Schrödinger's Equation — Question 1

Question 1

Substitution into the wave equation fixes the time evolution of a plane wave

A particle of mass m moves in constant potential $V_0$. Consider $\Psi=Ae^{i(kx-\omega t)}$. Use $i\hbar\partial_t\Psi=-(\hbar^2/2m)\partial_x^2\Psi+V\Psi$.

1. Substitute the wave into the equation and determine $\omega$.

2. Find its probability density and current, and check the continuity equation. The probability current is $j=(\hbar/m)\operatorname{Im}(\Psi^*\partial_x\Psi)$.

Question 1 - Solution

Substitution into the wave equation fixes the time evolution of a plane wave

(a) Method and calculation

$$\partial_t\Psi=-i\omega\Psi,\quad\partial_x^2\Psi=-k^2\Psi.$$$$\boxed{\hbar\omega=\hbar^2k^2/(2m)+V_0}.$$

(b) Interpretation and further calculation

$$\boxed{\rho=|A|^2,\qquad j=(\hbar k/m)|A|^2}.$$$$\boxed{\partial_t\rho+\partial_xj=0+0=0}.$$

Physical check

A plane wave is an idealized flux state rather than a square-normalizable state on the full real line. A localized packet can be assembled from such components.