Semi-Relativistic QM clarifies Bound-Free Transtns?

[Widdekind]

Q: In classical QM, free particles have positive energy & frequency (plane waves), whilst bound particles have negative energy & frequency (Hydrogen wave functions). Is this transition, from positive to negative frequency, significant (e.g., given that anti-matter has inverted frequencies) ??

 

A (??): No. Technically, the classical (scalar) Schrodinger Wave Equation (SWE) is the classical limit, of the (scalar) Klein-Gordon Wave Equation (KGWE), which incorporates the rest-mass-energy of the quantum 'particles'. Accounting for that rest-mass-energy, the SWE would be modified as follows:

 

[math]\hat{E} \Psi = \hat{H} \Psi \; \; \; \rightarrow \; \; \; \hat{E} \Psi = \left( \hat{H} + m c^2 \right) \Psi[/math]

Now, it can be shown, by simple substitution, that the addition of a spatially uniform & constant potential energy (V), to the Hamiltonian, multiplies the wave function by a corresponding time-varying phase factor (e^{-i V t / h}). And, mathematically, one can consider the 'particle' rest-mass-energy as precisely such a spatially uniform & constant potential energy (V = m c²)*. Thus, when one accounts for the rest-mass-energy, the 'particle' frequency increases markedly, to a "new baseline", centered around the rest-mass-frequency, which is that corresponding, and complementary, to the 'particle' Compton Wavelength (h f = h c / l = m c²).

 

*

Could considering a quantum 'particle' rest-mass-energy, as a spatially constant & uniform potential energy, correspond to the Higgs field (interaction with which supposedly gives mass to particles) ???

For example, for an electron, m c² = 511 KeV, so that the mere transition, from a free state, to a Hydrogen 1S state (say), with E = -13.7 eV, makes little "dent" in the electron's "hyper-fast" phase frequency. Indeed, the lowest energy, 1S, Hydrogenic bound states, for the first (and, hence, unscreened) electrons added to a bare nucleus, having Z protons, is - Z² x 13.7 eV. Since an electron's rest-mass-energy is nearly 40,000 times larger (m c² / E₀ = 37300), than the (magnitude of) the Hydrogen 1S state's binding energy, one would need an ultra-trans-uranic nucleus, with nearly 200 protons (> 193), to create electro-static potential wells deep enough to actually "invert" the electron's phase-frequency*.

 

*

The largest naturally occurring elements have ~

100

protons. One could construe such coincidences (

200

roughly equals, but safely exceeds,

100

) as an argument for "fine tuning" [

cf

. Bernard Haisch.

Purpose-Guided Universe

].

Thus, using the standard, un-mass-energy-modified, SWE, amounts, mathematically, to a "Frequency Modulation (FM) signal analysis", where you "skim off" the "carrier frequency" (h f₀ = m c²), and consider only the relative phase frequency (f - f₀). So, simply b/c the relative phase frequency becomes negative, in bound states, need not imply that the electron's overall "actual" phase frequency has been "inverted", by becoming bound.

Edited September 22, 2010 by Widdekind