Final answer:
To represent the given potential energy operator in momentum space, one must use the Fourier transform to transition from the coordinate representation to the momentum representation, where the momentum operator in the x-direction is −iħ (d/dx), and the variables are expressed in terms of momentum instead of position.
Step-by-step explanation:
The question concerns the representation of a potential energy operator in quantum mechanics in the momentum space. Given that in coordinate space the potential is represented as “⟦x|V|x'⟧ = δ(x - x′)V(x)”, to find its representation in momentum space, we should express it using the momentum eigenstates. This can be done by using the Fourier transform properties of the delta function and the relation between position and momentum eigenstates.
The momentum operator in the x-direction is often denoted by ℓ and is expressed as −iħ (d/dx). To find the expectation value of the momentum, we integrate the product of the momentum operator acting on a wave function ψ(x) and the complex conjugate of ψ(x), over the entire space. This yields the expectation value of the momentum in the x-direction.
Translating the potential ⟦x|V|x'⟧ to momentum space involves integrating it with respect to the momentum eigenstates. This process effectively replaces the position variables with momentum variables by utilizing Fourier transforms, allowing the potential to be expressed as a function of momentum.