Klein-Gordon linear potential solution

In summary, the speaker has an exact solution to the Klein-Gordon equation with linear potential, which is a relativistic analog to a simple solution in the Schrodinger case. The speaker is an amateur physics enthusiast and is unsure if the solution is of interest to others. The solution is not given in terms of energy eigenstates and may not be useful. The speaker is willing to share the solution with someone interested in further development or research. The solution is also not the same as a free particle viewed from a constantly accelerating frame. The equivalence principle holds experimentally for the Schrodinger case, but not necessarily for the Klein-Gordon case due to the lack of a probabilistic interpretation for K-G waves.
  • #1
pellman
684
5
I have an exact solution to the Klein-Gordon equation with linear potential. But I am only an amateur physics enthusiast with no incentive (or time) to do anything with it, nor familiarity enough with the physics to know if it is interesting and, if so, interesting to whom. It has been sitting on my desk for a couple of years now.

It is a quite simple solution but not given in terms of energy eigenstates. It is the relativistic analog to a very simple solution of the linear potential Schrodinger case--which itself is so simple you could write it on the palm of your hand, but which appears to be generally unknown since it also is not in terms of energy eigenstates and so (I presume) not useful.


If you are interested in having it to further develop it into a paper or incorporate it into your research, let me know.

If you are reading this any time after a week of posting this, send me a PM since I probably won't see replies to the discussion thread.

Todd
 
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  • #2
pellman said:
I have an exact solution to the Klein-Gordon equation with linear potential.

Todd

The solution(s) should simply be wavefunctions of accelerating particles.
See for instance. "The Lorentz force from the Klein Gordon equation"

http://www.physics-quest.org/Book_Lorentz_force_from_Klein_Gordon.pdf

Which should become more evident if you take the charge-current density of your solution.

[tex]
\begin{aligned}
&j^o ~~=~~~~ &\frac{i\hbar e}{2m}\left(~\psi^*\frac{\partial \psi}{\partial x^o}-\frac{\partial \psi^*}{\partial x^o}\psi ~ \right) ~~-~~ &\frac{e}{c}~\Phi~\psi^*\psi
\\
&j^i ~~=~~ - &\frac{i\hbar e}{2m}\left(~\psi^*\frac{\partial \psi}{\partial x^i}-\frac{\partial \psi^*}{\partial x^i}\psi ~ \right) ~~-~~ &e\,A^i~\psi^*\psi
\end{aligned}
[/tex]


Regards, Hans
 
  • #3
Hans de Vries said:
The solution(s) should simply be wavefunctions of accelerating particles.

A point of interest is that in both the Schrodinger and K-G cases the explicit solutions are NOT the same as a free particle viewed from a constantly accelerating frame. In other words, you can't make the solution for a linear potential -Fx look like the free particle solution by substituting x --> x - F/m .

In the Schrodinger case though it is true that any expectation value is the same as the free particle case viewed from an accelerated frame, so the equivalence principle would still hold experimentally.

I can't make the same claim for the Klein-Gordon case because ... I don't know what the probabilistic interpretation of a K-G wave is! (I think its still an open question.)
 

Related to Klein-Gordon linear potential solution

1. What is the Klein-Gordon linear potential solution?

The Klein-Gordon linear potential solution is a mathematical equation that describes the behavior of a scalar particle in a potential field. It is based on the Klein-Gordon equation, which is a relativistic wave equation that describes the motion of spinless particles.

2. How does the Klein-Gordon linear potential solution differ from other potential solutions?

The Klein-Gordon linear potential solution differs from other potential solutions in that it is specifically used to describe the behavior of scalar particles, while other potential solutions may be used for particles with different properties such as spin.

3. What is the significance of the linear potential in the Klein-Gordon equation?

The linear potential in the Klein-Gordon equation represents an external force acting on the scalar particle. It is a key component in the equation and is necessary for accurately describing the behavior of the particle in a potential field.

4. How is the Klein-Gordon linear potential solution used in physics?

The Klein-Gordon linear potential solution is used in many areas of physics, including quantum field theory, particle physics, and solid state physics. It is particularly useful in studying the behavior of scalar particles in potential fields.

5. What are some limitations of the Klein-Gordon linear potential solution?

One limitation of the Klein-Gordon linear potential solution is that it does not take into account the effects of quantum fluctuations, which can have a significant impact on the behavior of particles at very small scales. Additionally, it does not account for interactions between particles, which can be important in certain scenarios.

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