My book mentions that if you have an infinite (or even very deep) square well, a particle trapped in it will have the expectation value for momentum $p_x$ zero. How does one infer this using the uncertainty principle? Is it because we are sure that the particle is located inside the well and hence it has no uncertainty in it position?
[Physics] Why is the expectation value of momentum of a particle in an infinite potential well zero
heisenberg-uncertainty-principlemomentumpotentialquantum mechanicsschroedinger equation
Best Answer
Not via uncertainty principle but via Ehrenfest theorem --
If the particle is stuck at the bottom of the well, that means its expected position is constant, so $\frac{d}{dt}\langle x\rangle = 0$. Ehrenfest theorem says $ \frac{d}{dt}<x> = \frac{<p>}{m}$ so $\langle p \rangle$ must be zero.