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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

23

0

8

e

me

Rhc

∞

=

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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