sb635

Equating the "raw" time dilation to the Dirac time dilation allows for a derivation of a "Dirac velocity" equation as [math]{{{\left( {1 - {{\left( {\frac{v}{c}} \right)}^2}} \right)}^{ - 1/2}} = {{\left[ {1 + \frac{{{\alpha ^2}}}{{{{\left[ {n - j - 1/2 + \sqrt {{{(j + 1/2)}^2} - {\alpha ^2}} } \right]}^2}}}} \right]}^{1/2}}}[/math] [math]{v = c{{\left[ {1 + {{\left( {1 + \frac{{{\alpha ^2}}}{{{{\left[ {n - j - 1/2 + \sqrt {{{(j + 1/2)}^2} - {\alpha ^2}} } \right]}^2}}}} \right)}^{ - 1}}} \right]}^{1/2}}}[/math] These Dirac radii and velocity equations are "quantized" because time dilation itself is quantized in a bound system the size of an atom, such as hydrogen. The equations show the fundamental role quantized time dilation plays into modifying the nonrelativistic mechanics in a relativistic mechanics. The mechanics is special relativity, which on first glance, seems to me "incomplete." The assumption in special relativity is that no effective fields are present. The mechanical motion is "force free" and unaccelerated. If these were the actual conditions inside of hydrogen, the electron would not be bound. A central field is present, the electronic field generated by the proton. The definition of time dilation is a special case of a more general time dilation, the one existent in a field described by a nonEuclidean geometry. Perhaps a nonEuclidean Schroedinger mechanics could be developed, where the more general time dilation (assumed to be larger than in special relativity) plays even a more important role.

Numerical computations show the above is the time dilation in basic Dirac theory. The total relativistic orbital energy Dirac used in extending Schroedinger's nonrelativistic theory was [math]E = \sqrt {{c^2}{p^2} + {{(\mu {c^2})}^2}} + V( r )[/math] This can be written as (exposing one location in the equations where time dilation is present) [math]E = \mu {c^2}\left( {\frac{{dt}}{{d\tau }} - 1} \right) + V( r )[/math] The Dirac time dilation listed above is the correct value, but only if a specific value of r is used in the system potential. This value is [math]{r} = {n^2}\frac{{4\pi {\varepsilon _0}{\hbar ^2}}}{{{e^2}\mu }}\frac{{d\tau }}{{dt}}[/math] This radius equals the nonrelativistic Bohr radius times the inverse time dilation. This "Dirac radius" is smaller than the corresponding Bohr radius. The above listed energy equation results in exactly the same value as compared to using the straight Dirac equation, but only if the radius entering the system potential equation is relativistically shrunk. It is the inverse time dilation factor, which is a constant, which propagates through the probability theory as simply a constant, and shrinks the mode and mean of the relativistic radial pdfs.

What about the "heuristic velocities"? It would seem these would "be" larger. Special relativity gives time dilation as [math]\frac{{dt}}{{d\tau }} = {\left( {1 - {{\left( {\frac{v}{c}} \right)}^2}} \right)^{ - 1/2}}[/math] Is time dilation in Dirac theory understood to be [math]\frac{{dt}}{{d\tau }} = {\left[ {1 + \frac{{{\alpha ^2}}}{{{{\left[ {n - j - 1/2 + \sqrt {{{(j + 1/2)}^2} - {\alpha ^2}} } \right]}^2}}}} \right]^{1/2}}[/math] The reason why I used a positive square root on the outer brackets in the above (negative in Dirac equation) is because time dilation is > 1. The factor with -1/2 is < 1.

Thanks for the advice. I guess I should have checked my computations a billion plus one times <g>. I found my error, a simple sign flip. When corrected, the prediction agrees very well with the observed. It should, not only because all the theory is correct, but the actual values of the rest mass of the electron, proton, etc., used those exact observed values for their update. This is the way NIST does things, which is about the only way to do it. But then it really doesn't make good "scientific method" sense to predict the value when it was used in the update. That had better work, and it does quite well. The best thing to do is get a completely independently derived observed data set, and see how the theory performs. I have one, and I'll try to do this independent comparison. The usual basic Dirac equation, without QED corrections is given as [math]{E_n} = \mu {c^2}{\left[ {1 + \frac{{{\alpha ^2}}}{{{{\left[ {n - j - 1/2 + \sqrt {{{(j + 1/2)}^2} - {\alpha ^2}} } \right]}^2}}}} \right]^{ - 1/2}} - \mu {c^2}[/math] This is given on p. 607 of Merzbacher's Quantum Mechanics, 2nd. ed. The positive first term is what Merzbacher gives, but if you follow his derivation, the total orbital energy (negative valued for bound orbitals) needs the [math] - \mu {c^2}[/math] term. For spherical orbits, j = 1/2, so that [math]{E_n} = \mu {c^2}{\left[ {1 + \frac{{{\alpha ^2}}}{{{{\left[ {n - 1 + \sqrt {1 - {\alpha ^2}} } \right]}^2}}}} \right]^{ - 1/2}} - \mu {c^2}[/math] This is the equation I used to do the predictions. The simple error I made was to the subtract a QED correction when it was suppose to be added, and visa versa. You can see the kind of "mislogic" in comparing this prediction to an observed "input" data point (the 1S1/2 - 2S1/2 value, for example). This data point (and many others) were used to "update" the values of the electron and proton masses, etc. Those values therefore "parameterize" the Dirac theory to conceptually "work perfectly." That's ok, this is really the best way to do it, assuming the theory is correct. But is the theory absolutely correct? To question Dirac theory is blasphemous, but there are some aspects of it that confuse me. Consider hydrogen's spherical orbitals. Without special relativity, the Bohr theory results in simple quantized orbital radii and velocities. Of course, the rest mass of the electron is in these equations. When special relativity is introduced, the electron mass "on or in" the orbital time dilates and is bigger. The relativistic increase in the electron's mass increases the total orbital energy, resulting in the above equations. Do the orbital radii decrease and the velocities increase? The probability theory would suggest they do, because the ground state radial pdf, for example, has a mean and mode closer to the proton after relativistic effects are introduced.

I obtained a modern set of constants (c, rest mass of electron and proton, etc.) from NIST. I then used their hydrogen energy level equation and performed a prediction of the 1S1/2 - 2S1/2 transition energy and its frequency. NIST also lists the observed frequency for this transition. A prediction using only the basic Dirac energy equation (as posted by NIST) produces an amount of nonzero prediction error. This is to be expected, nothing is "perfect." Then I added in the QED corrections for these two levels, and did another prediction. The prediction error got worse. I would assume this is not suppose to happen. Dirac + QED should work better than Dirac alone, but it does not look like it does. Of course, I could have made an error in my computations, but it is actually very simple to do, and I've checked it about a billion times, with the same result. Any help on what might have gone wrong?