Quantum Mechanics: Concepts and Applications · Problem 3.14

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · ehrenfest-free-particle-fluctuations

Problem

Consider a one-dimensional free particle of mass

m

whose position and momentum at time

t = 0

are given by

x_0

and

p_0

, respectively. (a) Calculate

\langle \hat{P} \rangle(t)

and show that

\langle \hat{X} \rangle(t) = p_0 t/m + x_0

. (b) Show that

d\langle \hat{X}^2 \rangle/dt = 2\langle \hat{P}\hat{X} \rangle/m + i\hbar/m

and

d\langle \hat{P}^2 \rangle/dt = 0

. (c) Show that the position and momentum fluctuations are related by

d^2(\Delta x)^2/dt^2 = 2(\Delta p)^2/m^2

and that the solution to this equation is given by

(\Delta x)^2 = (\Delta p)_0^2 t^2/m^2 + (\Delta x)_0^2

where

(\Delta x)_0

and

(\Delta p)_0

are the initial fluctuations.