Question

Show explicitly that the eigenfunctions\\ $\psi_0(x) = \left(\frac{\alpha}{\pi}\right)^{1/4} e^{-\alpha x^2/2}$ \\ $\psi_1(x) = \left(\frac{4\alpha^3}{\pi}\right)^{1/4} x e^{-\alpha x^2/2}$ \\ $\psi_2(x) = \left(\frac{\alpha}{4\pi}\right)^{1/4} (2\alpha x^2 - 1) e^{-\alpha x^2/2}$ \\ satisfy the time independent Schrödinger equation for the harmonic oscillator.

          Show explicitly that the eigenfunctions\\
$\psi_0(x) = \left(\frac{\alpha}{\pi}\right)^{1/4} e^{-\alpha x^2/2}$ \\
$\psi_1(x) = \left(\frac{4\alpha^3}{\pi}\right)^{1/4} x e^{-\alpha x^2/2}$ \\
$\psi_2(x) = \left(\frac{\alpha}{4\pi}\right)^{1/4} (2\alpha x^2 - 1) e^{-\alpha x^2/2}$ \\
satisfy the time independent Schrödinger equation for the harmonic oscillator.

Show explicitly that the eigenfunctions

ψ0(x) = ((α)/(π))^1/4 e^-α x^2/2

ψ1(x) = ((4α^3)/(π))^1/4 x e^-α x^2/2

ψ2(x) = ((α)/(4π))^1/4 (2α x^2 - 1) e^-α x^2/2

satisfy the time independent Schrödinger equation for the harmonic oscillator.

Added by Ramon M.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Ameer Said

10/21/2021

Step 1

Step 1: The time-independent Schrödinger equation for a harmonic oscillator is given by: $-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi(x) = E \psi(x)$ where $\alpha = \frac{m\omega}{\hbar}$. Show more…

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Show explicitly that the eigenfunctions $\psi_0(x) = (\frac{\alpha}{\pi})^{1/4} e^{-\frac{\alpha x^2}{2}}$ $\psi_1(x) = (\frac{4\alpha^3}{\pi})^{1/4} x e^{-\frac{\alpha x^2}{2}}$ $\psi_2(x) = (\frac{\alpha}{\pi})^{1/4} (2\alpha x^2 - 1) e^{-\frac{\alpha x^2}{2}}$ satisfy the time independent Schrödinger equation for the harmonic oscillator.