate
--------(13)
which is Fermi-Golden rule.
Transition probability for harmonic perturbation
Perturbation depend harmonically on time
---------(14)
Transition probability
-------(15)
Neglecting crossed terms
-----------(16)
Using
, we get
---------(17)
In the limit
For non-zero transition rate
For continuum case
For hermitian perturbation
We have
Which is detailed balancing equation.
Problem: