Introduction to Quantum Mechanics · Problem 2.51

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · free_fall

Problem

Free fall. Show that Ψ(x,t)=Ψ0(x+12gt2,t)exp⁡[−imgtℏ(x+16gt2)]\Psi(x, t) = \Psi_0\left(x + \frac{1}{2}gt^2, t\right)\exp\left[-i\frac{mgt}{\hbar}\left(x + \frac{1}{6}gt^2\right)\right] satisfies the time-dependent Schrödinger equation for a particle in a uniform gravitational field, V(x)=mgx,V(x) = mgx, where Ψ0(x,t)\Psi_0(x, t) is the free gaussian wave packet (Equation 2.111). Find ⟨x⟩\langle x \rangle as a function of time, and comment on the result.

Related theory

Browse the problem library