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Quantum Mechanical Tunneling

The ability of Cooper pairs to tunnel through potential barriers, enabling lossless current flow in superconductors.

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The statement of the theorem

Consider a particle with mass

m

encountering a potential barrier

V(x)

of width

L

and height

V_0

, where

E < V_0

. The transmission probability

T

through the barrier is approximated by the WKB method:\n

T \approx e^{-2 \int_{x_1}^{x_2} \sqrt{\frac{2m}{\hbar^2} (V(x) - E)} dx}

\nFor Cooper pairs, the tunneling current

I

is proportional to

T

and the pair density.