An electron sits in a uniform field that gives a linear potential energy V(x) = x, =…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Position, momentum and the origin of the canonical commutator

Problem

An electron sits in a uniform field that gives a linear potential energy V(x)=αx,α=5.0 eV nm−1,V(x) = \alpha x,\qquad \alpha = 5.0\ \text{eV nm}^{-1}, and its position-space wave function is a Gaussian centred at x0=0.20x_0=0.20 nm, ψ(x)=(πσ2)−1/4exp⁡(−(x−x0)22σ2),σ=0.30 nm.\psi(x) = (\pi\sigma^2)^{-1/4}\exp\left(-\dfrac{(x-x_0)^2}{2\sigma^2}\right),\qquad \sigma = 0.30\ \text{nm}. ((a)) Using the position representation, in which V^=αx^\hat V=\alpha\hat x acts by multiplication, compute ⟨V^⟩\langle\hat V\rangle in electronvolts. ((b)) In the momentum representation x^\hat x acts as the derivative operator iℏ ∂pi\hbar\,\partial_p (the mirror image of ⟨x∣p^∣ψ⟩=−iℏ∂xψ(x)\langle x|\hat p|\psi\rangle=-i\hbar\partial_x\psi(x), with xx and pp exchanged). State which representation is easier for computing ⟨V^⟩\langle\hat V\rangle here, and why.

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