Introduction to Quantum Mechanics · Problem 6.35

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · galilean_transformation

Problem

A Galilean transformation performs a boost from a reference frame S\mathcal{S} to a reference frame S′\mathcal{S}' moving with velocity −v-v with respect to S\mathcal{S} (the origins of the two frames coincide at t=0t = 0). The unitary operator that carries out a Galilean transformation at time tt is Γ^(v,t)=exp⁡[−iℏv(t p^−m x^)].\hat{\Gamma}(v, t) = \exp\left[-\frac{i}{\hbar} v \left(t\,\hat{p} - m\,\hat{x}\right)\right]. (a) Find x^′=Γ^†x^ Γ^\hat{x}' = \hat{\Gamma}^\dagger \hat{x}\, \hat{\Gamma} and p^′=Γ^†p^ Γ^\hat{p}' = \hat{\Gamma}^\dagger \hat{p}\, \hat{\Gamma} for an infinitesimal transformation with velocity δ\delta. What is the physical meaning of your result? (b) Show that Γ^(v,t)=exp⁡[iℏ(m x v−12m v2t)]T^(vt)=T^(vt)exp⁡[iℏ(m x v+12m v2t)].\hat{\Gamma}(v, t) = \exp\left[\frac{i}{\hbar}\left(m\,x\,v - \frac{1}{2} m\,v^2 t\right)\right] \hat{T}(vt) = \hat{T}(vt) \exp\left[\frac{i}{\hbar}\left(m\,x\,v + \frac{1}{2} m\,v^2 t\right)\right]. where T^\hat{T} is the spatial translation operator (Equation 6.3). You will need to use the Baker–Campbell–Hausdorff formula (Problem 3.29). (c) Show that if Ψ\Psi is a solution to the time-dependent Schrödinger equation with Hamiltonian H^=p^22m+V(x)\hat{H} = \frac{\hat{p}^2}{2m} + V(x) then the boosted wave function Ψ′=Γ^(v,t) Ψ\Psi' = \hat{\Gamma}(v, t)\,\Psi is a solution to the time-dependent Schrödinger equation with the potential V(x)V(x) in motion: H^′=p^22m+V(x−v t).\hat{H}' = \frac{\hat{p}^2}{2m} + V(x - v\,t). *Note:* (d/dt) eA^=eA^ (dA^/dt)(d/dt)\,e^{\hat{A}} = e^{\hat{A}}\,(d\hat{A}/dt) only if [A^, (dA^/dt)]=0\left[\hat{A},\,(d\hat{A}/dt)\right] = 0. (d) Show that the result of Problem 2.50(a) is an example of this result.

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