Quantum Mechanical Tunneling

Tunnelling through step barier

\(V(x) = V_0\) for \(0\lt x\lt a\) and \(V(x) = 0\) for all other \(x\).

For regions I, III, the solutions will be oscillatory exponentials. At the boundary the function and the derivative are continuous.

\(\kappa = \sqrt{\frac{2mE}{\hbar^2}}\), \(\Psi_{I,III} = A_{I,III}\exp(i\kappa x) + B_{I,III}\exp(-i\kappa x)\).

\(A\) is the incoming wave, from \(-\) to \(+\). \(B\) is going the opposite direction.

Initial Condition

Incoming from \(-\infty\) and no reflective wall at \(+\infty\). Then \(B_{III}=0\).

Energy below Barrier

In region II it will be a decaying exponential. \(k=\sqrt{2m(V_0-E)}/\hbar\).

Energy above Barrier

In region II it will be a oscillatory exponential. \(k=\sqrt{2m(E-V_0)}/\hbar\).

Probability Flux

Region I. \(j=\frac{\hbar}{m}\Im\left[\Psi^*\frac{d}{dx}\Psi\right] = \frac{\hbar}{m}\kappa\left(|A|^2-|B|^2\right)\). \(j_{inc}=\frac{\hbar\kappa}{m}|A|^2\) is the incoming wave velocity. \(j_{ref}=\frac{\hbar\kappa}{m}|B|^2\) is the reflected wave velocity.

Region III. \(j=\frac{\hbar \kappa}{m}|C|^2\) is the transmitted wave velocity.

Reflection Coefficient. \(R=\frac{j_{ref}}{j_{inc}} = \frac{|B|^2}{|A|^2}\).

Transmission Coefficient. \(T=\frac{j_{trans}}{j_{inc}} = \frac{|C|^2}{|A|^2}\).

\(R+T=1\) for a non-absorbing wall.

Classical Expectations

If \(E\lt V_0\) then we expect \(R=1\) and \(T=0\). If \(E\gt V_0\) then we expect \(T=1\) and \(R=0\).

For \(E\lt V_0\), \(T=\left[1+\frac{V_0^2\sinh^2 ka}{4E(V_0-E)}\right]^{-1}\).

For \(E\gt V_0\), \(T=\left[1+\frac{V_0^2\sin^2 ka}{4E(E-V_0)}\right]^{-1}\).

Author: Christian Cunningham

Created: 2023-06-25 Sun 02:33