# Klein-Gordon Eq., Anti-Matter, & Huge Potentials ?

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The Klein-Gordon wave equation is the logical, Relativistic, extrapolation of the Schrodinger wave equation, from applying the standard Quantum Mechanical operator ansatz $( \hat{E} \rightarrow i \hbar \partial_t, \; \hat{p} \rightarrow - i \hbar \partial_x)$, to Einstein's mass-energy relation, $E^2 = (m c^2)^2 + (c p)^2$. The KGWE, like the Dirac Equation, predicts the presence of antimatter:

does not admit a positive definite conserved probability density. Still, with the appropriate interpretation, it does describe the quantum amplitude for finding a point particle in various places, the relativistic wave function, but the particle propagates both forwards and backwards in time. Any solution to the Dirac equation is automatically a solution to the Klein–Gordon equation, but the converse is not true.

The antimatter prediction may not have been realized at first (before Dirac):

Just one year after Schrodinger published his original equation, its reformulation with these [special Relativistic] changes was made, independently, by Oskar Klein & Walter Gordon, as well as Schrodinger himself. But the new equation had a rather serious problem: the quantum probabilities it predicted from its wave function could be negative. What on Earth could it mean, to say that an electron had a minus twenty percent chance of being somewhere ?

Jim Al-Khalili. Quantum, pg. 193.

Now, a previous version of the Wikipedia article said, that probabilities, in the KGWE, dip below zero, near where the wave function encounters relativistic potentials (e.g., millions of volts for electrons, billions of volts for baryons). Apparently, the "tunneling wave" (my words), which extends out into the classically forbidden region, "inverts" and turns negative, representing the particle "traveling backwards in time", to wit, interpreted as antimatter moving forward in time.

QUESTION: If this is so, couldn't you "slam" charged matter particles, against relativistic electrostatic potentials (millions of volts, or more), and "invert them" into antimatter ??? (Something along the lines of an "matter isospin flip" ???)

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There are other conservation laws at play, e.g. baryon number or lepton number.

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...QUESTION: If this is so, couldn't you "slam" charged matter particles, against relativistic electrostatic potentials (millions of volts, or more), and "invert them" into antimatter ??? (Something along the lines of an "matter isospin flip" ???)

Yes, if the energy is sufficient, hitting a particle with a strong potential barrier may produce particle-antiparticle pairs with high probability, just like in particle-particle high energy collisions.

Edited by Bob_for_short
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Yes, if the energy is sufficient, hitting a particle with a strong potential barrier may produce particle-antiparticle pairs with high probability, just like in particle-particle high energy collisions.

That's in addition to the original particle, though. It does not simply convert matter to antimatter.

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That's in addition to the original particle, though. It does not simply convert matter to antimatter.

You are right, of course. I did not mean "inverting" matter into anti-matter but producing additional globally neutral pairs.

Edited by Bob_for_short
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There are other conservation laws at play, e.g. baryon number or lepton number.

Is Lepton number (always) a "good" quantum number ?

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I know neutrino oscillations violate it, but I'm not aware of other issues.

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I know neutrino oscillations violate it, but I'm not aware of other issues.

I presume, that the point of "inversion" in space, would correspond to the "zeros" of the Energy Operator (as modified for the presence of electro-static potentials), $\hat{E} \rightarrow i \hbar \partial_t - e V$, as per the SWE & KGWE. (This parallels the modification of the Momentum Operator, for the presence of electro-dynamic potentials, $\hat{p} \rightarrow - i \hbar \vec{\partial}_x - e \vec{A}$.) But, unless such a simple process, of 'slamming' an electron against a high-voltage cathode, involved the Weak Force, then otherwise I would guess, that (as per Bob_for_Short) the KGWE would evolve the incident high-energy electron wave, into a low-energy "stalled out" electron wave, plus an electron-positron pair (presumably appearing near the point of such "stalling out").

If so, the positron would be attracted to the cathode, where it could -- w/ suitably sophisticated "something something" -- be captured and "bottled up" in a "fuel tank" (as it were). The two electrons would "bounce back", towards the sending source. Is this a physically plausible sort of scenario ?

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I presume, that the point of "inversion" in space, would correspond to the "zeros" of the Energy Operator (as modified for the presence of electro-static potentials), $\hat{E} \rightarrow i \hbar \partial_t - e V$, as per the SWE & KGWE. (This parallels the modification of the Momentum Operator, for the presence of electro-dynamic potentials, $\hat{p} \rightarrow - i \hbar \vec{\partial}_x - e \vec{A}$.) But, unless such a simple process, of 'slamming' an electron against a high-voltage cathode, involved the Weak Force, then otherwise I would guess, that (as per Bob_for_Short) the KGWE would evolve the incident high-energy electron wave, into a low-energy "stalled out" electron wave, plus an electron-positron pair (presumably appearing near the point of such "stalling out").

If so, the positron would be attracted to the cathode, where it could -- w/ suitably sophisticated "something something" -- be captured and "bottled up" in a "fuel tank" (as it were). The two electrons would "bounce back", towards the sending source. Is this a physically plausible sort of scenario ?

Why do you color your texts?

There are practical ways of extracting (separating) positrons from the reaction area and using them in accelerators. It has already been done.

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Why do you color your texts?

There are practical ways of extracting (separating) positrons from the reaction area and using them in accelerators. It has already been done.

(the color is for emphasis)

So, such is the standard means, of making anti-matter ?

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(the color is for emphasis)

So, such is the standard means, of making anti-matter ?

Yes, it is. You can find out about it on Wikipedia, I guess. There is a plenty of experiments with different antiparticles and their beams. Soon one will be able to deal with anti-Hydrogen which is especially interesting in many respects.

A pale color is not good for emphasizing, in my eyes. Use better bold, italic, and underlined texts of a black color for this purpose.

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One wonders if anti-Hydrogen is anti-combustible, given enough surrounding anti-Oxygen and ignited with anti-radiation.

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One wonders if anti-Hydrogen is anti-combustible, given enough surrounding anti-Oxygen and ignited with anti-radiation.

In my understanding, the only difference between a "photon" and an "anti-photon", is a pi phase shift (so that they add in deconstructive interference). Photons are their own anti-particles. I would guess, that "anti-combustion" would look allot like regular combustion.

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