Suppose the total energy of the particle is equal to its potential energy. Then its kinetic energy should be zero, speaking non-relativistically. But the Kinetic energy operator is hatT= hatp²/2m (where hatp=-i hbar frac partial partial x). So clearly since the Kinetic energy is 0 here, the momentum eigenvalue will also vanish. Now, putting E=V in the time-independent Schrodinger equation (1D) we get, frac partial² psi partial x²= frac2m(E-V) hbar² psi implies fracd² psid x²=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 can't be zero in quantum mechanics or did I just go wrong somewhere?