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?
It’s possible to describe the self-force in classical electrodynamics as well as in quantum electrodynamics.
The Schrödinger equation for a quantum particle describes the effects of non-fluctuating fields on a particle in equilibrium with quantum fluctuations.
I think I figured it out. Suppose the system is an electron in a central force problem and consider the following sequence of events. The electron is initially prepared to be in some excited state. The E&M environment is now turned on for a time duration T. During this time, the electron state is evolving via Schrödinger evolution (deterministic evolution dictated by Schrödinger's equation). During this evolution, the electron state is a gradually changing superposition of states. At the end of the time duration T, an energy measurement is performed and the electron state abruptly changes from a superposition to a definite energy state. If this energy state is lower than the initial state, the electron state change is accompanied by the emission of a photon. This abrupt change in state, which is statistical instead of deterministic, is measurement evolution. It is measurement evolution, not Schrödinger evolution, that created the emitted photon.
I complained that I did not expect a self force to be predicted (because Schrödinger's equation contains only external fields) but a self force is predicted (emitted photon energy is all of the energy lost by the electron). The reconciliation is that there is no self force (no emitted photon) during the gradual Schrödinger evolution. The emitted photon is produced by the abrupt measurement evolution.
Read articles. Everything about it is said there.
Article Classical-field model of the hydrogen atom
Article Nonlinear Schrodinger equation and semiclassical description...
Article Nonlinear Schrödinger equation and classical-field descripti...
Article Nonlinear Schrödinger equation and semiclassical description...
Chapter Classical Field Theory of the Photoelectric Effect
- Badges
- Science topic
- Similar topics
- Mathematics