Starting from the equation for the matter wave of a free particle, derive a plausible form of operator for the momentum

The wave function of a free particle is given by the de Broglie relation:

ψ = A * exp(i(kx - ωt))

where k is the wave number, ω is the angular frequency, x is the position, t is the time, and i is the imaginary unit.

The momentum of the particle is given by the de Broglie relation:

p = h * k

where h is Planck's constant.

We can express the wave number k in terms of the momentum p and Planck's constant h:

k = p / h

Substituting this into the wave function gives:

ψ = A * exp(i(px/h - ωt))

Now, we want to find an operator that, when applied to the wave function, gives the momentum of the particle. The derivative of the wave function with respect to x is:

dψ/dx = i * p/h * A * exp(i(px/h - ωt))

= i * p/h * ψ

Solving for p gives:

p = h/i * dψ/dx

This suggests that a plausible form for the momentum operator is:

P = -i * h * d/dx

This operator, when applied to the wave function, gives the momentum of the particle.