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The Two Relativistic Rydberg Formulas of Suto and Haug: Further Comments

Authors:
Journal of Modern Physics, 2020, 11, 1938-1949
https://www.scirp.org/journal/jmp
ISSN Online: 2153-120X
ISSN Print: 2153-1196
DOI:
10.4236/jmp.2020.1112122 Dec. 10, 2020 1938
Journal of Modern Physics
The Two Relativistic Rydberg Formulas of Suto
and Haug: Further Comments
Espen Gaarder Haug
Norwegian University of Life Sciences, Ås, Norway
Abstract
In a recent paper, we [1] discussed that Suto [2] has pointed out an interest-
ing relativistic extension of Rydberg’s formula. In that paper,
we had slightly
misunderstood Suto’s approach,
something we will comment on further here.
The relativistic Suto formula is actually
derived from a theory where the
standard relativistic momentum relation is changed. The relativistic Rydberg
formula we presented and mistakenly thought was the same as Suto’s for-
mula is, on the other hand, derived to be fully consistent with the standard
relativistic energy-momentum relation. Here we will point out the differ-
ences between the formulas and correct some errors in our previous pa
per.
The paper should give deeper and better intuition about the Rydberg formula
and what it represents.
Keywords
Rydberg’s Formula, Relativistic Extension, Compton Wavelength
1. Introduction
A considerable number of papers have been published on relativistic corrections
of the Rydberg states, see for example [3] [4]. However, the main focus of these
papers tends to be corrections based on relativistic quantum mechanics. Even if
relativistic quantum mechanics is very powerful, this seems to give a limited in-
tuition on why the Rydberg formula is non-relativistic, and how we can adjust
the Rydberg formula to make it relativistic. Here we will look directly at the
Rydberg formula and how it can be modified based on special relativistic effects
without looking into relativistic quantum mechanics. The relativistic quantum
mechanical approach may be considerably better in prediction power, but the
main advantage with the approach here is that it gives energy and additional in-
tuition about the non-relativistic versus relativistic Rydberg states.
How to cite this paper:
Haug, E.G. (2020
)
The Two Relativistic Rydberg Formulas of
Suto and Haug: Further Comments
.
Jou
r-
nal of Modern Physics
,
11
, 1938-1949.
https://doi.org/10.4236/jmp.2020.1112122
Received:
October 9, 2020
Accepted:
December 7, 2020
Published:
December 10, 2020
Copyright © 20
20 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1939
Journal of Modern Physics
Since we use a series of variables and parameters, we will start by providing a
list of symbols (Table 1), as a preface to our paper.
The well-known Rydberg [5] formula is given by
222
12
1 11
RZ nn
λ

= −


(1)
where
R
is Rydberg’s constant, which has a value of 10,973,731.568160(21)
m−1 (NIST CODATA 2018 value). Even though the formula is very simple, it is
hard to gain much intuition from it. The Rydberg constant can be rewritten as
4
7
23
0
110
8
e
cc
Rhc
α
λ




=

( )
32
27
3
23
27
110
1
84 10
ec
R
hc
c
α
λ
=

π

Table 1. Symbol list.
Symbol
Represents
h
Planck constant
reduced Planck constant
c
speed of light
v
velocity
of the electron
Z
atomic number
R
Rydberg constant
1
n
the principal quantum number of an energy level
2
n
the principal quantum number of an energy level for the atomic electron transition
0
vacuum permittivity
α
Fine structure constant
e
elementary charge
λ
Photon wavelength
e
λ
Compton wavelength electron
e
λ
Reduced Compton wavelength electron
e
m
rest mass of
electron
P
m
rest mass of
proton
p
momentum
E
energy
k
E
kinetic energy
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1940
Journal of Modern Physics
32
3
3
24
1
1
816
e
c
Rhc
c
α
λ
=
π
2
11
2
e
Rh
α
λ
=
2
11
2
2
e
h
Rh
α
λ
π
=
2
2
e
R
α
λ
=
(2)
And we can rewrite this as
2
2
2
2
e
e
mc
Rhh
mc
α
α
= =
(3)
(In the former paper, we had incorrectly used the Compton wavelength rather
than the reduced Compton wavelength in the beginning of this derivation and
thus we incorrectly got
2
2
4
2
e
e
mc
Rh
mc
α
α
= = π
. However, we made another error
further down that canceled this error out and therefore we obtained the right
result with respect to the Rydberg formula.)
This is standard knowledge, so we have shown nothing new so far. Let us now
replace this in Rydberg’s formula, which gives
2222
12
1 11
2
e
mcZ
hnn
α
λ

= −


22 22
222
12
11
22
ee
c cc
hZm m
nn
αα
λ

= −


(4)
where
22
2
1
c
n
α
can be seen as
2
1
v
and
22
2
2
c
n
α
as
2
2
v
. In other words, we can
write this as
222
12
11
22 2
ee
cZ
E h mv mv
λ

= =

π
(5)
Since
2
1
2mv
is the well-known approximation of the kinetic energy when
vc
(the first term of a Taylor series approximation), the Rydberg formula is
clearly non-relativistic. Even though this is known, we have not seen any relati-
vistic extension of the formula before Suto’s paper [2]. However, before we dis-
cuss his formula, we will briefly show how we arrived at our relativistic version
of the Rydberg formula. Since
2
1
2mv
is the approximation for
vc
, we
simply replaced this approximation by the full relativistic kinetic energy
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1941
Journal of Modern Physics
22
k
E mc mc
γ
= −
, where, as usual,
22
1
1vc
γ
=
. This gives
22
22
22 22
12
22
22 22
12
11
11
ee
ee
ee
mc mc
E mc mc
vc vc
mc mc
vc vc


= −− −

−−



= −

−−

(6)
where
11
v Z cn
α
=
and
22
v Z cn
α
=
, and we also have that
1
ee
h
mc
λ
=
, where
e
λ
is the Compton [6] wavelength of the electron. The equation can then be
rewritten as
22 22
22
12
11
ee
hh
cc
c
hZZ
nn
λλ
λαα



= −


−−


22 22
22
12
11 1
11
ee
ZZ
nn
λαα
λλ



= −


−−


(7)
We mistakenly thought that the formula we presented in the last paper was
the same as Suto’s relativistic Rydberg formula. However, Suto’s [2] relativistic
formula is (his equation 48)
22
22
21
11 1
11
ee
nn
λαα
λλ



= −


++


(8)
where
λ
is the photon wavelength, and
e
λ
is the Compton wavelength of the
electron.
While our new relativistic formula should be consistent with the standard energy
momentum relation
2 22 24
E pc mc= +
, where
p
is the momentum, the Suto for-
mula is not consistent with this, but it is consistent with the modified energy
momentum relation that he presented in the same paper as
24 22 24
en n
mc pc mc−=
.
The Taylor expansion of our relativistic formula is
22 44 66
24 6
11 1
22 44 66
24 6
22 2
11 3 5
12 8 16
35
12 8 16
e
ZZZ
nn n
ZZZ
nn n
ααα
λλ
ααα

= −+ + +



−− + + +


(9)
And the Taylor series expansion of Suto’s formula is
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1942
Journal of Modern Physics
246 246
24 6 24 6
11 1 22 2
11 35 35
11
2 8 16 2 8 16
e
nn n nn n
ααα ααα
λλ


= −−+ +−−+ +






(10)
We incorrectly pointed out that Suto might have made a mistake and there
was a sign error in his series expansion, but this is actually not the case. This was
because we thought his formula was identical to the one that we had derived1.
When expanded to hold for any atom, the Suto formula, based on his energy
momentum assumption, must likely be:
22 22
22
21
11 1
11
ee
ZZ
nn
λαα
λλ



= −


++


(11)
while our relativistic extension of the Rydberg formula is
22 22
22
12
11 1
11
ee
ZZ
nn
λαα
λλ



= −


−−


(12)
In other words, we have recently gotten two relativistic Rydberg formulas; one
consistent with the standard relativistic energy momentum relation (the Haug
formula) and one consistent with what we can call a somewhat alternative theory
of Suto.
In general, we would think the formula that is consistent with the standard
relativistic energy momentum relation is more correct and consistent. However,
it is not necessarily easy to test out which one is superior, as the hydrogen atom
is known to be best described by the relativistic Dirac [7] wave equation.
2. Length Contraction and Length Expansion of the Compton
Wavelength
In the Suto formula, the Compton wavlength of the electron looks like it is ex-
tended in length due to velocity of the electron, since in the denominator we have
22
2
2
1
e
Zn
α
λ
+
and
22
2
1
1
e
Zn
α
λ
+
, while in our relativistic formula we have
standard relativistic length contraction,
22
2
2
1
eZn
α
λ
and
22
2
1
1
eZn
α
λ
of
the Compton wavelength of the electron, as observed from the laboratory frame
that the electron is moving relative to. We would find it strange if the Compton
wave should undergo length expansion because of motion and not length contrac-
tion as measured with Einstein-Poincaré synchronized clocks. Einstein-Poincaré
synchronized clocks are simply indicating we assume the one-way speed of light
is the same as the round trip speed of light when we synchronize the clocks over
1We apologize for missing this, but we are taking steps to correct that here.
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1943
Journal of Modern Physics
distance.2
It seems that the methodology we have laid out basically only takes into ac-
count that the electron moves parallel to the laboratory frame. In other words,
our model seems to be 2-dimensional. However, if in reality the reduced Comp-
ton wavelength represents a radius of a sphere rather than a length, then this
length contraction will likely be correct for electrons moving in any direction as
observed from the laboratory frame with Einstein-Poincaré synchronized clocks.
There could naturally be a series of other corrections needed to fit observations,
such as relativistic quantum mechanical effects. There should also be several in-
teresting angles to investigate further: how close we can get to predicting obser-
vations with non-quantum mechanical models, for example.
3. Table Calculations
In Table 2, we look at the Lyman series. This is for a hydrogen atom where we
hold
1
1n=
and in this table let
2
n
vary from 2 to 7. We see that the Haug
formula predicts a slightly shorter wavelength than the non-relativistic Rydberg
formula, while the Suto formula predicts a slightly longer wavelength than the
non-relativistic formula. Table 3 shows predictions from the three formulas for
the Balmer series, where we have
1
2n=
and let
2
n
vary from 3 to 7. The first
column in Table 3 is from real observations. The real observations in this case
have been done in air, so we have adjusted all the three formulas by the refrac-
tion index in air; this simply means we need to divide the formulas by the refrac-
tion index in air, which is about 1.00029. One can find observation studies done
in a vacuum and in air; when comparing theoretical predictions against observa-
tions it is naturally important to know how the observations have been done the
medium matters.
Table 2. The table shows the Lyman series calculated from the non-relativistic formula,
the Haug relativistic formula, and the Suto relativistic formula. The difference-column
shows the difference in percent between the relativistic formula predictions and the
non-relativistic formula predictions for the Haug and Suto formulas. The Haug relativis-
tic formula predicts a slightly shorter wavelength than the non-relativistic formula, and
the Suto formula predicts a slightly longer wavelength. The Haug formula seems to be
consistent with relativistic length contraction (also of waves).
n
2
Non-Relativistic
Haug
Difference
Suto
Difference
2 121.568 121.562 0.0050% 121.575 0.0050%
3 102.573 102.569 0.0044% 102.578 0.0044%
4 97.255 97.251 0.0042% 97.259 0.0042%
5 94.975 94.971 0.0042% 94.979 0.0042%
6 93.781 93.778 0.0041% 93.785 0.0041%
7 93.076 93.072 0.0041% 93.080 0.0041%
2Whether or not such synchronization is fully valid is an ongoing discussion, see for example [8] [9]
,
but that is outside the scope of this paper.
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1944
Journal of Modern Physics
Table 3. The table shows the Balmer series calculated from the non-relativistic formula,
the Haug relativistic formula, and the Suto relativistic formula. The difference column is
the difference in percent between the relativistic formula predictions and the non-relativistic
formula predictions for the Haug and Suto formulas. The Haug relativistic formula pre-
dicts a slightly shorter wavelength than the non-relativistic formula, and the Suto formula
predicts a slightly longer wavelength. The Haug formula seems to be consistent with rela-
tivistic length contraction (also of waves). The observations are from the Atomic Spectra
NIST Standard Reference Database 78 Version 5.7, and are done in air, so we have made
an adjustment based on the refraction index in all formulas based on air. If this adjust-
ment is not done, our prediction is far off, as expected.
n
2
Observed
in air
Non-relativistic
in air
Difference
Haug in air
Difference
Suto in air
Difference
3 656.28 656.279 0.0001% 656.270 0.0016% 656.289 0.0013%
4 486.13 486.133 0.0006% 486.127 0.0007% 486.139 0.0018%
5 434.05 434.047 0.0007% 434.042 0.0018% 434.052 0.0005%
6 410.17 410.175 0.0011% 410.170 0.0000% 410.179 0.0022%
7 397.005 397.008 0.0009% 397.004 0.0002% 397.013 0.0020%
Since Table 2 and Table 3 are covering hydrogen atoms, we have also ad-
justed all of the formulas by multiplying by one divided by the adjusted mass
P
Pe
m
mm+
, where
P
m
is the proton mass, and
e
m
is the electron mass. First of
all, observations for the hydrogen atom are likely not accurate enough to distin-
guish between the non-relativistic and relativistic formulas, but we leave that for
future discussion. For the hydrogen atom, the Lyman series is where the differ-
ences between the three formulas are the greatest, so there is no reason to look at
the Paschen, Brackett, Humphreys, or Pfund series in addition, as the differences
between non-relativistic and relativistic predictions would be even smaller. To
test out the formulas, one would likely need to look at much heavier hydro-
gen-like atoms, as the electrons move much faster, in general, and therefore rela-
tivistic effects would play a bigger role. Still, one likely needs a relativistic wave
equation to include all necessary adjustments, another issue to consider in this
framework.
In our original table, there was a typo in the spreadsheet that resulted in in-
correct values from our relativistic Rydberg formula. Below, we present the cor-
rected tables (Table 4 and Table 5); here we have not adjusted the formulas
based on the reduced mass, so the difference in values will be the same in any of
the formulas, even if we multiply each predicted wavelength with one divided by
the reduced mass:
P
Pe
m
mm+
.
4. Conclusion
We have shown that the relativistic extensions of the Rydberg formula given by
Suto and Haug are two different formulas. The Haug relativistic Rydberg formula
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1945
Journal of Modern Physics
Table 4. The table shows the Rydberg formula predictions and the relativistic predictions for the first 137 elements. As we can see,
the difference increases between the two models. Here we are just looking at the case
1
1n=
and
2
2n=
.
Atomic
#
Rydberg
formula
Relativistic
formula
Diff.
Diff.
%
Atomic
#
Rydberg
formula
Relativistic
formula
Diff.
Diff.
%
1 121.5023 121.4962 (0.0061) 0.005% 71 0.0241 0.0181 (0.0060) 32.9%
2 30.3756 30.3695 (0.0061) 0.020% 72 0.0234 0.0175 (0.0060) 34.2%
3 13.5003 13.4942 (0.0061) 0.045% 73 0.0228 0.0168 (0.0060) 35.4%
4 7.5939 7.5878 (0.0061) 0.080% 74 0.0222 0.0162 (0.0060) 36.8%
5 4.8601 4.8540 (0.0061) 0.125% 75 0.0216 0.0156 (0.0060) 38.1%
6 3.3751 3.3690 (0.0061) 0.180% 76 0.0210 0.0151 (0.0060) 39.5%
7 2.4796 2.4736 (0.0061) 0.245% 77 0.0205 0.0145 (0.0060) 41.0%
8 1.8985 1.8924 (0.0061) 0.320% 78 0.0200 0.0140 (0.0060) 42.5%
9 1.5000 1.4940 (0.0061) 0.406% 79 0.0195 0.0135 (0.0060) 44.0%
10 1.2150 1.2090 (0.0061) 0.502% 80 0.0190 0.0130 (0.0059) 45.6%
11 1.0042 0.9981 (0.0061) 0.607% 81 0.0185 0.0126 (0.0059) 47.3%
12 0.8438 0.8377 (0.0061) 0.724% 82 0.0181 0.0121 (0.0059) 49.0%
13 0.7189 0.7129 (0.0061) 0.9% 83 0.0176 0.0117 (0.0059) 50.8%
14 0.6199 0.6138 (0.0061) 1.0% 84 0.0172 0.0113 (0.0059) 52.7%
15 0.5400 0.5339 (0.0061) 1.1% 85 0.0168 0.0109 (0.0059) 54.6%
16 0.4746 0.4686 (0.0061) 1.3% 86 0.0164 0.0105 (0.0059) 56.6%
17 0.4204 0.4144 (0.0061) 1.5% 87 0.0161 0.0101 (0.0059) 58.6%
18 0.3750 0.3689 (0.0061) 1.6% 88 0.0157 0.0098 (0.0059) 60.8%
19 0.3366 0.3305 (0.0061) 1.8% 89 0.0153 0.0094 (0.0059) 63.0%
20 0.3038 0.2977 (0.0061) 2.0% 90 0.0150 0.0091 (0.0059) 65.3%
21 0.2755 0.2695 (0.0061) 2.2% 91 0.0147 0.0087 (0.0059) 67.7%
22 0.2510 0.2450 (0.0061) 2.5% 92 0.0144 0.0084 (0.0059) 70.2%
23 0.2297 0.2236 (0.0061) 2.7% 93 0.0140 0.0081 (0.0059) 72.8%
24 0.2109 0.2049 (0.0061) 3.0% 94 0.0138 0.0078 (0.0059) 75.5%
25 0.1944 0.1884 (0.0061) 3.2% 95 0.0135 0.0075 (0.0059) 78.4%
26 0.1797 0.1737 (0.0061) 3.5% 96 0.0132 0.0073 (0.0059) 81.3%
27 0.1667 0.1606 (0.0061) 3.8% 97 0.0129 0.0070 (0.0059) 84.4%
28 0.1550 0.1489 (0.0061) 4.1% 98 0.0127 0.0067 (0.0059) 87.6%
29 0.1445 0.1384 (0.0060) 4.4% 99 0.0124 0.0065 (0.0059) 91.0%
30 0.1350 0.1290 (0.0060) 4.7% 100 0.0122 0.0062 (0.0059) 94.6%
31 0.1264 0.1204 (0.0060) 5.0% 101 0.0119 0.0060 (0.0059) 98.3%
32 0.1187 0.1126 (0.0060) 5.4% 102 0.0117 0.0058 (0.0059) 102.2%
33 0.1116 0.1055 (0.0060) 5.7% 103 0.0115 0.0056 (0.0059) 106.3%
34 0.1051 0.0991 (0.0060) 6.1% 104 0.0112 0.0053 (0.0059) 110.6%
35 0.0992 0.0931 (0.0060) 6.5% 105 0.0110 0.0051 (0.0059) 115.1%
36 0.0938 0.0877 (0.0060) 6.9% 106 0.0108 0.0049 (0.0059) 119.9%
37 0.0888 0.0827 (0.0060) 7.3% 107 0.0106 0.0047 (0.0059) 125.0%
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1946
Journal of Modern Physics
Continued
38 0.0841 0.0781 (0.0060) 7.7% 108 0.0104 0.0045 (0.0059) 130.4%
39 0.0799 0.0738 (0.0060) 8.2% 109 0.0102 0.0043 (0.0059) 136.1%
40 0.0759 0.0699 (0.0060) 8.6% 110 0.0100 0.0041 (0.0059) 142.1%
41 0.0723 0.0662 (0.0060) 9.1% 111 0.0099 0.0040 (0.0059) 148.6%
42 0.0689 0.0628 (0.0060) 9.6% 112 0.0097 0.0038 (0.0059) 155.4%
43 0.0657 0.0597 (0.0060) 10.1% 113 0.0095 0.0036 (0.0059) 162.8%
44 0.0628 0.0567 (0.0060) 10.6% 114 0.0093 0.0035 (0.0059) 170.7%
45 0.0600 0.0540 (0.0060) 11.2% 115 0.0092 0.0033 (0.0059) 179.1%
46 0.0574 0.0514 (0.0060) 11.7% 116 0.0090 0.0031 (0.0059) 188.3%
47 0.0550 0.0490 (0.0060) 12.3% 117 0.0089 0.0030 (0.0059) 198.1%
48 0.0527 0.0467 (0.0060) 12.9% 118 0.0087 0.0028 (0.0059) 208.9%
49 0.0506 0.0446 (0.0060) 13.5% 119 0.0086 0.0027 (0.0059) 220.6%
50 0.0486 0.0426 (0.0060) 14.1% 120 0.0084 0.0025 (0.0059) 233.4%
51 0.0467 0.0407 (0.0060) 14.8% 121 0.0083 0.0024 (0.0059) 247.4%
52 0.0449 0.0389 (0.0060) 15.4% 122 0.0082 0.0022 (0.0059) 263.0%
53 0.0433 0.0372 (0.0060) 16.1% 123 0.0080 0.0021 (0.0059) 280.4%
54 0.0417 0.0357 (0.0060) 16.9% 124 0.0079 0.0020 (0.0059) 299.9%
55 0.0402 0.0342 (0.0060) 17.6% 125 0.0078 0.0018 (0.0059) 321.9%
56 0.0387 0.0327 (0.0060) 18.3% 126 0.0077 0.0017 (0.0059) 347.1%
57 0.0374 0.0314 (0.0060) 19.1% 127 0.0075 0.0016 (0.0060) 376.2%
58 0.0361 0.0301 (0.0060) 19.9% 128 0.0074 0.0015 (0.0060) 410.2%
59 0.0349 0.0289 (0.0060) 20.8% 129 0.0073 0.0013 (0.0060) 450.8%
60 0.0338 0.0278 (0.0060) 21.6% 130 0.0072 0.0012 (0.0060) 500.2%
61 0.0327 0.0267 (0.0060) 22.5% 131 0.0071 0.0011 (0.0060) 562.0%
62 0.0316 0.0256 (0.0060) 23.4% 132 0.0070 0.0009 (0.0060) 642.0%
63 0.0306 0.0246 (0.0060) 24.3% 133 0.0069 0.0008 (0.0061) 751.3%
64 0.0297 0.0237 (0.0060) 25.3% 134 0.0068 0.0007 (0.0061) 912.6%
65 0.0288 0.0228 (0.0060) 26.3% 135 0.0067 0.0005 (0.0061) 1184.2%
66 0.0279 0.0219 (0.0060) 27.3% 136 0.0066 0.0003 (0.0062) 1794.2%
67 0.0271 0.0211 (0.0060) 28.4% 137 0.0065 0.0001 (0.0064) 11,232.7%
68 0.0263 0.0203 (0.0060) 29.5%
69 0.0255 0.0195 (0.0060) 30.6%
70 0.0248 0.0188 (0.0060) 31.7%
Table 5. The table shows the Rydberg formula predictions and the relativistic predictions for the first 137 elements. As we can see,
the difference increases between the two models. Here we are just looking at the case
1
1n=
and
2
2n=
.
Atomic
#
Rydberg
formula
Relativistic
formula
Diff.
Diff.
%
Atomic
#
Rydberg
formula
Relativistic
formula
Diff.
Diff.
%
1 121.5023 121.4962 0.0061 0.0050% 71 0.0241 0.0144 0.0097 67.6%
2 30.3756 26.0315 4.3440 16.7% 72 0.0234 0.0139 0.0096 68.9%
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1947
Journal of Modern Physics
Continued
3 13.5003 11.0414 2.4589 22.3% 73 0.0228 0.0134 0.0094 70.2%
4 7.5939 6.0710 1.5229 25.1% 74 0.0222 0.0129 0.0093 71.6%
5 4.8601 3.8329 1.0272 26.8% 75 0.0216 0.0125 0.0091 73.0%
6 3.3751 2.6374 0.7377 28.0% 76 0.0210 0.0121 0.0090 74.5%
7 2.4796 1.9247 0.5549 28.8% 77 0.0205 0.0116 0.0088 76.0%
8 1.8985 1.4659 0.4326 29.5% 78 0.0200 0.0112 0.0087 77.6%
9 1.5000 1.1533 0.3467 30.1% 79 0.0195 0.0109 0.0086 79.2%
10 1.2150 0.9308 0.2842 30.5% 80 0.0190 0.0105 0.0085 80.9%
11 1.0042 0.7668 0.2373 31.0% 81 0.0185 0.0101 0.0084 82.6%
12 0.8438 0.6425 0.2013 31.3% 82 0.0181 0.0098 0.0083 84.4%
13 0.7189 0.5460 0.1729 31.7% 83 0.0176 0.0095 0.0082 86.2%
14 0.6199 0.4696 0.1503 32.0% 84 0.0172 0.0092 0.0081 88.1%
15 0.5400 0.4081 0.1319 32.3% 85 0.0168 0.0088 0.0080 90.1%
16 0.4746 0.3579 0.1168 32.6% 86 0.0164 0.0085 0.0079 92.2%
17 0.4204 0.3163 0.1042 32.9% 87 0.0161 0.0083 0.0078 94.3%
18 0.3750 0.2815 0.0935 33.2% 88 0.0157 0.0080 0.0077 96.5%
19 0.3366 0.2521 0.0845 33.5% 89 0.0153 0.0077 0.0076 98.8%
20 0.3038 0.2270 0.0768 33.8% 90 0.0150 0.0075 0.0075 101.2%
21 0.2755 0.2054 0.0701 34.1% 91 0.0147 0.0072 0.0075 103.7%
22 0.2510 0.1867 0.0643 34.5% 92 0.0144 0.0070 0.0074 106.3%
23 0.2297 0.1704 0.0593 34.8% 93 0.0140 0.0067 0.0073 109.0%
24 0.2109 0.1561 0.0548 35.1% 94 0.0138 0.0065 0.0073 111.8%
25 0.1944 0.1436 0.0509 35.4% 95 0.0135 0.0063 0.0072 114.7%
26 0.1797 0.1324 0.0473 35.8% 96 0.0132 0.0061 0.0071 117.7%
27 0.1667 0.1225 0.0442 36.1% 97 0.0129 0.0058 0.0071 120.9%
28 0.1550 0.1136 0.0414 36.5% 98 0.0127 0.0056 0.0070 124.2%
29 0.1445 0.1056 0.0389 36.8% 99 0.0124 0.0054 0.0070 127.7%
30 0.1350 0.0984 0.0366 37.2% 100 0.0122 0.0053 0.0069 131.3%
31 0.1264 0.0919 0.0346 37.6% 101 0.0119 0.0051 0.0068 135.1%
32 0.1187 0.0860 0.0327 38.0% 102 0.0117 0.0049 0.0068 139.1%
33 0.1116 0.0806 0.0310 38.4% 103 0.0115 0.0047 0.0067 143.3%
34 0.1051 0.0757 0.0294 38.8% 104 0.0112 0.0045 0.0067 147.7%
35 0.0992 0.0712 0.0280 39.3% 105 0.0110 0.0044 0.0067 152.3%
36 0.0938 0.0671 0.0267 39.7% 106 0.0108 0.0042 0.0066 157.2%
37 0.0888 0.0633 0.0254 40.2% 107 0.0106 0.0040 0.0066 162.4%
38 0.0841 0.0598 0.0243 40.7% 108 0.0104 0.0039 0.0065 167.9%
39 0.0799 0.0566 0.0233 41.2% 109 0.0102 0.0037 0.0065 173.7%
40 0.0759 0.0536 0.0223 41.7% 110 0.0100 0.0036 0.0065 179.8%
41 0.0723 0.0508 0.0214 42.2% 111 0.0099 0.0034 0.0064 186.3%
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1948
Journal of Modern Physics
Continued
42 0.0689 0.0483 0.0206 42.7% 112 0.0097 0.0033 0.0064 193.3%
43 0.0657 0.0459 0.0199 43.3% 113 0.0095 0.0032 0.0064 200.8%
44 0.0628 0.0436 0.0191 43.9% 114 0.0093 0.0030 0.0063 208.8%
45 0.0600 0.0415 0.0185 44.4% 115 0.0092 0.0029 0.0063 217.3%
46 0.0574 0.0396 0.0178 45.1% 116 0.0090 0.0028 0.0063 226.6%
47 0.0550 0.0378 0.0172 45.7% 117 0.0089 0.0026 0.0062 236.6%
48 0.0527 0.0360 0.0167 46.3% 118 0.0087 0.0025 0.0062 247.4%
49 0.0506 0.0344 0.0162 47.0% 119 0.0086 0.0024 0.0062 259.2%
50 0.0486 0.0329 0.0157 47.7% 120 0.0084 0.0023 0.0062 272.1%
51 0.0467 0.0315 0.0152 48.4% 121 0.0083 0.0021 0.0062 286.3%
52 0.0449 0.0301 0.0148 49.1% 122 0.0082 0.0020 0.0061 302.0%
53 0.0433 0.0289 0.0144 49.8% 123 0.0080 0.0019 0.0061 319.5%
54 0.0417 0.0277 0.0140 50.6% 124 0.0079 0.0018 0.0061 339.1%
55 0.0402 0.0265 0.0136 51.4% 125 0.0078 0.0017 0.0061 361.3%
56 0.0387 0.0255 0.0133 52.2% 126 0.0077 0.0016 0.0061 386.6%
57 0.0374 0.0244 0.0130 53.0% 127 0.0075 0.0015 0.0061 415.8%
58 0.0361 0.0235 0.0126 53.8% 128 0.0074 0.0013 0.0061 450.0%
59 0.0349 0.0226 0.0123 54.7% 129 0.0073 0.0012 0.0061 490.7%
60 0.0338 0.0217 0.0121 55.6% 130 0.0072 0.0011 0.0061 540.2%
61 0.0327 0.0209 0.0118 56.6% 131 0.0071 0.0010 0.0061 602.1%
62 0.0316 0.0201 0.0115 57.5% 132 0.0070 0.0009 0.0061 682.3%
63 0.0306 0.0193 0.0113 58.5% 133 0.0069 0.0008 0.0061 791.8%
64 0.0297 0.0186 0.0111 59.5% 134 0.0068 0.0006 0.0061 953.2%
65 0.0288 0.0179 0.0108 60.6% 135 0.0067 0.0005 0.0062 1224.9%
66 0.0279 0.0173 0.0106 61.7% 136 0.0066 0.0003 0.0062 1835.0%
67 0.0271 0.0166 0.0104 62.8% 137 0.0065 0.0001 0.0064 11,273.7%
68 0.0263 0.0160 0.0102 63.9%
69 0.0255 0.0155 0.0101 65.1%
70 0.0248 0.0149 0.0099 66.3%
is consistent with the standard relativistic energy momentum relation, and the
Suto formula is based on an alternative theory, with a modified relativistic
energy momentum formula. It is too early to say whether or not these relativistic
extensions of the Rydberg formula can tell us anything new that is consistent
with observations, as it is likely that relativistic quantum mechanical corrections
would be needed for that. We encourage others to look further into this, and we
hope to do so some time in the future as well.
Acknowledgements
Thanks to Victoria Terces for helping me edit this manuscript.
E. G. Haug
DOI:
10.4236/jmp.2020.1112122 1949
Journal of Modern Physics
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication
of this paper.
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The results of an extended series of high-precision variational calculations for all states of helium up to {ital n}=10 and {ital L}=7 (excluding {ital S} states above {ital n}=2) are presented. Convergence of the nonrelativistic eigenvalues ranges from five parts in 10{sup 15} for the 2{ital P} states to four parts in 10{sup 19} for the 10{ital K} states. Relativistic and quantum electrodynamic corrections of order {alpha}{sup 2}, {alpha}{sup 3}, {alpha}{sup 2}{mu}/{ital M}, {alpha}{sup 2}({mu}/{ital M}){sup 2}, and {alpha}{sup 3}{mu}/{ital M} are included and the required matrix elements listed for each state. For the 1{ital s}2{ital p} {sup 3}{ital P}{sub {ital J}} states, the lowest-order spin-dependent matrix elements of the Breit interaction are determined to an accuracy of three parts in 10{sup 9}, which, together with higher-order corrections, would be sufficient to allow an improved measurement of the fine-structure constant. Methods of asymptotic analysis are extended to provide improved precision for the relativistic and relativistic-recoil corrections. A comparison with the variational results for the high-angular-momentum states shows that the standard-atomic-theory'' and long-range-interaction'' pictures discussed by Hessels {ital et} {ital al}. (Phys. Rev. Lett. 65, 2765 (1990)) come into agreement, thereby resolving what appeared to be a discrepancy. The comparison shows that the asymptotic expansions for the total energies are accurate to better than {plus minus}100 Hz for {ital L}{gt}7, and results are presented for the 9{ital L}, 10{ital L}, and 10{ital M} states (i.e., angular momentum {ital L}=8 and 9). Significant discrepancies with experiment persist for transitions among the {ital n}=10 states, which cannot be easily accommodated by supposed higher-order corrections or additional terms.
  • G E Haug
Haug, G.E. (2020) Journal of Modern Physics, 110, 528-534.
  • J R Rydberg
Rydberg, J.R. (1890) Philosophical Magazine, 29, 331-337.