In quantum mechanics the angular momentum operator is defined as $$ \mathbf{\hat L}=\mathbf{\hat x} \times \mathbf{ \hat p} $$ This definition explicitly depends on the choice of the origin of the coordinate system. For a state $|\psi\rangle$ that does not move, i.e. has a vanishing expectation value of momentum, it is easy to show that $\langle \psi | \mathbf{\hat L} |\psi\rangle$ does not depend on the choice of the origin. The expectation value of $\mathbf{\hat L^2}$, however, does depend on the choice of the origin point. My question: is there a simple transformation law of $\langle \psi | \mathbf{\hat L^2} |\psi\rangle$ under translations. To be specific, in the hydrogen atom problem what is the expectation value of the angular momentum square measured with respect to a point away from the nucleus.
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