An electron is confined in an infinite square well placed symmetrically about the…

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 is confined in an infinite square well placed symmetrically about the origin:

V = 0

for

-L/2 < x < L/2

and

V = \infty

outside, with

L = 0.60

nm. Take

\hbar^2/2m_e = 0.0381

eV nm

^2

. ((a)) Starting from the time-independent Schrödinger equation inside the well,

\psi'' = -k^2\psi

with

k = \sqrt{2m_eE}/\hbar

, and the conditions

\psi(-L/2) = \psi(L/2) = 0

, derive the allowed energies and the normalised eigenfunctions, showing that they fall into a cosine family and a sine family and stating which is which. ((b)) Evaluate

E_1

and

E_2

in eV for the stated width. ((c)) The rigid walls are now replaced by a well of finite depth:

V = -V_0

for

|x| < a

with

a = L/2 = 0.30

nm and

V_0 = 5.0

eV, and

V = 0

outside. The chapter's even bound states satisfy

\tan z = \sqrt{(z_0/z)^2 - 1}

, where

z = ka

and

z_0 = (a/\hbar)\sqrt{2m_eV_0}

. Find

z_0

and, by considering where the two curves can cross, state how many even bound states this well has.

Related theory

Quantum Mechanics

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