130_notes.dvi

table of integrals), so the result is simple.

∫∞

−∞

d∆f(∆)g(∆) = f(∆ = 0)

2 π

t

g(∆) =

2 π

t

δ(∆)

[

4 sin^2 ((ωni+ω)t/2)

(ωni+ω)^2

]

= 2πt δ(ωni+ω)

Q.E.D.

28.5 Homework Problems

1. A hydrogen atom is placed in an electric field which is uniform in space and turns on att= 0
   then decays exponentially. That is,E~(t) = 0 fort <0 andE~(t) =E~ 0 e−γtfort >0. What is
   the probability that, ast→∞, the hydrogen atom has made a transition to the 2pstate?


2. A one dimensional harmonic oscillator is in its ground state. It is subjected to the additional
   potentialW=−eξxfor a a time intervalτ. Calculate the probability to make a transition to
   the first excited state (in first order). Now calculate the probability to make a transition to
   the second excited state. You will need to calculate to second order.

28.6 Sample Test Problems

1. A hydrogen atom is in a uniform electric field in thezdirection which turns on abruptly at
   t= 0 and decays exponentially as a function of time,E(t) =E 0 e−t/τ. The atom is initially
   in its ground state. Find the probability for the atom to have made a transition to the 2P
   state ast→∞. You need not evaluate the radial part of the integral. Whatzcomponents of
   orbital angular momentum are allowed in the 2Pstates generated by this transition?