Superconductivity Derivation

Last edited February 24, 2026 by HelloWorld. Created February 24, 2026 by HelloWorld.

Derivation of Superconductivity Equation

Derivation of the Superconductivity Equation

Superconductor: A material that exhibits zero electrical resistance and expels magnetic fields below a critical temperature.

Cooper Pair: A bound state of two electrons with opposite momentum and spin behaving as a single quantum particle.

Macroscopic Wavefunction \( \psi \): Describes the collective quantum state of all Cooper pairs.

\( n_s \): Density of superconducting carriers (number of Cooper pairs per unit volume).

\( \phi \): Quantum mechanical phase of the superconducting wavefunction.

\( q \): Charge of the superconducting carrier. For Cooper pairs, \( q = -2e \).

\( e \): Elementary charge (magnitude of electron charge).

\( m \): Effective mass of the Cooper pair.

\( \hbar \): Reduced Planck constant.

\( \mathbf{A} \): Magnetic vector potential.

\( \mathbf{B} \): Magnetic field.

\( \mu_0 \): Vacuum permeability.

\( \mathbf{J}_s \): Superconducting current density.

\[ \psi(\mathbf{r}) = \sqrt{n_s} e^{i\phi(\mathbf{r})} \]

\[ \mathbf{J} = \frac{q}{m} \Re \left[ \psi^* \left( -i\hbar\nabla - q\mathbf{A} \right) \psi \right] \]

\[ q = -2e \]

\[ \mathbf{J}_s = \frac{n_s q}{m} \left( \hbar \nabla \phi - q \mathbf{A} \right) \]

\[ \mathbf{J}_s = -\frac{n_s q^2}{m} \mathbf{A} \]

\[ \nabla \times \mathbf{J}_s = -\frac{n_s q^2}{m} \mathbf{B} \]

\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J}_s \]

\[ \nabla^2 \mathbf{B} = \frac{1}{\lambda_L^2} \mathbf{B} \]

\[ \lambda_L = \sqrt{\frac{m}{\mu_0 n_s q^2}} \]

\[ B(x) = B_0 e^{-x/\lambda_L} \]

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