Question

Confirm that the ground-state harmonic oscillator wavefunction is a solution of the Schrödinger equation \frac{d}{dx} N_0 e^{-x^2/\alpha^2} = -N_0 \frac{x}{\alpha^2} e^{-x^2/\alpha^2} \frac{d^2}{dx^2} N_0 e^{-x^2/2\alpha^2} = \frac{d}{dx} \left( -N_0 \left( \frac{x}{\alpha^2} \right) e^{-x^2/\alpha^2} \right) = -N_0 \frac{e^{-x^2/2\alpha^2}}{\alpha^2} + N_0 \left( \frac{x}{\alpha^2} \right)^2 e^{-x^2/\alpha^2} = -\frac{1}{\alpha^2} x \psi_0 + \left( \frac{x^2}{\alpha^4} \right) \psi_0 miltonian Schrödinger \frac{\hbar^2}{2m} \left( \frac{mk}{\hbar^2} \right) \psi_0 - \frac{\hbar^2}{2m^2} \left( \frac{mk}{\hbar^2} \right) \psi_0 + \frac{1}{2} kx^2 \psi_0 = E \psi_0 \frac{\hbar^2}{2} \left( \frac{k}{m} \right) \psi_0 - \frac{1}{2} kx^2 \psi_0 + \frac{1}{2} kx^2 \psi_0 = E \psi_0 E = \frac{\hbar}{2} \left( \frac{k}{m} \right)^{1/2}

          Confirm that the ground-state harmonic oscillator wavefunction is a
solution of the Schrödinger equation
\frac{d}{dx} N_0 e^{-x^2/\alpha^2} = -N_0 \frac{x}{\alpha^2} e^{-x^2/\alpha^2}
\frac{d^2}{dx^2} N_0 e^{-x^2/2\alpha^2} = \frac{d}{dx} \left( -N_0 \left( \frac{x}{\alpha^2} \right) e^{-x^2/\alpha^2} \right)
= -N_0 \frac{e^{-x^2/2\alpha^2}}{\alpha^2} + N_0 \left( \frac{x}{\alpha^2} \right)^2 e^{-x^2/\alpha^2}
= -\frac{1}{\alpha^2} x \psi_0 + \left( \frac{x^2}{\alpha^4} \right) \psi_0
miltonian Schrödinger
\frac{\hbar^2}{2m} \left( \frac{mk}{\hbar^2} \right) \psi_0 - \frac{\hbar^2}{2m^2} \left( \frac{mk}{\hbar^2} \right) \psi_0 + \frac{1}{2} kx^2 \psi_0 = E \psi_0
\frac{\hbar^2}{2} \left( \frac{k}{m} \right) \psi_0 - \frac{1}{2} kx^2 \psi_0 + \frac{1}{2} kx^2 \psi_0 = E \psi_0
E = \frac{\hbar}{2} \left( \frac{k}{m} \right)^{1/2}

Confirm that the ground-state harmonic oscillator wavefunction is a
solution of the Schrödinger equation
(d)/(dx) N0 e^-x^2/α^2 = -N0 (x)/(α^2) e^-x^2/α^2
(d^2)/(dx^2) N0 e^-x^2/2α^2 = (d)/(dx) ( -N0 ( (x)/(α^2) ) e^-x^2/α^2 )
= -N0 (e^-x^2/2α^2)/(α^2) + N0 ( (x)/(α^2) )^2 e^-x^2/α^2
= -(1)/(α^2) x ψ0 + ( (x^2)/(α^4) ) ψ0
miltonian Schrödinger
(ħ^2)/(2m) ( (mk)/(ħ^2) ) ψ0 - (ħ^2)/(2m^2) ( (mk)/(ħ^2) ) ψ0 + (1)/(2) kx^2 ψ0 = E ψ0
(ħ^2)/(2) ( (k)/(m) ) ψ0 - (1)/(2) kx^2 ψ0 + (1)/(2) kx^2 ψ0 = E ψ0
E = (ħ)/(2) ( (k)/(m) )^1/2

Added by Geoffrey W.

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