An electron ( ) is confined to an infinite square well of width

PHYSICS · Subjective · 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 (

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

) is confined to an infinite square well of width

L=0.80\,\mathrm{nm}

, with

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

and

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

. At

t=0

it is prepared in

\Psi(x,0)=\big[\psi_1(x)+\psi_2(x)\big]/\sqrt2

. (a) Using

c_n(t)=c_n(0)e^{-iE_nt/\hbar}

, write

\Psi(x,t)

as a global phase times

\big[\psi_1(x)+e^{i\alpha(t)}\psi_2(x)\big]/\sqrt2

, and identify

\alpha(t)

. (b) Using

\psi_1(L-x)=\psi_1(x)

and

\psi_2(L-x)=-\psi_2(x)

, find the first time

t^*>0

at which

|\Psi(x,t)|^2

is the mirror image of

|\Psi(x,0)|^2

about

x=L/2

, and evaluate

t^*

in femtoseconds. (c) Prepare instead

\big[\psi_1(x)+\psi_3(x)\big]/\sqrt2

. Show that its density is symmetric about

x=L/2

at every time, and find

|\Psi(L/2,t)|^2

to show that the density nonetheless changes.