To begin this analysis, consider again the energy component of [1], Eq.(2.1). If this is to represent the orbital energy of a bound electron in an atom, for that electron orbit to be stable, it is necessary for the bound energy to be constant over an entire single orbit, e.g. there can be no net energy loss or gain. Thus for such a case, if the energy in [1], Eq.(2.1) is to be constant, then in [1], Eq.(2.15), f_v must also be constant over an entire orbit. In turn, this means that in a simple re-arrangement of [1], Eqs.(2.24) and (2.26),

vl_sv =

c²

f_v

(2.1)

the product vl_sv must be similarly constant.

and where

v is the electron's orbital velocity.
l_sv is the electron's orbital spatial matter wave wavelength.
c is the velocity of light.
f_v is the electrons spatial - temporal matter wave frequency.

It is noted that (2.1) appears to invoke the fictitious spatial "phase velocity" of the electron matter wave, viz [1], Eq.(2.19). However, although the quantity c²/v appears in (2.1) it is only via a re-arrangement of the relationships between real parameters and is therefore acceptable as a mathematical descriptor.

In a stable elliptic orbit, the orbital velocity v cannot be constant because the distance of the orbiting electron from the central nucleus is continuously changing. Consequently for (2.1) to be constant for such an orbit, l_sv while being single valued over an entire orbit , must however vary proportionately in precise inverse harmony to the variation in v within the orbit. This then ensures that the elliptical orbit is stable. In circular orbits, the distance of the electron from the central nucleus is constant and so the orbital velocity is constant. Thus in this case l_sv is not only single valued over a complete orbit, but also exactly constant throughout it. The so called "pendulum orbits" are discussed at the end of this Section.

Now, to justify the criterion of quantisation, the single valuedness of l_sv over a complete orbit, inserting the component parts of v for a basic elliptic orbit in (2.1) gives after minor re-arrangement

l_sv =

c²

f_v

æ
è

+w_f² r²

ö
ø

-1/2

(2.2)

where

r is the radial distance of the electron from the nucleus.

×
r is the radial velocity of the electron in its orbit.

w_f is the angular rate of the electron in its orbit.

Now, in any elliptic orbit it is well known that

r =

( 1+ecosf )

so that

w_f Lesinf

( 1+ecosf )²

(2.3)

Where

f is the angular position of the radius vector from some axis origin.
L is the semi latus rectum of the elliptic orbit.
e is the eccentricity of the orbit.

Substitution of (2.3) into (2.2) gives after minor reduction

l_sv =

c²

f_v

m₀ L

M_f

( 1+e²+2ecosf )^-1/2

(2.4)

Where

M_f is the angular momentum of the rest mass of the electron and is constant by the law of conservation.
m₀ is the rest mass of the electron.

On the RHS of (2.4) the only variable is the angular position so that it is clear that over a complete orbit

l_sv |_{f = 2p} = l_sv |_{f = 0}

(2.5)

Eq.(2.5) states that over the orbital path length of a stable orbit the orbiting electron's matter wave wavelength is an integer number. Therefore the criterion of quantisation is now linked to the requirement that, within a stable orbit, the bound energy of the orbiting electron must be constant. Accordingly, the criterion of quantisation is thus proved to be a necessary and sufficient condition for the stability of a basic electron orbit. The variability of l_sv within the orbit is also clearly visible in (2.4).

If e = 0 then (2.4) becomes

l_sv =

rm₀ c²

f_v M_f

(2.6)

and is constant throughout the complete orbit. This is the electron matter wave wavelength for circular orbits, (r º L).

In the case of the pendulum orbits, in (2.4) L = 0 and therefore l_sv = 0. Accordingly from [1], Eq.() f_sv becomes infinite, which from [1], Eq.(2.26) requires that the orbital velocity v also becomes infinite. This contravenes the criterion of existence within the Relativistic Space-Time Domain D₀, and is a sufficient proof for the exclusion of the pendulum orbits. Note that for f_sv to be infinite would also necessitate infinite orbital energy.

In (2.4) if e = 1, it becomes

l_sv =

m₀ c²L

Ö2 f_v M_f

( secf )^1/2

(2.7)

The orbit is parabolic and f varies from -(p-d) to +(p-d). Quantisation does not apply because the orbit is not closed. The same comment applies to hyperbolic orbits, (e > 1).

2.3 Quantisation of the Bohr/Sommerfeld Atom.

In view of the results of the preceding Section, it is now permissible to apply the quantisation process to relativistically modified electron orbits to resurrect the Bohr/Sommerfeld theory of atomic structure.

Rewriting [1], Eq.(2.18) as

l_sv

= mv

(2.8)

where h is Planck's constant.

Because the wavelength is to be quantised for all orbit path lengths, both sides of (2.8) are integrated over the orbital path length to give

l_sv

ó
(ç)
õ

mvdl

(2.9)

where l is the length of the orbit path, and m the energy mass of the electron.

On the LHS if l is to be an integral number of wavelengths then (2.9) can be written

nh =

ó
(ç)
õ

mvdl

(2.10)

Where n is an integer, the principle quantum number of the orbit. In (2.9) and (2.10) the circled integral sign indicates integration over the complete path of the orbit. It is (2.10) which will later be used as the source equation for the sample quantisation of a number of permitted orbits.

To show that the above derivation leads to the same quantisation results as in the original Bohr theory, it is sufficient to demonstrate that (2.10) leads to the original quantisation rules as propounded by Niels Bohr.

With

dl = vdt

(2.11)

insertion of this into (2.10) and expansion of the RHS gives

nh =

ó
(ç)
õ

æ
è

+w_f ² r²

ö
ø

(2.12)

Splitting the RHS of (2.12) into two terms then gives

nh =

ó
(ç)
õ

dr+

ó
(ç)
õ

mw_f r²df

(2.13)

which can clearly be written

nh =

ó
(ç)
õ

M_r^* dr+

ó
(ç)
õ

M_f ^* df

(2.14)

where

M_r^* is the radial momentum of the relativistically mass corrected electron.
M_f ^* is the angular momentum of the relativistically mass corrected electron.

Eq.(2.14) is identical to the original quantisation rules of Niels Bohr with the minor exception that the momentum terms are corrected for the relativistic mass increase of the electron. Eq.(2.14) is perhaps the most elegant way of representing the quantisation process but simpler analysis results from the use of (2.10) as in Section 3 below.

2.4 Orbital Energy Levels.

In order to derive quantised orbital energy levels, an energy expression suitable for use in the process is required. To derive the form required here, use is made of the solution to the orbital equation of motion, which has been effected in [2]. The result however, requires considerable preliminary analysis before insertion into the orbital energy derivation process. For clarity, this preliminary analysis is relegated to Appendix A, Section A.1, the results of which are used in the following process.

Starting with Einstein's energy/momentum equation as stated at [1], Eq.(2.29), the bound energy of the electron is

E_or = m₀ c²

æ
è

M²

m₀² c²

ö
ø

1/2

- m₀ c² -

Ze²

(2.15)

where

E_or is the orbital or bound energy of the electron.
M is the spatial momentum of the electron in its orbit.
Z is the atomic number of the atom.
e is electronic charge.

It should be noted that although the atomic number has been included in this analysis, only hydrogen, (Z = 1), will be considered in detail when calculating spectra.

Via binomial expansion, retaining only second order relativistic terms, (2.15) reduces to

E_or =

m₀ v²

m₀ v⁴

c²

Ze²

( 1+ecosj )

(2.16)

where

L is the relativistically corrected orbit semi-latus rectum.
e is the relativistically corrected orbit eccentricity.

and where (A.3) has been inserted for r. An expression for v is now required.

From (A.3)

drdj

djdt

w_j Lesinj

( 1+ecosj )²

(2.17)

The angular velocity of the electron is w_f r which from (A.3) and (A.14) is

w_f r=

w_j L

( 1+ecosj )

æ
è

Z²e⁴

c²M_j ^*2

ö
ø

1/2

(2.18)

So that from (2.17) and (2.18)

v² =

+ w_f ² r² =

w_j ² L ²e ² sin²j

( 1+ecosj )⁴

w_j² L²

( 1+ecosj )²

æ
è

Z²e⁴

c²M_j ^*2

ö
ø

(2.19)

which with (A.14) reduces to

v² =

M_j ^*2

m₀² L²

ì
í
î

1 + 2ecosj + e² +

Z²e⁴

c²M_j ^*2

( 1+ecosj )²

ü
ý
þ

æ
è

v²

c²

ö
ø

(2.20)

Solving (2.20) for v² gives

v² =

M_j ^*2

m₀² L²

ì
í
î

1 + 2ecosj + e²+

Z²e⁴

c²M_j ^*2

( 1+ecosj )²

ü
ý
þ

é
ë

M_j ^*2

m₀² c²L²

ì
í
î

1 + 2ecosj + e²+

Z²e⁴

c²M_j ^*2

( 1+ecosj )²

ü
ý
þ

ù
û

(2.21)

Substitution of this into (2.16) gives for the orbital energy, after some reduction including binomial expansion to relativistic second order

E_or =

M_j ^*2

2m₀² L²

ì
í
î

1 + 2ecosj + e² +

Z²e⁴

c²M_j ^*2

( 1+ecosj )²

ü
ý
þ

M_j ^*4

8m₀³ c²L⁴

{ 1 + 2ecosj + e² }²-

Ze²

( 1+ecosj )

(2.22)

Now, substituting for L from (A.14) gives, again after some reduction

E_or =

Z²e⁴

2M_j ^*2

{ 1 + 2ecosj + e² }-

Z²e⁴

M_j ^*2

( 1+ecosj )

Z⁴e⁸

c²M_j ^*4

ì
í
î

( 1 + ecosj )² -

( ecosj + e² )( 1 - e² ) -

( 1 + 2ecosj + e² )²

ü
ý
þ

(2.23)

which finally reduces to

E_or = -

Z²e⁴m₀ ( 1-e² )

2M_j ^*2

æ
è

Z²e⁴

c²M_j ^*2

( 1-e² )

ö
ø

(2.24)

This expression for the orbital energy can now be quantised for all permitted orbits by inserting the appropriate expression for M_j^*2/( 1-e² ). Non-relativistically corrected orbits can be treated by letting c ® ¥, and circular orbits by putting e = 0. Eq.(2.24) leads directly to the expanded version of Sommerfeld's equation for relativistically mass corrected elliptic orbits as will be shown in Section 3.5.

P2 Version 2.0.1
Ó P.G.Bass, April 2008

2.0 Resurrection of the Bohr/Sommerfeld Theory of Atomic Structure.

2.1 Quantum Number Nomenclature.

2.2 Justification of the Quantization Criteria & Determination of Permitted Orbits.

2.3 Quantisation of the Bohr/Sommerfeld Atom.

2.4 Orbital Energy Levels.