2.I .9 F. 9 \mathrm{~F} \quad

Quantum Mechanics | Part IB, 2001

Consider a solution ψ(x,t)\psi(x, t) of the time-dependent Schrödinger equation for a particle of mass mm in a potential V(x)V(x). The expectation value of an operator O\mathcal{O} is defined as

O=dxψ(x,t)Oψ(x,t)\langle\mathcal{O}\rangle=\int d x \psi^{*}(x, t) \mathcal{O} \psi(x, t)

Show that

ddtx=pm,\frac{d}{d t}\langle x\rangle=\frac{\langle p\rangle}{m},


p=ix,p=\frac{\hbar}{i} \frac{\partial}{\partial x},

and that

ddtp=Vx(x)\frac{d}{d t}\langle p\rangle=\left\langle-\frac{\partial V}{\partial x}(x)\right\rangle

[You may assume that ψ(x,t)\psi(x, t) vanishes as x±.]x \rightarrow \pm \infty .]

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