The focal point in the early evolution of quantum theory.
(2023-03-04) Rydberg's Formula
Great synthetic formula describing the whole spectrum of hydrogen.
The lines of atomic spectra are best identified by their frequencies
(n in Hz or cycles per second). In modern times (since the work of
Planck in 1900 and
Einstein in 1905) this is known to
be strictly proportional to the energy per photon
n) the coefficient of proportionality being
is Planck's constant :
Equivalently, some theoretical physicists prefer to use the pulsatance,
defined as the rate at which the phase of a signal (in radians) changes with
time. Because there are 2p radians in a cycle,
we have w = 2pn. Using pulsatance avoids unsightly factors of
2p in explicit trigonometric formulas.
As it's equally desirable to have a terse formula for the energy per photon
(hw/2p) we then use the quantum of spin = h/2p so that:
h n = w
where =
1.054571817646156391262428...10-34 J.s/rad
In the nineteenth century, spectroscopist could measure lines with better precision than
they knew the speed of light (either in vacuum or in air, where it's about 0.03% slower).
They didn't yet know about photons or Planck's constant.
For the utmost in precision under those circumstances, they resolve to tabulate wavenumbers
which could be used to obtain a wavelength by dividing the speed of light
(in whichever medium was relevant) into the wavenumber.
Tables of wavelengths where understood to be in vacuum for UV light (200 nm and below)
because air is opaque to far UV. Above that measurements were made in air
(because that's more convenient) and that's what's more convenient.
If needed adjustments can be made using tables giving the refractive index of air
under observed conditions of temperature, pressure and humidity.
In practical spectroscopy
Nowadays, the formula is most often stated for wavenumbers in a vacuum,
using what's called the Rydberg constant for hydrogen; R = 10967.9 m-1 10973731.568160(21)
.
Wavenumbers of Hydrogenoid Series (in cm-1 )
Series
n = 2
n = 3
n = 4
n = 5
n = 6
n = 7
¥
Ly
1
82303
97544
102879
105348
106689
107498
109737
Ba
2
15241
20576
23045
24386
25195
27434
Pa
3
5334
7804
9145
9953
12193
Br
4
2469
3810
4619
6859
Pf
5
1341
2150
4389
Hu
6
809
3048
For hydrogen, the ratio M/(M+m) is less than unity so R = 109677.583 cm-1 ... / ...
The above series are the only named ones.
The seventh series of atomic hydrogen was first charted experimentally in 1972
at Amrherst by
Niels-Peter Hansen (1942-2008)
supervised by John D. Strong
(1905--1992).
Arguably the seventh series should be called the Hansen series.
Peter hansen (portrayed at right) was a lifelong spectrocopist with a degree in electrical engineering.
Energy spectrum for Protium (in eV)
Series
n = 2
n = 3
n = 4
n = 5
n = 6
n = 7
Ionize
Ly
1
10.1987
12.0874
12.7484
13.0544
13.2206
13.3208
13.5983
Ba
2
1.8887
2.5497
2.8556
3.0218
3.1221
3.3996
Pa
3
0.6610
0.9670
1.1332
1.2334
1.5109
Br
4
0.3060
0.4722
0.5724
0.8499
Pf
5
0.1662
0.2664
0.5439
Hu
6
0.1002
0.3777
The (n-m)-th line in the m-th named series may be denoted by the (abbreviated) name of that series followed by the (n-m)-th Greek letter.
Thus the Lyman-a has a vacuum wavenumber of ... corresponding to a wavelength of ...
(2023-03-07)
Positronium
Bound state of an electron and a positron.
Orthopositronium (equal spins) or parapositronium (opposite spins).
Neither component of positronium has a color charge, so neither feels the strong nuclear force
and the system is mostly governed by electromagnetism alone
(as the weeak force is about 25 orders of magniude weaker and gravity is utterly negligible).
(2023-03-04) Dirac equation as the basis for pertubation theory
Relativistic correction include spin-orbit coupling and Darwin term.
= h/2p so that:
