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Consider a spin-1/2 particle in a magnetic field (say in z direction) and in a harmonic potential. For the 3D harmonic oscillator component, The Hamiltonian $H_1= \frac{p^2}{2m}+\frac{1}{2}m\omega ^2r^2.$ For the spin component, the Hamiltonian $H_2=-\gamma B_z S_z$, where $\gamma$ is the gyroscopic ratio.

Question: Are the $H_1$ and $H_2$ simultaneously diagonalizable?

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  • $\begingroup$ Compute $[H_1,H_2]$. $\endgroup$
    – user111476
    Commented Nov 5, 2019 at 6:47

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