WOLFRAM|DEMONSTRATIONS PROJECT

Relativistic Energy Levels for Hydrogen Atom

​
n
1
2
3
1/c
0.75
schematic energy-level diagram
In nonrelativistic quantum mechanics, the energy levels of the hydrogen atom are given by the formula of Bohr and Schrödinger,
E
n
=-
1
2
2
n
, expressed in hartrees (assuming the appropriate correction for the reduced mass of the electron). The energy depends only on the principal quantum number
n=1,2,3,…
and is
2
2
n
-fold degenerate (including electron spin). In Dirac's relativistic theory, this degeneracy is partially resolved and the energy is found to depend as well on the angular-momentum quantum number
j
. To second order in the fine-structure constant
α=
2
e
/ℏc≈1/137
, the hydrogen energy levels are given by
E
nj
=-
1
2
2
n
1+
2
α
2
n
n
j+1/2
-
3
4
​
. In Dirac's theory, levels such as
2
s
1/2
and
2
p
1/2
remain degenerate. The discovery of the Lamb shift showed that these two levels were actually split by 1057.8 MHz. This was a major stimulus for the development of quantum electrodynamics in the 1950s. The Lamb shift, significant only for
l = 0
(s
-states), raises the energy by approximately
13
4
3
α
3
n
. The relativistic and radiative correction to hydrogen energy levels can therefore be written
Δ
E
rel
=-
2
α
2
4
n
n
j+1/2
-
3
4
+
13
4
3
α
3
n
δ
l,0
, to third order in
α
. In this Demonstration, you can conceptually vary the fine-structure constant from 0 to its actual value, or equivalently the speed of light
c
from
∞
to 1 (meaning
3×
8
10
m/s), to show the transition from nonrelativistic to relativistic energies for quantum numbers
n=1,2
, and
3
. The energies are expressed in MHz (1 hartree =
6.57966×
9
10
MHz).
A checkbox lets you view a schematic diagram of hydrogen energy levels for various levels of the theory. This includes also the hyperfine structure, from interaction between electron and nuclear magnetic moments. The
F=0←F=1
transition in the
1
s
1/2
level gives rise to the famous 1420 MHz (21 cm) microwave radiation in radio astronomy.