A wave packet is built not from a Gaussian but from a flat momentum amplitude

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Wave mechanics in one dimension

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Free particles, wave packets and tunnelling

Problem

A wave packet is built not from a Gaussian but from a flat momentum amplitude,

\phi(k) = A

for

k_0 - b < k < k_0 + b

and

\phi(k) = 0

outside, with

A

real. Take

k_0 = 10.0\ \mathrm{nm^{-1}}

and

b = 0.50\ \mathrm{nm^{-1}}

. ((a)) Find

A

from the normalisation

\int|\phi(k)|^2dk = 1

. ((b)) Use the Fourier pair to find

\Psi(x,0)

in closed form. ((c)) Evaluate

\Psi(0,0)

, the packet's peak amplitude.