Schrödinger's Equation — Question 8

Question 8

A finite potential step requires flux weighting of transmitted amplitude

A particle approaches a finite upward step with $E>V_0$. On the left $\psi=e^{ik_1x}+re^{-ik_1x}$; on the right $\psi=te^{ik_2x}$, where $k_1,k_2>0$. Both $\psi$ and $\psi\prime$ are continuous at zero. The probability current is $j=(\hbar/m)\operatorname{Im}(\Psi^*\partial_x\Psi)$.

1. Derive r and t from the two matching equations.

2. For $k_2=k_1/2$, find R and T from currents and verify conservation.

Question 8 - Solution

A finite potential step requires flux weighting of transmitted amplitude

(a) Method and calculation

$$1+r=t,\qquad k_1(1-r)=k_2t.$$$$\boxed{r=\frac{k_1-k_2}{k_1+k_2}},\qquad\boxed{t=\frac{2k_1}{k_1+k_2}}.$$

(b) Interpretation and further calculation

$$\boxed{R=|r|^2=1/9},\qquad\boxed{T=\frac{k_2}{k_1}|t|^2=\frac12\left(\frac43\right)^2=8/9}.$$$$\boxed{R+T=1},\qquad |t|^2=16/9\ne T.$$

Physical check

An amplitude larger than one does not violate probability conservation. The transmitted density must be multiplied by the transmitted velocity to obtain flux.