is turned on at t_i = -infinity and left on until t_f = infinity. Here H' is independent of time, and alpha is a constant. Show that at t_f = infinity, the probability that the system has evolved into the eigenstate | psi_f > with energy E_f is

If we denote the time dependent wave function of the system by |psi(t)> and the eigenstates with energy E_r by |r>, then we can write:

|psi> = sum over r of c_r (t) exp(-E_r t/hbar) |r>

The coefficients c_r(t) would not depend on time if there were no time dependent perturbation. If the system is initially in the state |i>, then that means that initially c_i = 1 and all other coefficients are zero. If we compute the coefficients in perturbation theory, we find ...

Solution Summary

We consider a time dependent perturbation of the form

V(t) = H'exp[-((alpha)^2)(t^2)]

and compute to first order in perturbation theory the transition probability for an eigenstate with energy E_1 at
t = -infinity to evolve to an eigenstate with energy E_2 at t = infinity.

Quantum Mechanics: Time dependent perturbation problem. ... explain how, using first order time dependent perturbation theory, one can ... to all orders in perturbation...

... exploration, is that the situations an autonomous system will en- first one integrates ... They can be ordered ac- Maslow's postulated order for emergence ...

... In this problem we have: ... then first order perturbation theory yields that ... We apply time dependent perturbation theory to compute the probability of a spin flip ...

... of each component of the CMG index, in descending order. ... depth, and US foreign holdings) Our first measure of ... market capitalization.9 We run five panel time-. ...

... to be minimized is continuous in its first derivative. ... calculated since calculation of second order information, using ... then apply the method one time for each ...

... the development toward a spatially ordered is not ... Since autonomous systems are First, an informational record can ... robust thanks to their fair order to create ...

... and why it's even important in the first place. ... 2 dry-land performance measures, swim time did not im ... of functions that the body will require in order for the ...