Use the derived hydrogen levels

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The hydrogen atom

Problem

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

, with

hc = 1240\ \mathrm{eV\,nm}

and

a_0 = 0.0529\ \mathrm{nm}

. The electron has spin one-half, with two values of

m_s

. The proton is held fixed; fine structure and the reduced-mass correction are ignored, and each transition emits one photon whose energy equals the difference of the two levels. The

2p

radial function is

R_{21}(r) \propto r\,e^{-r/2a_0}

. ((a)) Find the wavelength, in nm, of the line emitted in the transition

n = 7 \to n = 5

. ((b)) Evaluate

R_{21}(r)

r=0

. Is the probability density

|\psi_{21}|^2

at the origin larger than, smaller than, or equal to its value at

r=a_0

? Contrast this with the

1s

state, whose

|\psi_{100}|^2

is largest exactly at the origin, and explain the difference using how

R_{nl}(r)

behaves as

r\to0

. ((c)) How many states does the

n = 8

level contain, without spin and then with spin?

Related theory

Quantum Mechanics

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