An electron sits in an infinite square well on with nm

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

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Stationary states and bound states in one dimension

Problem

An electron sits in an infinite square well on

0 < x < a

with

a = 0.500

nm, rigid walls and

V = 0

inside, so

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

and

E_n = n^2\pi^2\hbar^2/2ma^2

. At

t = 0

it is prepared in

\Psi(x,0) = \frac{1}{\sqrt3}\left[\psi_1(x) + \psi_2(x) + \psi_3(x)\right].

Take

\hbar^2/2m_e = 0.0381

eV nm

^2

and

h = 4.136\times10^{-15}

eV s. ((a)) Give the probability of measuring the energy

E_2

. ((b)) Find

\langle H\rangle

in eV. ((c)) Predict the earliest time

t > 0

at which

|\Psi(x,t)|^2

has returned exactly to its shape at

t = 0

, in fs.