An electron of mass is confined to an infinite square well of width

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch The wave function is not the state

Problem

An electron of mass

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

is confined to an infinite square well of width

L = 1.20\,\mathrm{nm}

, with normalised eigenfunctions

\psi_n(x) = \sqrt{2/L}\sin(n\pi x/L)

and energies

E_n = n^2\pi^2\hbar^2/(2mL^2)

. At

t=0

the state is prepared as

\Psi(x,0) = \frac{1}{\sqrt2}\big[\psi_2(x) + \psi_3(x)\big].

((a)) Working in the energy description (the list of amplitudes

c_n

), give

P(E_2)

and

P(E_3)

. ((b)) From the energy list, give the numerical value of

E_3 - E_2

in joules, and hence the beat angular frequency

\omega_{32} = (E_3-E_2)/\hbar

at which the position density oscillates. ((c)) Evaluate the position probability density

|\Psi(x,0)|^2

x = L/2

, using the position description directly, and check that it equals

|c_3|^2|\psi_3(L/2)|^2

, saying why

c_2

contributes nothing at this point.