Schrödinger's Equation — Question 6

Question 6

Counterpropagating waves can form density fringes while carrying constant net current

A stationary free-particle state has spatial part $\psi=Ae^{ikx}+Be^{-ikx}$, with real $A=2B>0$ and $k>0$. The common time factor is $e^{-iEt/\hbar}$. The probability current is $j=(\hbar/m)\operatorname{Im}(\Psi^*\partial_x\Psi)$.

1. Derive the spatial probability density and its extrema.

2. Compute the current and explain why a fringed density need not imply zero transport.

Question 6 - Solution

Counterpropagating waves can form density fringes while carrying constant net current

(a) Method and calculation

$$\rho=A^2+B^2+2AB\cos2kx=B^2(5+4\cos2kx).$$$$\boxed{\rho_{\max}=9B^2,\qquad\rho_{\min}=B^2}.$$

(b) Interpretation and further calculation

$$\operatorname{Im}(\psi^*\psi\prime)=k(A^2-B^2).$$$$\boxed{j=\frac{\hbar k}{m}(A^2-B^2)=\frac{3\hbar kB^2}{m}>0}.$$

Physical check

The interference term modulates density but cancels from current. Unequal rightward and leftward fluxes leave a nonzero net flow.