An electron of mass is prepared at as a free Gaussian packet of position spread…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations

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

Problem

An electron of mass m=9.11×10−31 kgm = 9.11\times10^{-31}\,\mathrm{kg} is prepared at t=0t=0 as a free Gaussian packet of position spread (standard deviation) σ0=1.0 nm\sigma_0 = 1.0\,\mathrm{nm}. ((a)) Using the free propagator's known effect on a Gaussian, σ(t)=σ01+(ℏt2mσ02)2\sigma(t) = \sigma_0\sqrt{1+\left(\dfrac{\hbar t}{2m\sigma_0^2}\right)^2}, find σ(t)\sigma(t) at t=1.0×10−12 st=1.0\times10^{-12}\,\mathrm{s}. ((b)) The same packet has ⟨x^⟩=0\langle\hat x\rangle=0 and ⟨p^⟩=2.0×10−25 kg m s−1\langle\hat p\rangle = 2.0\times10^{-25}\,\mathrm{kg\,m\,s^{-1}} at t=0t=0. Using x^H(t)=x^+p^t/m\hat x_H(t)=\hat x+\hat p t/m, find ⟨x^⟩\langle\hat x\rangle at the same time tt.

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