In the frame moving with the group velocity,

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Quantum Mechanics · Wave mechanics in one dimension

Modes, Packets and Media, Ch Water waves and de Broglie waves

Problem

\xi = x - v_{\rm g}t

, the envelope

a(\xi,t)

of a wave packet centred on carrier wavenumber

k_0

obeys, with the time factor

e^{+i\omega t}

i\frac{\partial a}{\partial t} = \frac{\omega''}{2}\frac{\partial^2 a}{\partial \xi^2}, \qquad \omega'' \equiv \left.\frac{d^2\omega}{dk^2}\right|_{k_0}.

((a)) Using the free-particle de Broglie dispersion relation

\omega(k) = \hbar k^2/2m

, find

\omega''

and show that the envelope equation becomes

i\hbar\frac{\partial a}{\partial t} = \frac{\hbar^2}{2m}\frac{\partial^2 a}{\partial x^2},

the free-particle Schrödinger equation with this sign convention. ((b)) Such a packet, initially of width

\sigma_0

, has width

\sigma(t) = \sigma_0\sqrt{1+(t/t_{\rm s})^2}

with spreading time

t_{\rm s} = 2m\sigma_0^2/\hbar

. A neutron of mass

m = 1.675\times10^{-27}\,\mathrm{kg}

is localised to

\sigma_0 = 5.0\,\mathrm{nm}

. Find

t_{\rm s}

, using

\hbar = 1.055\times10^{-34}\,\mathrm{J\,s}