Is the expectation value of momentum always zero for real wavefunctions?

[dyn]

When calculating the expectation value of momentum of a real wavefunction is it always zero ? The momentum operator introduces an i into the integral and with real wavefunctions there is no other i to cancel and all Hermitian operators have real expectation values.

 

[jtbell]

For a stationary state (e.g. any energy eigenstate of a bound system like the infinite square well), <p> must indeed be zero.

However, for a non-stationary state, e.g. a linear combination of energy eigenstates of a bound system, in general <p> ≠ 0. Consider for example $$\Psi(x,t) = \frac{1}{\sqrt{2}} \left[ \psi_1(x)e^{-iE_1 t / \hbar} + \psi_2(x)e^{-iE_2 t / \hbar} \right]$$