Relativistic Quantum Wave Equation ?

[Widdekind]

Why doesn't the following formula work, for fully relativistic Quantum phenomena ? It seems to have the correct Classical, low-energy, expansion:

 

[math]\left( E - V \right)^2 = (m c^2)^2 + (p c)^2[/math]

 

For, when V << E, pc << mc², the aforesaid formula forms the familiar Classical equation. Seeking the square-root:

 

[math]E - V = (m c^2) \sqrt{1 + \frac{p}{mc}^2 } \approx (m c^2) + \frac{p^2}{2 m} [/math]

 

[math]E \approx (m c^2) + \frac{p^2}{2 m} + V[/math]

 

This, in turn, reduces to the regular Schrodinger Wave Equation with the standard substitutions (Hughes. Elementary Particles, pg. 48):

 

[math] \hat{E} \to i \hbar \partial_t \;\;\;\;\;\;\;\; \hat{\vec{p}} \to - i \hbar \vec{\nabla} [/math]

 

save, of course, for an overall phase factor proportional to [math]\hbar \omega_0 = m c^2[/math].

 

Indeed, is it not true, that, in the presence of electro-magnetic fields, those classical quantities, and their corresponding Quantum Mechanical operators, transform as (?):

 

[math] \hat{E} \to \hat{E} - q V \;\;\;\;\;\;\;\; \hat{\vec{p}} \to \hat{\vec{p}} - q \vec{A} [/math]

 

Then, one could extend the analogy further, allowing the q's & V's to represent other kinds of charges (e.g. Weak Charge, Color Charge) and their corresponding potentials.

Edited May 30, 2010 by Widdekind