A free electron of mass has Heisenberg-picture position operator

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch Time evolution as a unitary group, and the three pictures

Problem

A free electron of mass m=9.11×10−31 kgm=9.11\times10^{-31}\,\mathrm{kg} has Heisenberg-picture position operator x^H(t)=x^+p^ t/m\hat x_H(t)=\hat x+\hat p\,t/m, obtained by solving the Heisenberg equation of motion for the free particle. ((a)) Show that [x^H(t),x^H(0)]=−iℏt/m[\hat x_H(t),\hat x_H(0)] = -i\hbar t/m. ((b)) Using the generalised uncertainty relation ΔA ΔB≥12∣⟨[A^,B^]⟩∣\Delta A\,\Delta B \ge \frac12\left|\langle[\hat A,\hat B]\rangle\right| (from the Schwarz inequality), find the lower bound on Δx(t) Δx(0)\Delta x(t)\,\Delta x(0) at t=1.0×10−9 st=1.0\times10^{-9}\,\mathrm s.

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