 #1
Diracobama2181
 74
 2
 Homework Statement:

Consider a rigid rotator (i.e. a bar shaped system of fixed separation) of moment of inertia I about an axis through its center perpendicular to the direction of the bar, with Hamiltonian $$H_0 = \frac{L_2}{2I} and electric dipole moment d. Suppose that while it is in its ground state
it is subjected to a perturbation V (t) = −d · E(t) due to a timedependent external electric field
$$E(t) = zˆE0e^{t/τ}$$ which points in the zdirection and which is switched on at time t = 0.
Here E0 is a timeindependent constant. Determine to which of its
excited states the rotator can make transitions in lowest order in V (t) ,
and calculate the transition probabilities for finding the rotator in each
of these states at time t → ∞.
 Relevant Equations:
 $$d_f=\frac{i}{\hbar}\int_{0}^{T'} e^{iw_{fi}t}v_{ni} dx$$
Since E_i=0 for the ground state, and $$E_f=\frac{(\hbar)^2l(l+1)}{2I}$$, $$w_{fi}=\frac{E_fE_i}{\hbar}=\frac{(\hbar)l(l+1)}{2I}$$.
So, $$d_f(\infty)=\frac{i}{\hbar}\int_{\infty}^{\infty}<fE_od_z0>e^{\frac{i\hbar l(l+1)t}{2I}+\frac{t}{\tau}}dt$$
My question is in regards to $$<fE_0d_z0>$$. Does d_z have parity? Also, how can I apply the selection rules to determine which eigenstates (ie, spherical harmonics) will not give $$<fE_0d_z0>=0$$? Also, is my integral set up correctly, because it seems like it would diverge.
So, $$d_f(\infty)=\frac{i}{\hbar}\int_{\infty}^{\infty}<fE_od_z0>e^{\frac{i\hbar l(l+1)t}{2I}+\frac{t}{\tau}}dt$$
My question is in regards to $$<fE_0d_z0>$$. Does d_z have parity? Also, how can I apply the selection rules to determine which eigenstates (ie, spherical harmonics) will not give $$<fE_0d_z0>=0$$? Also, is my integral set up correctly, because it seems like it would diverge.