Question

I've been studying the postulates of QM and seeing how to derive important ideas from them. One thing that I haven't been able to derive from them, however, is the identity of the momentum operator.

For simplicity, I'm only thinking about no relativistic effects, no spin, no time-dependent potentials, and one spatial dimension. Also I'm assuming the position operator is simply multiplication by x, as in, I'm in position space. So the Hamiltonian operator is H=−(h²/2m)∇²+V.
I know that the momentum operator is p=−ih(∂/∂x).
But how do I get there from the postulates?

Answer 1

To derive the position-space representation of the momentum operator, start with the Hamiltonian operator in one spatial dimension without relativistic effects, spin, time-dependent potentials, and assume the position operator is multiplication by x.

Step-by-step explanation:

To derive the position-space representation of the momentum operator, we need to consider the wave function and the Hamiltonian operator. In one spatial dimension without relativistic effects, spin, time-dependent potentials, and assuming the position operator is multiplication by x, the momentum operator in the x-direction is given by p = -iħ(∂/∂x), where ∂/∂x represents the partial derivative with respect to x.

To derive this, you can start with the definition of the Hamiltonian operator, H, which is equal to -((ħ^2)/2m)∇^2+V. Then, use the postulates of quantum mechanics, consider the wave function acted upon by the momentum operator, and perform the necessary calculations to obtain the desired momentum operator in position space.