Phase and Coherence

An implication of this long range coherence is the ability to calculate phase and amplitude at any point on the wave's path from the knowledge of its phase and amplitude at any single point, combined with its wavelength and frequency. The wavefunction of the electron-pair wave in Equation 3.16 can be rewritten in the form of a one-dimensional wave as

$$\Psi_{P} = \Psi \sin 2\pi \left( \frac{\boldsymbol{x}}{\lambda} - vt \right)$$ (3.17)

If we take the wave frequency, $\nu$, as being related to the kinetic energy of the Cooper pair with a wavelength, $\lambda$, being related to the momentum of the pair by the relation $\lambda = \nicefrac{h}{P}$ then it is possible to evaluate the phase difference between two points in a current carrying superconductor.

If a resistanceless current flows between points X and Y on a superconductor there will be a phase difference between these points that is constant in time. The phase difference for such a plane wave is given by

$$(\Delta\phi)_{XY} = \phi_{X} - \phi_{Y} = 2\pi \int^{Y}_{X} \frac{\mathbf{\hat{x}}}{\lambda} \centerdot \mathrm{d}\boldsymbol{l}$$ (3.18)

where $\mathbf{\hat{x}}$ is a unit vector in the direction of the wave propagation, and $\mathrm{d}\boldsymbol{l}$ is an element of a line joining X to Y.[12]

The relation of $v$ to the supercurrent density, $\mathcal{J}_{s}$, is $\mathcal{J}_{s} = \nicefrac{1}{2}\,n_{s} . 2e . v$, where $n_{s}$ is the superelectron density, and $\nicefrac{1}{2}\,n_{s}$ is the electron-pair density. The wavelength can therefore be written as

$$\lambda = \frac{hn_{s}e}{2m\mathcal{J}_{s}}$$ (3.19)

and hence the phase difference between points X and Y can be written as

$$(\Delta\phi)_{XY} = \frac{2\pi m}{hn_{s}e} \int^{Y}_{X} \boldsymbol{J}_{s} \centerdot \mathrm{d}\boldsymbol{l}$$ (3.20)