An electron (mass ) moves in one dimension in a wide region where the potential…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension

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

Problem

An electron (mass

m_e

) moves in one dimension in a wide region where the potential energy has the constant value

V = 2.00

eV, the same at every time. The electron is in a stationary state of energy

E = 5.00

eV, so its wave function is the separated product

\Psi(x,t) = \psi(x)f(t)

. ((a)) Write down the time factor

f(t)

that separation of variables gives, and find the time in which its phase advances by

2\pi

. ((b)) Show that in this region the time-independent Schrödinger equation takes the form

\psi'' = -k^2\psi

, and evaluate

k

in nm

^{-1}

. ((c)) Hence give the wavelength

\lambda = 2\pi/k

of the sinusoidal solutions, in nm. Use

\hbar^2/2m_e = 0.0381

eV nm

^2

h = 6.626\times10^{-34}

J s and

1

= 1.602\times10^{-19}