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CAF123
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Homework Statement
Let the time evolution of a system be determined by the following Hamiltonian: $$\hat{H} = \gamma B \hat{L}_y$$ and let the system at t=0 be described by the wave function ##\psi(x,y,z) = D \exp(-r/a)x,## where ##r## is the distance from the origin in spherical polars. Find the state of the system at any time t. Deduce the expectation value of ##\hat{L}_z## in the state ##\psi##.
Homework Equations
In Cartesians, $$\hat{L}_y = -i\hbar \left(z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z}\right)$$
The Attempt at a Solution
Computing the Hamiltonian gives the new wave function ##-Di \hbar \gamma B z \exp(-r/a)##. To obtain the time dependence, by solving the spatial-independent Schrodinger equation, $$\psi(x,y,z,t) = \psi(x,y,z)\exp(-Et/\hbar).$$ I am not sure if this is correct since ##\psi(x,y,z)## is not an eigenstate of ##\hat{L}_y## and hence not of ##\hat{H}## either.
Thanks.