Chemistry Reference
In-Depth Information
thenwe can associate amomentum2
m
e
v
=
q
with each Cooper pair, with
the supercurrent density,
j
s
(
r
)
given by
n
s
2
(
−
q
2
m
e
)
j
s
(
r
)
=
2
e
2
e
m
e
=−|
ψ(
r
)
|
q
(8.54)
We must modify this definition of current density in the presence of an
applied magnetic field,
B
. We saw in Section 8.5 that the electromagnetic
momentum
p
for a particle (Cooper pair) of mass 2
m
e
and charge
−
2
e
is
given by eq. (8.6) as
p
=
2
m
e
v
−
2
e
A
(8.55)
where
A
is the magnetic vector potential, and
B
=∇×
A
. The kinetic
momentum is therefore given by
2
m
e
v
=
p
+
2
e
A
(8.56)
and the velocity
v
of a Cooper pair by
A
p
2
m
e
+
e
m
e
v
=
(8.57)
To calculate the velocity
v
in a quantum mechanical analysis, we replace
the total momentum
p
by the momentum operator
. If we assume
that this is also true for the superconducting order parameter, we might
deduce from eq. (8.54) that the superconducting current density
j
−
i
∇
(
r
)
in the
presence of a magnetic field
B
is given by
)
=
ψ
∗
(
j
(
r
r
)(
−
2
e
v
)ψ(
r
)
e
m
e
ψ
∗
(
=−
r
)(
−
i
∇+
2
e
A
)ψ(
r
)
(8.58)
This analysis is only partly correct. It is possible using eq. (8.58) to find
sensible wavefunctions (e.g.
for which the calculated
current density is imaginary. Amore complete analysis (see e.g. Hook and
Hall 1991) shows that in order for the local current density to be real (and
therefore a measurable quantity), we must define
j
ψ(
r
)
=
sin
(
k
·
r
))
(
r
)
as half the sum of
eq. (8.58) and its complex conjugate:
2
e
2
m
e
A
i
e
2
m
e
(ψ
∗
(
)
∇
ψ
∗
(
ψ
∗
(
j
(
r
)
=
r
)
∇
ψ(
r
)
−
ψ(
r
r
))
−
r
)ψ(
r
)
(8.59)
We can choose to write the superconducting order parameter at each point
as the product of a real number times a phase factor,
e
i
θ(
r
)
ψ(
r
)
=|
ψ(
r
)
|
(8.60)
