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 · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Hilbert space and Dirac notation

Problem

An electron of mass

m_e = 9.109\times10^{-31}\,\mathrm{kg}

is confined to an infinite square well of width

L=2.00\,\mathrm{nm}

, with impenetrable walls at

x=0

and

x=L

. The energy eigenstates

|n\rangle

satisfy

\langle x|n\rangle=\sqrt{2/L}\sin(n\pi x/L)

and

\hat H|n\rangle=E_n|n\rangle

with

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

, where

\hbar = 1.0546\times10^{-34}\,\mathrm{J\,s}

. The electron is prepared in the normalised state

|\psi\rangle=\frac{1}{\sqrt5}|1\rangle+\frac{2}{\sqrt5}|2\rangle.

((a)) Let

|\psi'\rangle=e^{i\pi/4}|\psi\rangle

. Compute the probabilities of finding energies

E_1

and

E_2

for both

|\psi\rangle

and

|\psi'\rangle

, and compute

\langle\hat H\rangle

for both. State what this shows about

|\psi\rangle

and

|\psi'\rangle

as physical states. ((b)) Insert the resolution of the identity

\sum_n|n\rangle\langle n|=\mathbb 1

into

\langle\psi|\hat H|\psi\rangle

to write the energy expectation value as a sum over

n

, then evaluate it numerically in electronvolts.