A Simple Trigonometric Formula for

A Relativistic Hydrogen Atom

 

© 2003 David Barwacz

2/23/2003

 

daveb@triton.net

http://members.triton.net/daveb

 

 

 

 

Abstract

 

In this paper will develop a relativistic, trigonometric version of the Bohr hydrogen atom. The energy levels (Lyman series) calculated will be virtually exact when compared to published measured values. This is far from a complete formulization, and it is included here only to lend credibility to the above space time geometry.

 

 

We start with the momentum representation shown in Figure 11

 

 

 

In figure 11 the “ruler” line is included and converted to momentum (). The ST vector is also converted to momentum and labeled as  although neither is necessary in the calculation.

 

Two assumptions shall now be made

 

 1)

 

2)     where

 

 

Assumption 1 simply defines the ground state space component of momentum. The value was selected to yield one line exactly.

 

Assumption 2 simply quantizes the values of in terms of the ground state.

 

c is the total energy of state n.

 

Clearly . Using the reduced mass of the electron in a hydrogen atom (510720.755 eV)  it is evident from assumption 1 above that c eV

 

From which it is clear that:        Equation  2.0

 

To get the difference in total energy to the rest energy, note that

 

From which the energy difference equation can be derived:

 

   or      Equation 2.1

 

 

Again using the reduced mass of an electron in a hydrogen atom the following energy states can be calculated. Equation 2.0 is used to find  and then equation 2.1 is used to find the energy.

 

 

 

Calculated transition energies of hydrogen

 

            N                                

 

1                                                         13.598442168

2                                                         3.3996444861

3                                                         1.5109558987

4                                                         0.8499293690

 

 

What is commonly measured is the absorption and or emission spectrum. The chart below shows the values calculated by this theory, and the actually measured values published in the 2002 edition of the CRC Handbook of Chemistry and Physics [3]. The values are converted to wavelength (angstroms).

 

The value of the initial angle was selected to yield the results for one line. 

 

 

 

Published value           This theory

 

                       

926.226                       926.225                                  

930.748                       930.748                                  

937.803                       937.803                                  

949.743                       949.743                                              

972.537                       972.537                                              

1025.722                     1025.723                                            

1215.674                     1215.674                                            

 

 

As one can see, this theory yields virtually exact results with a single input variable. 

Changing the value of   gives extremely accurate results for transitions from  where n>m.

 

 

 

 

 

 

 

 

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