An electron's position-space wave function at is a Gaussian envelope on a plane wave,…

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's position-space wave function at t=0t=0 is a Gaussian envelope on a plane wave, ψ(x)=N eik0x exp⁡(−x22σ2),k0=5.0×109 m−1, σ=2.0 nm,\psi(x) = N\,e^{ik_0x}\,\exp\left(-\dfrac{x^2}{2\sigma^2}\right),\qquad k_0 = 5.0\times10^{9}\ \text{m}^{-1},\ \sigma = 2.0\ \text{nm}, with NN a normalisation constant. ((a)) Using ⟨x∣p^∣ψ⟩=−iℏ ∂xψ(x)\langle x|\hat p|\psi\rangle=-i\hbar\,\partial_x\psi(x), derived from the infinitesimal translation, write ⟨x∣p^∣ψ⟩\langle x|\hat p|\psi\rangle as [function of x]×ψ(x)[\text{function of }x]\times\psi(x). ((b)) Evaluate the coefficient of ψ(x)\psi(x) at x=0x=0, in kg m s−1^{-1}, and say what this shows about the state near the peak of the envelope. ((c)) State how the same information appears in the momentum representation, where ψ~(p)\tilde\psi(p) is obtained from ψ(x)\psi(x) by a Fourier transform.

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