Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 15A (Quantum Mechanics)

For a test wavefunction $f$,Thus the canonical commutation relation is

The time-dependent Schrödinger equation is

Differentiating $⟨O⟩=⟨\psi |O|\psi ⟩$ and using the Schrödinger equation and its adjoint gives the Ehrenfest theorem

Using $[A,BC]=[A,B]C+B[A,C]$ and $[x,p_x]=i\hslash I$,

With $L=x{p}_y-y{p}_x$, the canonical commutators giveIt follows thatTherefore rotational symmetry of the isotropic oscillator gives

In polar coordinates, $z=r{e}^{i\varphi }$ andConsequentlyThe two states are orthogonal. They are degenerate energy eigenstates, so their coefficients acquire the same overall time-dependent phase; equivalently, $[L,H]=0$ makes $⟨L⟩$ constant. For the normalized statethe angular momentum of a complex-coordinate Gaussian state is therefore