 2D Harmonic Oscillator and Spin

Subject Physics Quantum Physics

Question

1. The potential function for the 2D harmonic oscillator is: V(x,y)=(1/2)mw²(x²+y²), where x and y are the 2D cartesian coordinates.
(a) Please write down the Schrodinger equation in x and y, then solve it using the separation of variables to derive the energy spectrum. Does it make sense relative to the 1D and 3D versions ([2,61] and [4.189])?
(b) Please find the degeneracy d(n) as a function of n for each energy En.

2. Spin. Please work on problem 4.49, but instead of the provided spinor X=A(1-i, 2) use the following spinor X=A(1,2-i).
(a) Please use normalization to find A.
(b) What values of Sz could you get? What is the probability of each? What is the expectation value of Sz?
(c) Same for Sx
(d) Same for Sy.

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