Question

The first excited state for a quantum harmonic oscillator is described by the wave function $\psi(x) = (\frac{\alpha}{\pi})^{1/4}\sqrt{(2\alpha)}xe^{-\alpha x^2/2}$ where $\alpha = m\omega_0/\hbar$ ($\omega_0$ is defined on your equation sheet). Plug this function into the time-independent Schrödinger equation $\frac{-\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$ and take the necessary derivatives to show that you get an energy of $\frac{3}{2}\hbar\omega_0$.

Name: the first excited state for a quantum harmonic oscillator is described by the wave function psix fracalphapi14sqrt2alphaxe alpha x22 where alpha momega0hbar omega0 is defined on your equat 17194
Uploaded: 2021-10-22T02:36:42-08:00
Duration: 5 min 34 s
Channel: Ameer Said
Description: the first excited state for a quantum harmonic oscillator is described by the wave function psix fracalphapi14sqrt2alphaxe alpha x22 where alpha momega0hbar omega0 is defined on your equat 17194

          The first excited state for a quantum harmonic oscillator is described by the wave
function $\psi(x) = (\frac{\alpha}{\pi})^{1/4}\sqrt{(2\alpha)}xe^{-\alpha x^2/2}$ where $\alpha = m\omega_0/\hbar$ ($\omega_0$ is defined on your
equation sheet). Plug this function into the time-independent Schrödinger equation
$\frac{-\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$ and take the necessary derivatives to show that you get an
energy of $\frac{3}{2}\hbar\omega_0$.

$the first excited state for a quantum harmonic oscillator is described by the wave function psix fracalphapi14sqrt2alphaxe alpha x22 where alpha momega0hbar omega0 is defined on your equat 17194$

Added by Lance L.

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