Quantum Mechanics - Prof. Eric R. Bittner - University of Houston

In other words:
We can also define a transition rate as
Ps(t) = 2πt
¯h |〈s| ˆ V |n〉| 2 ρ(Es) (7.111)
R(t) =
P (t)
t
thus, the Golden Rule Transition Rate from state n to state s is
(7.112)
Rn→s(t) = 2π
¯h |〈s| ˆ V |n〉| 2 ρ(Es) (7.113)
This is perhaps one of the most important approximations in quantum mechanics in that it
has implications and applications in all areas, especially spectroscopy and other applications of
matter interacting with electro-magnetic radiation.
7.6 Interaction between an atom and light
What I am going to tell you about is what we teach our physics students in the third
or fourth year of graduate school... It is my task to convince you not to turn away
because you don’t understand it. You see my physics students don’t understand it...
That is because I don’t understand it. Nobody does.
Richard P. Feynman, QED, The Strange Theory of Light and Matter
Here we explore the basis of spectroscopy. We will consider how an atom interacts with a
photon field in the low intensity limit in which dipole interactions are important. We will then
examine non-resonant excitation and discuss the concept of oscillator strength. Finally we will
look at resonant emission and absorption concluding with a discussion of spontaneous emission.
In the next section, we will look at non-linear interactions.
7.6.1 Fields and potentials of a light wave
An electromagnetic wave consists of two oscillating vector field components which are perpendicular
to each other and oscillate at at an angular frequency ω = ck where k is the magnitude
of the wavevector which points in the direction of propagation and c is the speed of light. For
such a wave, we can always set the scalar part of its potential to zero with a suitable choice in
gauge and describe the fields associate with the wave in terms of a vector potential, � A given by
�A(r, t) = Aoeze iky−iωt + A ∗ oeze −iky+iωt
Here, the wave-vector points in the +y direction, the electric field, E is polarized in the yz plane
and the magnetic field B is in the xy plane. Using Maxwell’s relations
�E(r, t) = − ∂A
∂t = iωez(Aoe i(ky−ωt) − A ∗ oe −i(ky−ωt) )
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