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cscott
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Homework Statement
Consider a particle of mass [itex]m[/itex] moving in a 3D potential
[tex]V(\vec{r}) = 1/2m\omega^2z^2,~0<x<a,~0<y<a[/tex].
[tex]V(\vec{r}) = \inf[/tex], elsewhere.
2. The attempt at a solution
Given that I know the solutions already for a 1D harmonic oscillator and 1D infinite potential well I'm going to combine them [itex]E_x + E_y + E_z = E[/itex], and [itex]\psi=\psi(z)\psi(x)\psi(y)[/itex] as for separation of variables of 3D Schrodinger.
Therefore,
[tex]E = (n_z+1/2)\hbar\omega + \frac{\pi^2\hbar^2}{2ma^2}(n_x^2+n_y^2)[/tex]
[tex]\psi=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\frac{1}{\sqrt{2^{n_z}n_z!}}H_{n_z}(\zeta)e^{-\zeta^2/2}\sqrt{\frac{2}{a}}\sin\left(\frac{n_x\pi}{a}x\right)\sqrt{\frac{2}{a}}\sin\left(\frac{n_y\pi}{a}y\right)[/tex]
where [itex]H[/itex] are the Hermite polynomials and [tex]\zeta=\sqrt{m\omega/\hbar}z[/tex].
Is this a correct approach? I couldn't see how going all through the separation of variables for 3D Schrodinger would give me a different answer.
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