An electron moves in one dimension in the potential V(x) = cases & x 0 0 & 0 < x < L…

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 moves in one dimension in the potential

V(x) = \begin{cases}\infty & x \le 0\\ 0 & 0 < x < L\\ V_0 & x \ge L\end{cases}

with

L = 1.2

nm and

V_0 = 2.00

eV. The infinite wall forces

\psi(0) = 0

. A numerical solution gives exactly three bound states, at

E_1 = 0.21

eV,

E_2 = 0.83

eV and

E_3 = 1.78

eV above the floor of the well. Sketch the three eigenfunctions, using node counting and the curvature rule, and read off the following. Take

\hbar^2/2m_e = 0.0381

eV nm

^2

. ((a)) How many nodes does each of

\psi_1

\psi_2

and

\psi_3

have in

x > 0

(the wall at

x = 0

is not counted)? Give the three numbers in order. ((b)) In the region

x > L

each eigenfunction falls off as

e^{-\kappa x}

with

\kappa = \sqrt{2m(V_0 - E)}/\hbar

. Calculate the decay length

1/\kappa

for

n = 1

and for

n = 3