Page 142 - Principles and Applications of NanoMEMS Physics

130 Chapter 3

2 G 1 § * q 2 G ' ·
Ψ v = ¨− = i Ψ ∇ Ψ − Ψ A ¸
M © c ¹
. (161)
G
= 1 § Ψ * ∇ Ψ − 2e Ψ A ' · ¸
2
¨− = i
2m © c ¹
e
Now, writing the complex wave function as Ψ = Ψ e , where χ is a
χ i
space-dependent phase, and substituting into (161) we obtain,
G
G
v = = ∇ χ − e A . (162)
'
s
2m m c
e e
∇
This equation reveals that, even if χ = 0, current flow may be excited by
G G G
the vector potential. In fact, since B = ∇ × A , we may redefine A to
'
G
include the phase, without changing B , i.e.,
G G c =
A = A ' + ∇ χ , (163)
e 2
from where we get,

G e G
v = − A . (164)
'
s
m c
e
The supercurent, then, is given by,
G e 2 n G
J = − s A . (165)
'
s
m c
e
The effects of a superconductor on a magnetic field inside its bulk follow
from from substituting (164) into the equation (165),
G 4π
∇ × B = J , (166)
s
c

and taking its curl, i.e.,

G 4π 4π e 2 n G
∇ × ∇ × B = ∇ × J = − s B . (167)
c s m c 2
e