The Hydrogen atom
© 2003 David Barwacz
http://members.triton.net/daveb
Abstract
In this paper I will use the previously developed geometry and scalar energy equation to develop an equation for the energy levels of hydrogenic atoms. The equation developed is identical in form to the energy equation developed by Dirac.
The general form of
the energy equation
We start with the scalar energy formula developed previously, (Vector and Scalar Energy © 2003 David Barwacz).
where 
.
Since 
and 
we can write the
energy equation as
The first term is the measurable energy and is all we are concerned with.
In Figure 15 below, all vectors are depicted as momentum and
the 
vectors have been
added. 
is
just the vector representation of 
In the following discussion, I will simply not use the arrow when referring to the magnitude of a vector. From figure 15 we can write the following equations:
eq
6.4
eq. 6.5
eq 6.6
Let 
eq 6.61
In general, 
can be any real
number.
Next we define two numbers 
and 
as follows:
and
eq. 6.7
Combining 6.7 with 6.6, 6.61 and 6.5 and substituting into 6.4 yields:
Multiplying each term by 
and simplifying
yields:
Dividing every term by 
and rearranging
yields:
Multiplying by c and taking the square root results in
eq. 6.10
The above formula is the exact form of the Dirac energy equation for hydrogen.
This equation is a general energy equation for any relativistic system and is not limited to quantum mechanics.
Placing the following specific restrictions on the values of , and 
reduces this to Diracs solution for hydrogen.
and are limited to
positive integer values and 
.
Some insight into just what these quantum conditions mean can be gained by representing the quantum equation graphically as shown below:
From the above diagram it can be seen that 
and 
must be equal and that
The quantum conditions can be written in terms of the angles as:
and
Summary
It is very interesting that this equation can be arrived at with absolutely no assumptions of fields. Dirac added a scalar and vector field to his differential equations to arrive at an equation of this form. It may be possible that, just as Dirac found intrinsic spin to be a consequence of his matrices, fields may be a consequence of this space/time geometry.