Eigenvalues of a "unusual" Hamiltonian of a harmonic oscillator

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Homework Statement
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for d), I am a bit confused. I have two trains of thoughts here

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any thoughts on which answer is correct, and why the other one is incorrect? Both seem like valid solutions to me. Or is the question ambiguous?

thanks
 

Thread 'Help needed with Polarisation (Rotated Waveplates)'

(a) The polarisation pattern is elliptical with maximum (1,1) and minimum (-1,-1), and anticlockwise in direction. (b) I know the solution is a quarter-wave plate oriented π/4, and half-wave plate at π/16, but don't understand how to reach there. I've obtained the polarisation vector (cos π/8, isin π/8) so far. I can't find much online guidance or textbook material working through this topic, so I'd appreciate any help I can get. Also, if anyone could let me know where I can get more...
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