Question

If a particle in the harmonic oscillator potential is initially in a momentum eigenstate can the expectation value of an operator then be time dependent? I get that i doesen't happen if the particle is initially in an energy eigenstate but that it may happen if it is initially in a superposition of different energy eigenstates.

Answer 1

We write time–independent Schrodinger equation as $\mathcal{H}$ | $\psi$  $>=$ $E_n\mid\psi$ to this point. Since the Hamiltonian is the energy operator, the eigenvalues are necessarily energy eigenvalues. The state vector is assumed to be a linear combination of all energy eigenvectors. If we specifically measure the eigenvalue En , then the state vector is necessarily the associated eigenvector which can be written $\mid E_n>$ . The time–independent Schrodinger equation written as $\mathcal{H}\mid E_n>=E_n\mid E_n>$ is likely a better expression for the development..