Question

find Approximation transmission probability in quantum mechanics E<U

Answer 1

Answer

Let for electron has wave function

$\psi\propto e^\frac{ikx}{\hbar}$

The effective momentum of electron is given by

$\frac{\hbar^2 k^2}{2m*}=-q\phi$

$k=i\sqrt{\frac{2m*q\phi}{\hbar^2}}$

So wavefunction become

$\psi\propto e^{i\sqrt{\frac{2m*q\phi}{\hbar^2}}x}$

At x=0 to x=d the ratio of wave functions is given

$\frac{\psi (x=d) }{\psi(x=0) }\propto \frac{ e^{-\sqrt{\frac{2m*q\phi}{\hbar^2}}.d}}{e^{-\sqrt{\frac{2m*q\phi}{\hbar^2}}.0}}\propto e^{-\sqrt{\frac{2m*q\phi}{\hbar^2}}d}$

Let J_I

 be the incident current and J_T

be the transmitted current. Then

Transmission probability

$T\propto \frac{J_T}{J_0}\propto (\frac{\psi(x=d) }{\psi(x=0) }) ^2\\\propto e^{-2\sqrt{-\frac{2m*q\phi}{\hbar^2}}d}$