A single neutron inside a nucleus is modelled in two crude ways

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · 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

A single neutron inside a nucleus is modelled in two crude ways. Model A is a cubic box of side

6.0

fm with impenetrable walls (

V = 0

inside, infinite outside). Model B is an isotropic three-dimensional harmonic oscillator,

V = \tfrac12 m_n\omega^2(x^2+y^2+z^2)

, with

\hbar\omega = 10

MeV. Ignore spin and all interactions with the other nucleons. ((a)) For model A, find the energy of the level with

n_x^2+n_y^2+n_z^2 = 11

, in MeV, and the number of states it contains. ((b)) For model B, find the energy of the

N = 3

shell, in MeV, and the number of states it contains. ((c)) One state of that

N = 3

shell has the angular factor

Y_3^{3}(\theta,\phi)

. Give the eigenvalues of

\hat L^2

and

\hat L_z

for it, as multiples of

\hbar^2

and of

\hbar

respectively. Take

\hbar^2/2m_n = 20.7

MeV fm

^2

Related theory

Quantum Mechanics

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