Schrödinger's Equation — Question 7

Question 7

Probability conservation follows from the wave equation and its complex conjugate

A normalizable particle evolves in a real potential $V(x,t)$. Define $\rho=\Psi^*\Psi$ and $j=\hbar(\Psi^*\partial_x\Psi-\Psi\partial_x\Psi^*)/(2mi)$. Use $i\hbar\partial_t\Psi=-(\hbar^2/2m)\partial_x^2\Psi+V\Psi$.

1. Derive the local continuity equation.

2. Express the rate of change of probability in $[a,b]$ in terms of boundary currents and evaluate it for $j(a)=.030\,\mathrm{s}^{-1}$ and $j(b)=.010\,\mathrm{s}^{-1}$.

Question 7 - Solution

Probability conservation follows from the wave equation and its complex conjugate

(a) Method and calculation

$$\partial_t\rho=\frac{i\hbar}{2m}\left(\Psi^*\partial_x^2\Psi-\Psi\partial_x^2\Psi^*\right).$$$$\partial_xj=\frac{\hbar}{2mi}\left(\Psi^*\partial_x^2\Psi-\Psi\partial_x^2\Psi^*\right)\quad\Rightarrow\quad\boxed{\partial_t\rho+\partial_xj=0}.$$

(b) Interpretation and further calculation

$$\frac{d}{dt}\int_a^b\rho\,dx=-\int_a^b\partial_xj\,dx=j(a)-j(b).$$$$\boxed{\dot P_{[a,b]}=.020\,\mathrm{s}^{-1}}.$$

Physical check

The potential terms cancel because V is real. Positive accumulation means inward flux exceeds outward flux; it does not create probability globally.