Can momentum never be zero in quantum mechanics?

Question

I have seen Zetilli's QM book deals with $E>V$ and $E< V$ (tunnelling) in case of the potential wells deliberately avoiding the E=V case,So I thought maybe something is intriguing about this and made this up.

Suppose the total energy of the particle is equal to its potential energy.Then its kinetic energy should be zero,(Speaking non relativistically). But Kinetic energy operator is $\hat{T}=\hat{p}^2/2m$ (where $\hat{p}=-i\hbar\frac{\partial}{\partial x}$), So clearly since Kinetic energy is 0 here,momentum eigenvalue will also vanish.

Now, Putting $E=V$ in time-independent Schrodinger equation (1D) we get, $$\frac{\partial^2\psi}{\partial x^2}=\frac{2m(E-V)}{\hbar^2}\psi\implies\frac{d^2\psi}{d x^2}=0\implies\psi=Ax+B$$ where $A$ and $B$ are arbitrary constants. Since,the Wave function must vanish at $\pm\infty$, $A=0$,hence the wave function equals a constant=$B$ and is not normalizable. So,a particle with no momentum(or kinetic energy),gives a physically unrealizable wave function!

Does this imply $E=V$ is a restricted critical case or momentum cant be zero in quantum mechanics or did i just go wrong somewhere?

Answer 1

You are not wrong, but it is worth noting that the same thing is true of any momentum eigenstate (or closely related unbound eigenstate of a Hamiltonian with a potential well in it). Explicitly $$ -i\hbar\frac{\partial}{\partial x} \psi(x) = p\,\psi(x) $$ then $$ \psi = A e^{i \frac{p}{\hbar}x} $$ which is not normalisable either. This means that we can never truly realise a momentum eigenstate, but we can still use them as a basis for physically realisable states usng the rigged Hilbert space formalism.

So yes we cannot realise a state with exactly zero momentum, but this is not a special property of the zero momentum state.

Answer 2

I would like to add two points to the accepted answer:

• If you use the periodic boundary conditions trick to normalize the momentum eigenstates, then all the momentum eigenstates become normalizable, including the zero momentum eigenstate.

• The discussion of tunneling in QM books is usually within the quasi-classical approximation, which breaks down when the difference $V - E$ is small. Thus, one usually treats the cases when the particle energy is well below or well above the barrier.