I am a second year undergrad and I was thinking about a scattering problem between a photon and a point charge. I was planning to use simply the maxwell's equations for the same. However the problem is that I can't decide the boundary conditions. Although I have been recommended to use Field Theory (which I don't know yet), I wanted to simply use Maxwell's equations with proper Boundary conditions to study the interaction. Since the electric field lines can not both enter and leave a point charge, I am having trouble keeping the notion of point charge. Is it okay to consider the charged particle to have a constant radius and then say that my boundary conditions evolve with time so that the electric field lines either enter(for negative charge) or leave (for positive charge) the system at any instant of time.
I would highly recommend Charles Joachain's book "Quantum Collision Theory" - it's quite comprehensive and deals with boson scattering as well, which turns out to be probably slightly more complex than would first meet the eye.
Since a photon is a discrete particle consisting of the electromagnetic field, you can derive the equations from energy relations (see http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/compeq.html#c1). This is known as Compton scattering, while the scattering of electromagnetic waves by a spherical object can be treated with Maxwell's equations. The case where the particle's radius (R) is much smaller than the wavelength of the radiation is known as Mie scattering. The differential cross-section (dσ/dΩ) here is,
R2(kR)4[(5/8)(1 + cos2θ) - cosθ]
See Classical Electrodynamics 3rd edition by Jackson for more info (or Google).
I understand that if I want to use Maxwell's Equations for the problem I will face troubles regarding boundary conditions because the Maxwell's Equations do not recognize what a charge is. They are treated as mysterious objects whose interactions are known in full detail. Thus boundary conditions on a charge make no sense. However, is there no way If I want to say even approximately understand what might happen and how the electric field lines will behave in the vicinity of the charged particle?
Abhishek,
See my previous response in regards to treating a charged particle as a finite radii object. This is the best approximation you will get relative to applying Maxwell's equations. A search brings up several links that provide derivations for Mie scattering, http://www.comsol.com/paper/download/181101/crompton_paper.pdf or http://www.altasimtechnologies.com/electromagnetics/scattering-of-electromagnetic-waves-by-particles/
I wouldn't necessarily say the problem is due to boundary conditions either, but instead properly describing the charged particle. Realistically, a charged particle has infinite extent in regards to its field and thus it is impossible to place any real boundaries on it. It is the interaction between a charged particles motion and incident electromagnetic radiation that causes scattering. For example, a plane wave will cause the electron to accelerate back and forth in space, where some of the incoming EM energy is converted into the resulting dipole radiation. Since energy must be conserved you can work this out via Thompson scattering, where one must once again assume a finite radii of the particle (classical electron radius for this case).
http://farside.ph.utexas.edu/teaching/em/lectures/node96.html
Thus you should instead ask, what happens to a charged particle when it is in the vicinity of electromagnetic radiation (and how is the underlying field energy exchanged via Maxwell's equations and electrodynamics)?
Sir,
I stand corrected. When I say massive I mean that the particles motion is not much affected by the incident em wave which will cause acceleration in the particle. My problem is considering the scattering of em waves by a charged particle of infinite mass. Under such circumstances I can neglect the dipole radiation. My problem is not specifically related to electrons but to massive spin half particles.
As mentioned, if you want an approximation you can apply scattering calculations with the classical electron radius (or any other classical radius of preferred particle). Massive refers to any particle with mass, not that of infinite mass. I’m having trouble comprehending the question in terms of a particle of infinite mass beyond modelling it as a simple sphere, because such situation would not exist in nature and thus becomes incalculable with conventional physics. It’s also impossible to neglect the effect of for example, an electromagnetic plane wave upon a charged particle, as this is part of the solution. If the electric field causes the particle to accelerate back and forth, the resulting radiation from electrodynamics (dipole) + incident radiation = scattered radiation. Thus you can consider the energy radiated by the dipole radiation as the actual scattered radiation from the incident wave. If you want to consider very close to the particle's classical position, then you have no choice but to apply a classical radius. My point however was to consider the conservation of field energy and the resulting electrodynamics rather than problem of point-like sources versus classical radius, as this is what actually happens.
@ Abhishek Pal : u said ##scattering problem between a photon and a point charge##, ok, using classical el-dyn I can understand what u mean for point charge... but I cant figure out what U mean for Photon. R u saying classical-monocromatic em wave? Or maybe r u saying about a massless-particle with hv E?
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