Chemistry Reference
In-Depth Information
and we can choose
A
so that
=−
∂
A
∂
E
(8.11)
t
The change in kinetic momentum of the particle between times
t
1
and
t
2
is
the impulse of the force
F
=
q
E
acting on it during this interval:
t
2
q
t
2
t
1
∂
A
∂
t
t
1
=
t
t
1
[
M
v
]
q
E
d
t
=−
t
d
t
=[−
q
A
]
(8.12)
t
1
Rearranging eq. (8.12) we find
M
v
1
+
q
A
1
=
M
v
2
+
q
A
2
(8.13)
so that
p
q
A
is conserved at all times
t
, as the current decays. This
is then the appropriate definition of total momentum for a charged particle
in the presence of a magnetic field.
We saw in Chapter 1 that the transition from classical to quantum
mechanics is made by replacing the momentum
p
by the operator
−
=
M
v
+
∇
i
, with
p
ψ
=−
i
∇
ψ
(8.14)
As
p
is defined in terms of the gradient operator, this implies that
p
is com-
pletely determined by the geometry of the wavefunction (a more rapidly
varying wavefunction implies larger total momentum).
In isolated atoms, the electronic wavefunction is rigid, and unchanged
to first order in an applied magnetic field,
B
. We will see below that the
same must also be true for superconducting electrons. The total momen-
tum
p
is therefore conserved in an applied magnetic field. The average
electron velocity must be zero
(
v
=
0
)
when the applied vector potential
A
=
0, giving
p
=
0. The conservation of total momentum then requires
M
v
+
q
A
=
0, so that for an electron of mass
m
e
and charge
−
e
, we can
write
e
A
m
e
v
=
(8.15)
in an applied vector potential
A
. The resulting induced current density
j
is
then given by
ne
2
m
e
A
j
=
n
(
−
e
)
v
=−
(8.16)
where
n
is thedensityof electrons per unit volume. Byassuming thatwe can
associate a rigid (macroscopic) wavefunction with the
n
s
superconducting
electrons per unit volume in a superconductor we derive that
n
s
e
2
m
e
=
−
j
A
(8.17)
