Model a neutron in a small nucleus as a particle in an infinite spherical well of…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Central potentials and canonical three-dimensional systems · Central potentials and canonical three-dimensional systems

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The Schrödinger equation in three dimensions and central potentials

Problem

Model a neutron in a small nucleus as a particle in an infinite spherical well of radius

a

: the potential energy is zero for

r < a

and infinite for

r \ge a

. The levels are

E = \dfrac{\hbar^2\beta^2}{2m_n a^2}

with

\beta = n\pi

for

l = 0

and

\beta = 4.49

for the lowest

l = 1

level. Take

\dfrac{\hbar^2}{2m_n} = 20.7

MeV fm

^2

. ((a)) Compute the lowest level of a neutron in a well of radius

a = 4.0

fm. ((b)) Compute the lowest

l = 1

level in the same well. ((c)) Predict, from the form of the level formula, whether the lowest level rises or falls when the radius is increased to

5.0

fm, and then compute its value.

Related theory

Quantum Mechanics

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