Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 3 / 7G / Solution

For a state in the domain of the operator, its expectation value isFor a normalized state the denominator is one. Use an orthonormal energy eigenstate basis, choosing orthogonal vectors within degenerate eigenspaces. Inserting an expansion into the time-dependent Schrödinger equation gives $i\hslash {\dot{a}}_n(t)=E_n{a}_n(t)$. ConsequentlyThe phases preserve normalization. Projection onto a specified basis eigenstate gives probability $\boxed{|a_p{|}^2}$, independent of time. If the measurement distinguishes only the energy and $E_p$ is degenerate, sum $|a_n{|}^2$ over all $n$ with $E_n=E_p$.

Using orthonormality to eliminate all cross terms,This is time independent whenever the expectation is defined. The time-independent Hamiltonian changes only relative phases, not the energy probabilities.