Barrier Tunneling — Question 6

Question 6

Two evanescent components can carry current although either alone cannot

Inside a finite forbidden region let $\psi(x)=Ce^{-\kappa x}+iDe^{\kappa x}$, with real positive C, D and $\kappa$. Use $j=(\hbar/m)\operatorname{Im}(\psi^*\psi\prime)$. This is a local solution on a finite interval, not on the entire real line.

1. Calculate the probability current and show that it is independent of x.

2. Find the current if either C or D is zero and explain why a pure decaying exponential cannot by itself describe nonzero steady transmission.

Question 6 - Solution

Two evanescent components can carry current although either alone cannot

(a) Method and calculation

$$\psi\prime=-\kappa Ce^{-\kappa x}+i\kappa De^{\kappa x}.$$$$\operatorname{Im}(\psi^*\psi\prime)=2\kappa CD\quad\Rightarrow\quad\boxed{j=2\hbar\kappa CD/m,\quad\partial_xj=0}.$$

(b) Interpretation and further calculation

$$C=0\ \text{or}\ D=0:\quad\boxed{j=0}.$$$$T=j_{\mathrm{trans}}/j_{\mathrm{inc}}>0\quad\Rightarrow\quad j_{\mathrm{barrier}}=j_{\mathrm{trans}}>0\quad\Rightarrow\quad\boxed{CD\ne0}.$$

Physical check

Boundary matching generally requires both exponential components inside a finite barrier. The growing term is harmless on a finite interval; only a component diverging at spatial infinity must be excluded.