For a free electron of mass

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

For a free electron of mass m=9.11×10−31 kgm=9.11\times10^{-31}\,\text{kg}, consider the improper momentum eigenstates ∣p⟩|p\rangle with position representative ⟨x∣p⟩=12πℏ eipx/ℏ.\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}}\,e^{ipx/\hbar}. ((a)) Using the derived relation ⟨x∣p^∣ψ⟩=−iℏ ∂xψ(x)\langle x|\hat p|\psi\rangle = -i\hbar\,\partial_x\psi(x), verify that ⟨x∣p⟩\langle x|p\rangle is an eigenfunction of p^\hat p with eigenvalue pp, and show it is also an eigenfunction of H^=p^2/2m\hat H=\hat p^2/2m with eigenvalue E=p2/2mE=p^2/2m. ((b)) The free-particle Hamiltonian satisfies [H^,p^]=0[\hat H,\hat p]=0 because it is translation invariant. Explain, using the shared-eigenbasis result for compatible observables, why ∣p⟩|p\rangle can be chosen as a simultaneous eigenstate of H^\hat H and p^\hat p without extra work, and explain why H^\hat H alone is not a complete set of commuting observables (CSCO) for this system even though {H^,p^}\{\hat H,\hat p\} is. ((c)) Confirm that these states obey the delta-normalisation ⟨p∣p′⟩=δ(p−p′)\langle p|p'\rangle = \delta(p-p'), using ∫−∞∞12πℏei(p−p′)x/ℏ dx=δ(p−p′)\displaystyle\int_{-\infty}^{\infty}\frac{1}{2\pi\hbar}e^{i(p-p')x/\hbar}\,dx = \delta(p-p'). ((d)) For this electron with p=1.5×10−25 kg m s−1p = 1.5\times10^{-25}\,\text{kg m s}^{-1}, evaluate the kinetic energy E=p2/2mE=p^2/2m in electronvolts.

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