An electron sits in a one-dimensional infinite square well of width nm occupying

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 a one-dimensional infinite square well of width

a = 1.2

nm occupying

0 < x < a

, with

V = 0

inside and

V

infinite outside; the walls are impenetrable and the electron is non-relativistic. Let

\psi_n

be the normalised energy eigenfunctions and

E_n

the energy eigenvalues. At

t = 0

the wave function is the equal superposition

\Psi(x,0) = \tfrac{1}{\sqrt2}\big[\psi_1(x) + \psi_2(x)\big].

((a)) Find

E_1

and

E_2

in eV, using

\hbar^2/2m_e = 0.0381

eV nm

^2

. ((b)) Evaluate the expectation value

\langle x\rangle

t = 0

, in nm. (You will need

\int_0^a x\,\psi_1\psi_2\,dx

; obtain it with a product-to-sum identity and integration by parts.) ((c)) Find the period, in fs, with which

\langle x\rangle

oscillates as the state evolves. ((d)) The

n = 2

eigenfunction is a standing matter wave of wavelength

\lambda_2 = a

. Use the de Broglie relation to find the corresponding momentum in kg m s

^{-1}

, and check that

p^2/2m_e

reproduces your

E_2