Why Must the Expectation Value of Momentum Squared Be Strictly Positive?

[c299792458]

Homework Statement

Why is \langle p^2\rangle >0 where p=-i\hbar{d\over dx}, (noting the ***strict*** inequality) for all normalized wavefunctions? I would have argued that because we can't have \psi=constant, but then I thought that we can normalize such a wavefunction by using periodic boundary conditions... So I don't how to argue that the inequality should be strict... Is it that otherwise it would be trivial?

Homework Equations

p=-i\hbar{d\over dx}

The Attempt at a Solution

clearly, \langle \psi|p^2|\psi\rangle = \langle p\psi|p\psi\rangle \geq 0 since p is Hermitian. But why the strict inequality??

 

[Thaakisfox]

When would that inner product with zero? and what would happen then? ;)

 

[dextercioby]

Well, if there exists \psi \in D(p), so that \langle \psi,p^2 \psi\rangle = 0, then ||p\psi|| has 0 norm. Which vector has 0 norm ? Is it normalizable ?