An electron moves in one dimension in the delta-function well with eV nm

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 delta-function well

V(x) = -\alpha\delta(x)

with

\alpha = 0.250

eV nm. Its single bound state has energy

E = -m_e\alpha^2/2\hbar^2

and normalised wave function

\psi(x) = \sqrt{\kappa}\,e^{-\kappa|x|}

with

\kappa = m_e\alpha/\hbar^2

. ((a)) Find the bound-state energy, in eV. ((b)) Find the decay length

1/\kappa

, in nm. ((c)) Find the probability of finding the electron farther from the origin than one decay length, that is at

|x| > 1/\kappa

.Use

\hbar^2/2m_e = 0.0381

eV nm

^2