Physics 6210/Spring 2007/Lecture 19
Lecture 19
Relevant sections in text:
§
2.6
Charged particle in an electromagnetic field
We now turn to another extremely important example of quantum dynamics. Let us
describe a nonrelativistic particle with mass
m
and electric charge
q
moving in a given
electromagnetic field. This system has obvious physical significance.
We use the same position and momentum operators (in the Schr¨
odinger picture)
X
and
P
(although there is a subtlety concerning the meaning of momentum, to be mentioned
later). To describe the electromagnetic field we need to use the electromagnetic scalar and
vector potentials
φ
(
x, t
)
, A
(
x, t
).
They are related to the familiar electric and magnetic
fields (
E, B
) by
E
=
∇
φ

1
c
∂A
∂t
,
B
=
∇ ×
A.
The dynamics of a particle with mass
m
and charge
q
is determined by the Hamiltonian
H
=
1
2
m
P

q
c
A
(
X, t
))
2
+
qφ
(
X, t
)
.
This Hamiltonian takes the same form as the classical expression in Hamiltonian mechanics.
We can see that this is a reasonable form for
H
by computing the Heisenberg equations of
motion, and seeing that they are equivalent to the Lorentz force law, which we shall now
demonstrate.
For simplicity we assume that the potentials are time independent, so that the Heisen
berg and Schr¨
odinger picture Hamiltonians are the same, taking the form
H
=
1
2
m
P

q
c
A
(
X
))
2
+
qφ
(
X
)
.
For the positions we get (exercise)
d
dt
X
(
t
) =
1
i
¯
h
[
X
(
t
)
, H
] =
1
m
{
P
(
t
)

q
c
A
(
X
(
t
))
}
.
We see that (just as in classical mechanics) the momentum – defined as the generator of
translations – is not necessarily given by the mass times the velocity, but rather
P
(
t
) =
m
dX
(
t
)
dt
+
q
c
A
(
X
(
t
))
.
As in classical mechanics we sometimes call
P
the
canonical momentum
, to distinguish it
from the
mechanical momentum
Π =
m
dX
(
t
)
dt
=
P

q
c
A
(
X
(
t
))
1
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Physics 6210/Spring 2007/Lecture 19
Note that the mechanical momentum has a direct physical meaning, while the canonical
momentum depends upon the nonunique form of the potentials. We will discuss this in
detail soon.
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 Spring '07
 M
 mechanics, Charge, Mass, φ, mechanical momentum

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