Megascopic Quantum Phenomena
359
London [66]. The macroscopic wave function obtains as
(r) = | 0 (r)| exp(iθ (r)) =
√
n s exp(iθ (r))
(14.1)
with the phase θ and the amplitude 0 characterizing the density, n s , of the superconducting carriers. From this equation, using the well-known relation between the
kinetic and the canonical momentum, one gets the expression for the velocity of
superconducting carriers
v s =
m s
∇.θ −
e s
m s c
A
(14.2)
and after taking the curl of the velocity
∇ × v s =
−e s
m s c
B
(14.3)
and substituting from (9.1) one finally obtains the phenomenological London
Eq. (9.5). It is interesting to note that from London’s macroscopic wave function
(14.1) one can derive Josephson’s equations [143]. Hence in this formulation one does
not need any microscopic concepts of superconductivity, as was already shown by
Feynman [144]. Considering the solution of an easier case of a SIS (SuperconductorInsulator-Superconductor) junction, both “superconductor bodies” start to ‘feel’ each
other at a certain distance, e.g. of the order of nanometres, because of the estimated coherence length consistent with their wave functions. Feynman proposed
two coupled time dependent equations
i
∂∂ 1
∂t
= E 1 1 + K 2 ; i
∂∂ 2
∂t
= E 2 2 + K 1
(14.4)
where K is a phenomenological parameter which describes the properties of the
insulating barrier. From Eqs. (14.1) and (14.4) one simply obtains the first Josephson
equation
I (t) = I c sin(θ 2 (t) − θ 1 (t))
(14.5)
and the second Josephson equation
U (t) =
2e
∂(θ 2 (r) − θ 1 (r))
∂t
(14.6)
where U(t) and I(t) are the voltage across and the current through the Josephson junction, θ 2 − θ 1 is the phase difference across the junction, and I c is a phenomenological
constant, representing the “critical current” of the junction.
One should observe that the ground and the excitation states of the superconductors have a microscopic explanation, as well as all solids etc. Previously we
359
London [66]. The macroscopic wave function obtains as
(r) = | 0 (r)| exp(iθ (r)) =
√
n s exp(iθ (r))
(14.1)
with the phase θ and the amplitude 0 characterizing the density, n s , of the superconducting carriers. From this equation, using the well-known relation between the
kinetic and the canonical momentum, one gets the expression for the velocity of
superconducting carriers
v s =
m s
∇.θ −
e s
m s c
A
(14.2)
and after taking the curl of the velocity
∇ × v s =
−e s
m s c
B
(14.3)
and substituting from (9.1) one finally obtains the phenomenological London
Eq. (9.5). It is interesting to note that from London’s macroscopic wave function
(14.1) one can derive Josephson’s equations [143]. Hence in this formulation one does
not need any microscopic concepts of superconductivity, as was already shown by
Feynman [144]. Considering the solution of an easier case of a SIS (SuperconductorInsulator-Superconductor) junction, both “superconductor bodies” start to ‘feel’ each
other at a certain distance, e.g. of the order of nanometres, because of the estimated coherence length consistent with their wave functions. Feynman proposed
two coupled time dependent equations
i
∂∂ 1
∂t
= E 1 1 + K 2 ; i
∂∂ 2
∂t
= E 2 2 + K 1
(14.4)
where K is a phenomenological parameter which describes the properties of the
insulating barrier. From Eqs. (14.1) and (14.4) one simply obtains the first Josephson
equation
I (t) = I c sin(θ 2 (t) − θ 1 (t))
(14.5)
and the second Josephson equation
U (t) =
2e
∂(θ 2 (r) − θ 1 (r))
∂t
(14.6)
where U(t) and I(t) are the voltage across and the current through the Josephson junction, θ 2 − θ 1 is the phase difference across the junction, and I c is a phenomenological
constant, representing the “critical current” of the junction.
One should observe that the ground and the excitation states of the superconductors have a microscopic explanation, as well as all solids etc. Previously we