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 First order transition probability will be ---------(4) ---------(5) ---------(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 We write ---------(9) ---------(10)

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