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5.74 Introductory Quantum Mechanics II
Spring 2009
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4.1. INTERACTION OF LIGHT WITH MATTER
One of the most important topics in timedependent quantum mechanics for chemists is the
description of spectroscopy, which refers to the study of matter through its interaction with light
fields (electromagnetic radiation). Classically, lightmatter interactions are a result of an oscillating
electromagnetic field resonantly interacting with charged particles.
Quantum mechanically, light
fields will act to couple quantum states of the matter, as we have discussed earlier.
Like every other problem, our starting point is to derive a Hamiltonian for the lightmatter
interaction, which in the most general sense would be of the form
HH
H H
=
M
+
L
+
LM
.
(4.1)
The Hamiltonian for the matter
H
M
is generally (although not necessarily) time independent,
whereas the electromagnetic field
H
L
and its interaction with the matter
H
LM
are timedependent.
A quantum mechanical treatment of the light would describe the light in terms of photons for
different modes of electromagnetic radiation, which we will describe later.
We will start with a common semiclassical treatment of the problem. For this approach we
treat the matter quantum mechanically, and treat the field classically. For the field we assume that
the light only presents a timedependent interaction potential that acts on the matter, but the matter
doesn’t influence the light.
(Quantum mechanical energy conservation says that we expect that the
change in the matter to raise the quantum state of the system and annihilate a photon from the
field. We won’t deal with this right now).
We are just interested in the effect that the light has on
the matter. In that case, we can really ignore
H
L
, and we have a Hamiltonian that can be solved in
the interaction picture representation:
H t
+
( )
≈
M
LM
HV
+ ()
(4.2)
=
0
t
Here, we’ll derive the Hamiltonian for the lightmatter interaction, the Electric Dipole
Hamiltonian. It is obtained by starting with the force experienced by a charged particle in an
electromagnetic field, developing a classical Hamiltonian for this system, and then substituting
quantum operators for the matter:
Andrei Tokmakoff, MIT Department of Chemistry, 2/7/2008
42
p
→−
i
=
∇
ˆ
(4.3)
x
→
x
ˆ
In order to get the classical Hamiltonian, we need to work through two steps: (1) We need
to describe electromagnetic fields, specifically in terms of a vector potential, and (2) we need to
describe how the electromagnetic field interacts with charged particles.
Brief summary of electrodynamics
Let’s summarize the description of electromagnetic fields that we will use.
A derivation of the
plane wave solutions to the electric and magnetic fields and vector potential is described in the
appendix. Also, it is helpful to review this material in Jackson
1
or CohenTannoudji, et al.
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.
 Spring '04
 RobertField

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