An electron (mass ) sits in the energy eigenstate of a one-dimensional harmonic…

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Compatible observables and uncertainty as a theorem

Problem

An electron (mass me=9.109×10−31 kgm_e = 9.109\times10^{-31}\,\mathrm{kg}) sits in the n=1n=1 energy eigenstate of a one-dimensional harmonic oscillator of angular frequency ω=3.0×1014 rad s−1\omega = 3.0\times10^{14}\,\mathrm{rad\,s^{-1}}. For any energy eigenstate of this oscillator ⟨x^⟩=⟨p^⟩=0\langle\hat x\rangle=\langle\hat p\rangle=0, and the nnth eigenstate satisfies ⟨x^2⟩=(n+12)ℏmeω,⟨p^2⟩=(n+12)ℏmeω.\langle\hat x^2\rangle = \left(n+\tfrac12\right)\frac{\hbar}{m_e\omega}, \qquad \langle\hat p^2\rangle = \left(n+\tfrac12\right)\hbar m_e\omega. (a) Compute Δx\Delta x and Δp\Delta p for n=1n=1, and evaluate Δx Δp\Delta x\,\Delta p numerically in J s\mathrm{J\,s}. (b) Write the full uncertainty relation (Δx)2(Δp)2≥14∣⟨[x^,p^]⟩∣2+14∣⟨{Δx^,Δp^}⟩∣2(\Delta x)^2(\Delta p)^2 \ge \tfrac14|\langle[\hat x,\hat p]\rangle|^2 + \tfrac14|\langle\{\Delta\hat x,\Delta\hat p\}\rangle|^2, evaluate the commutator term numerically, and confirm the inequality holds for this state. (c) Show that ⟨{Δx^,Δp^}⟩=0\langle\{\Delta\hat x,\Delta\hat p\}\rangle = 0 for this state, using that its wavefunction can be chosen real. (d) Given that the correlation term vanishes, decide whether the n=1n=1 state saturates the uncertainty bound, and justify the answer from the saturation condition rather than from the numbers of part (a) alone.

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