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.

An electron is in a strong, uniform, constant magnetic field with magnitude B_0 aligned in the +x direction. The electron is initially in the state |+> with x component of spin equal to hbar/2. A weak, uniform, constant magnetic field of B_1 (where B_1 << B_0) in the +z direction is turned on at t=0 and turned off at t=t_0. Let

A particle with mass m is in a one-dimensional infinite square-well potential of width a, so V(x)=0 for 0 <= x <= a, and there are infinite potential barriers at x=0 and x=a. Recall that the normalized solutions to the Schrodinger equation are
psi_n(x) = sqrt(2/a)sin[(n pi x)/a]
with energies
E_n = (hbar^2 (pi^2 n^2)/

I am trying to calculate the probability that the n = 3 state will be excited at t = infinity for the case where b<<< a. and to show that for any value of b only the odd n states will be excited at t = infinity.
My initial conditions and setup are:
V(x,t) = Vinit(x) exp(-lambda times t) where I have a negative potential

Consider an electron with spin magnetic moment u_s in a strong magnetic field B_z in the z direction. The potential for an electron with spin magnetic moment u_s in a magnetic field B is V=-u_s . B
where u_s = -((g_s)(u_B))/(hbar) . S
Thus the Hamiltonian is H_0 = ((g_S)(u_B))/(hbar) . B . S = ((g_s)/2)(u_B) . B . sigma

See attachment for better symbol representation.
When an atom is placed in a uniform external electric field Eext, the energy levels are shifted - a phenomenon known as the stark effect. In this problem we analyze the stark effect for the n=1 and n=2 states of hydrogen. Let the field point in the z direction, so the potent

Recall that a 1-D delta-function potential well of the form V (x) =
−B delta(x) had exactly one bound state, with a double-tailed exponential wave function.
(a) Apply a harmonic oscillator perturbation of the form V ′(x) = (m omega^2 x^2)/2. Calculate the
ground-state energy for this perturbed system to firstorder.
(b)

An electron is in a strong, static, homogenous magnetic field with magnitude B0 in the z direction. At time t=0, the spin of the electron is in the +z direction. At t=0 a weak, homogenous magnetic field with magnitude B1 (where B1 << B0) is turned on. At t=0 this field is pointing in the x direction, but it rotates counterclockw

In the hydrogen atom, the proton is not really a point charge but has a finite size. Assume that the proton behaves as a uniformly-charged sphere of radius R=10^(-15) m. Calculate the shift this produces in the ground-state energy of hydrogen.