A He ion ( ) has one electron in a state with

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Central potentials and canonical three-dimensional systems

Reading the Spectrum, Ch The Coulomb spectrum from wave mechanics: quantum numbers and degeneracy

Problem

A He

^+

ion (

Z=2

) has one electron in a state with

l=1

. Use the effective potential

V_{\rm eff}(r) = -\dfrac{Ze^2}{4\pi\epsilon_0 r} + \dfrac{l(l+1)\hbar^2}{2\mu r^2}

, with

e^2/4\pi\epsilon_0a_0 = 27.2\,\mathrm{eV}

and

\hbar^2/2\mu a_0^2 = 13.6\,\mathrm{eV}

, and the levels

E_n = -13.6\,\mathrm{eV}\times Z^2/n^2

with

n \gt l

. ((a)) Find the radius, in units of

a_0

, at which

V_{\rm eff}

has its minimum for this ion and this

l

, and the value of

V_{\rm eff}

at that minimum, in eV (report it as it comes out of the formula, negative). ((b)) Find the lowest level

E_n

of He

^+

that has an

l=1

state, and how far above the floor of the well it lies. ((c)) Show that for any

Z

and any allowed pair

n

l

with

l \ge 1

, the level

E_n

lies above the minimum of its own

V_{\rm eff}

, and say why a bound level could not sit exactly at that minimum.

Related theory

Quantum Mechanics

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