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