An electron of mass moves freely (no potential).

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch Time evolution as a unitary group, and the three pictures

Problem

An electron of mass

m = 9.11\times10^{-31}\,\mathrm{kg}

moves freely (no potential). ((a)) Using the free-particle propagator

K(x,t;x',t') = \sqrt{\dfrac{m}{2\pi i\hbar(t-t')}}\,e^{im(x-x')^2/2\hbar(t-t')}

, find the modulus

|K|

and the phase (mod

2\pi

) for

x-x' = 2.0\,\mathrm{nm}

and

t-t' = 1.0\times10^{-15}\,\mathrm{s}

. ((b)) A wave packet has, at

t=0

, mean position

\langle\hat x\rangle = 0

and mean momentum

\langle\hat p\rangle = 1.0\times10^{-25}\,\mathrm{kg\,m\,s^{-1}}

. Using the Heisenberg-picture operator

\hat x_H(t) = \hat x + \hat p t/m

, find

\langle\hat x\rangle

t=1.0\times10^{-15}\,\mathrm{s}