Derivation photon emission angle/frequency

[TimothypoeZ]

I've been obsessed with the derivation in the attachment for hours now, all rights & credits to the one who came up with it, but I completely can't follow it. How does one get rid of the e's and epsilons?! How to combine the momentum and energy equations? Squaring gives for me just onworkable equations, but somehow it has to make sense, but I can't see it, both steps.
Anyone who could help me out?

P.S.
omega=frequency
vector k = momentum
k = wavenumber
angle theta = the emission angle (<pi/2)

 

[Vailanor]

m = mass of the particlee= elementary chargeE= energy of the particleThe attachment: The derivation you are looking at is a simplification of the relativistic energy-momentum relation, which states that the total energy of a particle with mass m and momentum p is given byE^2=(pc)^2+(mc^2)^2 where c is the speed of light. The derivation then makes use of the fact that for a particle that travels in a straight line, the momentum is related to its wavelength (λ) and frequency (ω) via the equationp=h/λ=hω/cwhere h is Planck's constant. Substituting this into the energy-momentum equation givesE^2=(hω/c)^2+(mc^2)2Now, if we take the square root of both sides, we can simplify the expression toE=(hω/c) + (mc^2)Finally, we can substitute in a new variable, the emission angle θ, which is related to the wavelength and wavenumber (k) via the equationk = 2π/λ = ω/c sinθSubstituting this into the energy equation givesE=(h k/2π) + (mc^2)which is the expression you are looking at. To get rid of the e's and epsilons, simply combine the momentum and energy equations and solve for E in terms of k. This will give you an equation without any constants.

 

[astrokreng]

The derivation you have attached is most likely the derivation of the Planck-Einstein relation, which relates the energy of a photon to its frequency. This relation is given by E = hω, where h is Planck's constant and ω is the frequency of the photon. The e's and epsilons you mention may be related to the permittivity of free space, which is a constant used in electromagnetic theory. However, without seeing the specific derivation you are referring to, it is difficult to provide a detailed explanation.

In general, the derivation of the Planck-Einstein relation involves combining the equations for energy and momentum of a photon. The energy of a photon is given by E = pc, where p is the magnitude of the photon's momentum and c is the speed of light. The momentum of a photon can be expressed in terms of its wavenumber, k, and emission angle, θ, as p = ħk sinθ, where ħ is the reduced Planck's constant. By substituting this expression for momentum into the energy equation and using the relation E = hω, we can derive the Planck-Einstein relation.

The steps involved in this derivation may seem complicated, but it is important to carefully consider the definitions and equations being used. It may also be helpful to review the basic principles of electromagnetic theory and special relativity, as these concepts are fundamental to understanding the behavior of photons.

I recommend seeking assistance from a colleague or mentor who is knowledgeable in this area to help guide you through the derivation and address any specific questions or confusion you may have. Additionally, there are many resources available, such as textbooks and online tutorials, that can provide a more detailed explanation of this derivation. With persistence and a solid understanding of the underlying principles, you will be able to successfully follow the derivation and gain a deeper understanding of the relationship between photon energy and frequency.