Asked 20th Sep, 2022

How to get the conditions required to analyse the results?

I'm struggling to understand the method followed in the following analysis. Can someone please explain how the author got the values of Δ_1 and K_1 that justify its analysis?
I have tried to isolate "Δ" and "K" by setting Equation (B8) equal to zero. but I have failed to get similar conditions.
P.S: I'm new to mathematical modelling, so I really need to understand what's going on here. Thanks

All Answers (3)

23rd Sep, 2022
Steftcho P. Dokov
IUT, INHA University in Tashkent
You may attempt to collect some additional information about the other quantities involved in the equation - like "beta" and c-with-subscript-m.
27th Sep, 2022
Stam Nicolis
University of Tours
The RHS is a fraction, whose numerator and denominator are quadratic expressions in Δ. Therefore the fraction takes positive values when numerator and denominator are of the same sign...
1 Recommendation
Stam Nicolis This and you can't divide by zero.

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How do we justify the use of Noether’s theorem applied to local gauge transformations of the Klein-Gordon equation?
6 answers
  • L.D. EdmondsL.D. Edmonds
The topic considered here is the Klein-Gordon equation governing some scalar field amplitude, with the field amplitude defined by the property of being a solution of this equation. The original Klein-Gordan equation does not contain any gauge potentials, but a modified version of the equation (also called the Klein-Gordon equation in some books for reasons that I do not understand) does contain a gauge potential. This gauge potential is often represented in the literature by the symbol Ai (a four-component vector). Textbooks show that if a suitable transformation is applied to the field amplitude to produce a transformed field amplitude, and another suitable transformation is applied to the gauge potential to produce a transformed gauge potential, the Lagrangian is the same function of the transformed quantities as it is of the original quantities. With these transformations collectively called a gauge transformation we say that the Lagrangian is invariant under a gauge transformation. This statement has the appearance of being justification for the use of Noether’s theorem to derive a conservation law. However, it seems to me that this appearance is an illusion. If the field amplitude and gauge potential are both transformed, then they are both treated the same way as each other in Noether’s theorem. In particular, the theorem requires both to be solutions of their respective Lagrange equations. The Lagrange equation for the field amplitude is the Klein-Gordon equation (the version that includes the gauge potential). The textbook that I am studying does not discuss this but I worked out the Lagrange equations for the gauge potential and determined that the solution is not in general zero (zero is needed to make the Klein-Gordon equation with gauge potential reduce to the original equation). The field amplitude is required in textbooks to be a solution to its Lagrange equation (the Klein-Gordon equation). However, the textbook that I am studying has not explained to me that the gauge potential is required to be a solution of its Lagrange equations. If this requirement is not imposed, I don’t see how any conclusions can be reached via Noether’s theorem. Is there a way to justify the use of Noether’s theorem without requiring the gauge potential to satisfy its Lagrange equation? Or, is the gauge potential required to satisfy that equation without my textbook telling me about that?
Does the semi-classical treatment of radiation acting on matter imply a self-force when (I think) it should not?
4 answers
  • L.D. EdmondsL.D. Edmonds
Start with a purely classical case to define vocabulary. A charged marble (marble instead of a point particle to avoid some singularities) is exposed to an external electromagnetic (E&M) field. "External" means that the field is created by all charges and currents in the universe except the marble. The marble is small enough for the external field to be regarded as uniform within the marble's interior. The external field causes the marble to accelerate and that acceleration causes the marble to create its own E&M field. The recoil of the marble from the momentum carried by its own field is the self force. (One piece of the charged marble exerts an E&M force on another piece and, contrary to Newton's assumption of equal but opposite reactions, these forces do not cancel with each other if the emitted radiation carries away energy and momentum.) The self force can be neglected if the energy carried by the marble's field is negligible compared to the work done by the external field on the marble. Stated another way, the self force can be neglected if and only if the energy carried by the marble's field is negligible compared to the change in the marble's energy. Also, an analysis that neglects self force is one in which the total force on the marble is taken to be the force produced by external fields alone. The key points from this paragraph are the last two sentences repeated below:
(A) An analysis that neglects self force is one in which the total force on the marble is taken to be the force produced by external fields alone.
(B) The self force can be neglected if and only if the energy carried by the marble's field is negligible compared to the change in the marble's energy.
Now consider the semi-classical quantum mechanical (QM) treatment. The marble is now a particle and is treated by QM (Schrodinger's equation) but its environment is an E&M field treated as a classical field (Maxwell's equations). Schrodinger's equation is the QM analog for the equation of force on the particle and, at least in the textbooks I studied from, the E&M field is taken to be the external field. Therefore, from Item (A) above, I do not expect this analysis to predict a self force. However, my expectation is inconsistent with a conclusion from this analysis. The conclusion, regarding induced emission, is that the energy of a photon emitted by the particle is equal to all of the energy lost by the particle. We conclude from Item (B) above that the self force is profoundly significant.
My problem is that the analysis starts with assumptions (the field is entirely external in Schrodinger's equation) that should exclude a self force, and then reaches a conclusion (change in particle energy is carried by its own emitted photon) that implies a self force. Is there a way to reconcile this apparent contradiction?

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