Solution to 2D Quantum Mechanical Harmonic Oscillator
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A particle of mass m moves in two dimensions under the influence of the potential V(x,y)=1/2 m?^2 (((6x)^2)-2xy+(6y)^2 ). Using the rotated coordinates u=(x+y)/?2 and w=(x-y)/?2 show that the Schrödinger equation in the new coordinates (u,w) is
-(?^2)/2m ((d^2/du^2) +(d^2/dw^2))?(u,w)+V ?(u,w)?(u,w)=E?(u,w)
Where V ?(u,w) should be found.
Let ?(u,w)=U(u)W(w). Use the technique of separation of variables to show that U(u) and W(w) satisfy the Schrödinger equations for the one dimensional quantum harmonic oscillator. Construct the allowed energy levels E_(n,m) and write down the corresponding wavefunction ?_(m,n) (x,y).
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Solution Summary
This solution solves a version of the quantum mechanical harmonic oscillator in two dimensions, in which a coordinate transformation and separation of variables are used.
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