Hydrogen Atom: Fine Structure of Energy Levels

​
principal quantum number n
1
energy axis:
scale
0
shift
0
The energy levels of the hydrogen atom
(0)
E
, taking account only of the Coulomb interaction between the electron and proton, are shown on the left. Perturbed energy levels
(0)
E
+
(1)
E
, also including spin-orbit interaction and relativistic corrections, produce the so-called fine structure, as shown on the right.

Details

Atomic structure, specifically for the hydrogen atom, is determined principally by Coulomb interactions among electrons and the nucleus. This leads to the unperturbed energy
(0)
E
. There also exist smaller contributions to the energy, most notably from spin-orbit interactions. These are interactions between orbital and spin magnetic moments of the electron, represented by the Hamiltonian
H'
so
=
2
e
2
2
m
e
2
c
3
r

l
·
s
,
where

l
and
s
are the orbital and spin angular momenta, respectively. A secondary perturbation comes from relativistic corrections to electron kinetic energy, represented by a term in the Hamiltonian of the form
H'
rel
=-
4
p
8
2
m
e
2
c
.
The total fine structure is then represented by the perturbation
H'
α
=
H'
so
+
H'
rel
,
with a first-order energy correction
(1)
E
=〈ΨH'Ψ〉
.
This gives the total energy of the state
n
,
j
, to first order in perturbation theory,
E
nj
=
(0)
E
n
+
(1)
E
nj
=-
m
e
4
e
/
2
ℏ
2
2
n
1+
2
α
n
1
j+1/2
-
3
4n
,
where
n
is the principal quantum number,
j
is the total electronic angular momentum quantum number and
α
is the fine-structure constant
2
e
/ℏc≈1/137
.

External Links

Relativistic Energy Levels for Hydrogen Atom
Hydrogen Orbital Densities
The Fine-Structure Constant from the Old Quantum Theory

Permanent Citation

Lukás Rafaj
​
​"Hydrogen Atom: Fine Structure of Energy Levels"​
​http://demonstrations.wolfram.com/HydrogenAtomFineStructureOfEnergyLevels/​
​Wolfram Demonstrations Project​
​Published: May 23, 2017