jaejoon89
Oct30-09, 01:41 AM
a) Show that the Hamiltonian for the quantum harmonic oscillator in 3D is separable, b) calculate the energy levels.
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a) If it's separable H = H_x + H_y + H_z, so do I just re-arrange the kinetic and potential terms of the Hamiltonian in this case? that seems kind of trivial, as if I'm probably missing something...
b) I assume that if the Hamiltonian is separable, we can use the method of separation of variables to find the energy, i.e., Psi(x,y,z) = X(x)Y(y)Z(z), substitute and then divide by it to find the three energies. So I get, for example,
(1/(X(x))*(-hbar^2 / 2m d^2 / dx^2 + 1/2 k_x x^2)X
How does that correspond to the energy level of the oscillator for the x coordinate, i.e.
hbar*omega (n + 1/2) where n is an integer
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a) If it's separable H = H_x + H_y + H_z, so do I just re-arrange the kinetic and potential terms of the Hamiltonian in this case? that seems kind of trivial, as if I'm probably missing something...
b) I assume that if the Hamiltonian is separable, we can use the method of separation of variables to find the energy, i.e., Psi(x,y,z) = X(x)Y(y)Z(z), substitute and then divide by it to find the three energies. So I get, for example,
(1/(X(x))*(-hbar^2 / 2m d^2 / dx^2 + 1/2 k_x x^2)X
How does that correspond to the energy level of the oscillator for the x coordinate, i.e.
hbar*omega (n + 1/2) where n is an integer