Relation between Hamiltonian of light ray and that of mechanics

In summary: Momentum of a photon is unambiguous: hν/c.In summary, the "Hamiltonian of light" is defined by H = n-|\vec{p}| = 0, where n is the refractive index and \vec{p} is the canonical momentum. The canonical momentum is derived from Fermat's Principle and represents the momentum of the ray. Momentum of a photon can also be represented by hν/c. Snell's Law and specular reflection both conserve momentum in the ray model of light propagation. The Hamiltonian of light in classical mechanics is K+V=kinetic energy+potential energy, but in optics, it represents the perpendicularity of rays to wavefronts.
  • #1
genxium
141
2
I'm learning ray optics and feeling so confused by the definition of "Hamiltonian of light".

What I learned was that the "Hamiltonian of light" defined by [itex]H = n-|\vec{p}| = 0[/itex] indicates the momentum conservation, where [itex]n[/itex] is refractive index and [itex]\vec{p}[/itex] here is the canonical momentum. The canonical momentum is defined by [itex]\vec{p}=\frac{dL}{d\vec{r}'}=\frac{dL}{d(\frac{d\vec{r}}{ds})}[/itex] where [itex]\vec{r}[/itex] is the position vector, [itex]s[/itex] is the path length and [itex]L = n*|\vec{r}'| [/itex] is the Lagrangian.

My questions are

1. [itex]H[/itex] of light is conserved, but is momentum of light conversed? If so how is it indicated in the equations?

2. [itex]H[/itex] of classical mechanics is [itex]K+V[/itex]=kinective energy+potential energy, this is a clear physical meaning, but what does [itex]H[/itex] of light mean?

(Sorry for the long definition statement, I want to make sure that people hold the same definition of things otherwise they can point out where I went wrong)
 
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  • #3
genxium said:
I'm learning ray optics and feeling so confused by the definition of "Hamiltonian of light".

<snip>

Google is your friend:

http://en.wikipedia.org/wiki/Hamiltonian_optics

Buchdahl's book is good, too.

Short answers to your questions:

1) Yes (both Snell's law and specular reflection conserve momentum)
2) It means rays are perpendicular to wavefronts.
 
  • #4
@Andy, thank you so much for the reply! It's informative but may I ask for more of question 1? What do you mean by "Snell's Law conserves momentum"?

I'm actually not clear about the definition of light momentum here, is it "momentum of ray" or "momentum of photon"? What is the instance that "owns" momentum? When I learned the canonical momentum mentioned in my question it was derived from Fermat's Principle thus I take it as "momentum of ray".
 
  • #5
This thread is predicated on using the geometrical (ray) model of light propagation; momentum of the ray is represented by the wavevector k, with |k|= 2*pi/λ.
 

What is the Hamiltonian of a light ray and how does it relate to the Hamiltonian of mechanics?

The Hamiltonian of a light ray is a mathematical quantity that represents the total energy of the ray. It is related to the Hamiltonian of mechanics through the principle of least action, which states that the path taken by a ray of light will minimize the time or distance it takes to travel. The Hamiltonian of mechanics is derived from the kinetic and potential energy of a system, while the Hamiltonian of light ray takes into account the wave nature of light and its speed in a medium.

How is the Hamiltonian of a light ray used in optics and mechanics?

The Hamiltonian of a light ray is used in optics to describe the propagation of light and the formation of images through lenses and mirrors. In mechanics, the Hamiltonian is used to describe the motion of particles and systems, and can be used to derive equations of motion and predict the behavior of physical systems.

What are the key differences between the Hamiltonian of a light ray and that of mechanics?

One key difference is that the Hamiltonian of a light ray takes into account the wave nature of light, while the Hamiltonian of mechanics is based on classical mechanics principles. Another difference is that the Hamiltonian of a light ray is a function of the electric and magnetic fields, while the Hamiltonian of mechanics is a function of position and momentum.

How does the Hamiltonian of a light ray change in different media?

The Hamiltonian of a light ray can change in different media due to the variation in the speed of light. In a medium with a higher refractive index, the speed of light is slower and therefore the Hamiltonian will also be different. This can lead to changes in the path of the light ray and its behavior.

Can the Hamiltonian of a light ray be used to describe all properties of light?

No, the Hamiltonian of a light ray only describes the propagation of light and does not take into account other properties such as polarization, diffraction, or interference. These properties can be described by other mathematical quantities and principles, such as the Jones matrix or Huygens' principle.

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