Schrödinger's Equation — Question 2

Question 2

A real traveling sinusoid fails the time-dependent quantum equation

In a free region, compare $\Psi_1=A\cos(kx-\omega t)$ with $\Psi_2=A\cos(kx)e^{-i\omega t}$, where A is real, $k>0$, and $\omega=\hbar k^2/(2m)>0$. Use $i\hbar\partial_t\Psi=-(\hbar^2/2m)\partial_x^2\Psi+V\Psi$.

1. Test $\Psi_1$ by differentiating both sides.

2. Test $\Psi_2$ and determine whether its probability density moves.

Question 2 - Solution

A real traveling sinusoid fails the time-dependent quantum equation

(a) Method and calculation

$$i\hbar\partial_t\Psi_1=i\hbar\omega A\sin(kx-\omega t).$$$$-\frac{\hbar^2}{2m}\partial_x^2\Psi_1=\frac{\hbar^2k^2}{2m}A\cos(kx-\omega t)\quad\Rightarrow\quad\boxed{\Psi_1\text{ fails}}.$$

(b) Interpretation and further calculation

$$i\hbar\partial_t\Psi_2=\hbar\omega\Psi_2=\frac{\hbar^2k^2}{2m}\Psi_2\quad\Rightarrow\quad\boxed{\Psi_2\text{ satisfies}}.$$$$\boxed{|\Psi_2|^2=A^2\cos^2(kx),\quad\partial_t|\Psi_2|^2=0}.$$

Physical check

Taking only the real part of a complex solution is not generally allowed for this equation. The stationary standing wave combines opposite momenta at the same energy.