1. The problem statement, all variables and given/known data I apologize, this is not really a homework problem. I have an exam coming up, and I need to be able to explain the difference between a stationary/non-stationary quantum state in a qualitative way, and in what cases these states have time dependent probabilities. I am hoping someone can correct my understanding if it is wrong. Thank you! 2. Relevant equations 3. The attempt at a solution A stationary state is any quantum state which consists of only one eigenstate of the Hamiltonian H. For example, if a spin 1/2 system in the z basis with a magnetic field in the z-direction, a stationary state we may work with is |ψ(0)> = a|+>. In the same system, a non-stationary state would be |ψ(0)> = a|+> + b|->. The important distinction is that stationary states are composed of one energy eigenstate of the H, and non-stationary states are a superposition of n energy eigenstates of H (for a spin 1/2 system, this would only be up to n=2). Probabilities of any state are time independent if: The state we are measuring the probability in is stationary OR We are measuring the probability in a basis that commutes with the basis of the Hamiltonian. Probabilities of any state are time dependent if: The state is non-stationary and we are measuring the probability in a basis that does not commute with the basis of the Hamiltonian. Also sorry if this is a dumb question, but is it possible to have a state that is not made up of energy eigenvalues/eigenstates of the Hamiltonian? I don't think you can.