Chapter 6, Problem 6.35P

(a)

To determine

The value of ${\widehat{x}}^{\prime }={\widehat{\Gamma }}^{†}\widehat{x}\widehat{\Gamma }$ and ${\widehat{p}}^{\prime }={\widehat{\Gamma }}^{†}\widehat{p}\widehat{\Gamma }$ for an infinitesimal transformation with velocity $\delta$.

(b)

To determine

Show that $\widehat{\Gamma }\left(v,t\right)=\widehat{T}\left(vt\right)\exp \left[\frac{i}{\hslash }\left(mxv+\frac{1}{2}m{v}^{2}t\right)\right]$.

(c)

To determine

Show that the $\Psi$ is a solution to the time-dependent Schrödinger equation with Hamiltonian $\widehat{H}=\frac{{\widehat{P}}^2}{2m}+V\left(x\right)$ , then the boosted wave function ${\Psi }^{\prime }=\widehat{\Gamma }\left(v,t\right)\Psi$ is a solution to the time-dependent Schrödinger equation with the potential $V\left(x\right)$ in motion is $\widehat{H}=\frac{{\widehat{P}}^2}{2m}+V\left(x-vt\right)$.

(d)

To determine

Show that the boosted wave function ${\Psi }^{\prime }\left(x,t\right)=\frac{\sqrt{m\alpha }}{\hslash }{e}^{-m\alpha {\left|x-vt\right|}^2/{\hslash }^2}{e}^{-i\left[\left(E+m{v}^2/2\right)t-mvx\right]/\hslash }$.