1. The problem statement, all variables and given/known data A quantum-mechanical harmonic oscillator with frequency ω has Hamiltonian eigenstates |n with eigenvalues En = (n + 1/2) ħω. Initially, the oscillator is in the state (|0> + |1>)/√2. Write down how the state of the oscillator evolves as a function of time t. Calculate the first time for which the time-evolved state is orthogonal to the initial one. 3. The attempt at a solution I know how to evolve the states in time using the exponential solution to the TDSE. |ψ(t)> = (e-iωt/2|0> + e-3iωt/2|1>)/√2. This is fine. now I want the probability of finding the system in state |0> at time t. I bra through with <0| <0|ψ(t)> = (e-iωt/2<0|0> + e-3iωt/2<0|1>)/√2. <0|1> are orthogonal states so their inner product dissapears. <0|0> = 1 <0|ψ(t)> = e-iωt/2/√2 The probability of finding the state is |<0|ψ(t)>|2 P(|0>) = 1/2 Now here's the problem. When I take the mod square, the time dependence disappears, implying that the probability is constant for all time. This isn't true, at least I don't think it's true. Isn't the probability supposed to oscillate? I've forgotten how this oscillation comes about in bra-ket notation.