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Path integral formulation of wave-optics

  1. Apr 7, 2006 #1
    A few years back,I stumbled upon a nice idea which I am reporting below:-

    Just as classical mechanics is the [tex] h \rightarrow 0 [/tex] limit of quantum mechanics(rather action >> [tex]\hbar[/tex],from path integral formulation),so should it be possible to argue from a path integral approach, that ray optics is the [tex] \lambda\rightarrow 0[/tex] limit of wave optics.What would such a path integral be?Here we go:-

    The optical action may be written as
    [tex]
    S=\int \frac{ds}{v(x,y,z)}=\int dt,[/tex]
    where the symbols are self-explanatory.So the prob. amplitude for light/sound(or photon/..) to go from point A to point B may be written as
    [tex]
    \begin{equation}
    P=\sum_{all paths}\exp{\iota S/T_0},
    \end{equation}
    [/tex]
    where [tex]T_0[/tex] is the time period.

    As long as [tex] S=\int dt [/tex] is not very large compared to [tex] T_0 [/tex],all conceivable ray paths between any two fixed points are possible,so that there is an uncertainty in the ray path taken by light/sound in going from one point to the other.

    Specially interesting is the case of a photon.For a photon
    [tex]
    \begin{equation}
    \frac{1}{T_0}=\nu=\frac{E}{h}.
    \end{equation}
    [/tex]
    Hence for a photon,prob. amplitude to go from one point to the other is given as
    [tex]
    \begin{equation}
    P=\sum_{all paths}\exp{\iota ES/h},
    \end{equation}
    [/tex]
    i.e.
    [tex]
    \begin{equation}
    P=\sum_{all paths}\exp{\frac{\iota E\int dt}{h}}
    \end{equation}
    [/tex]

    P.S. I am not able to see the latex graphics that I've generated in my browser--hope others are able to view it!
     
    Last edited: Apr 7, 2006
  2. jcsd
  3. Apr 10, 2006 #2
    i think it should be correct..but the action would come from the Eikonal equation:

    [tex] (\nabla{S})^{2}=n^{2} [/tex] where n is the refraction index as a function of x,y,z,t...then the path integral for optics would be:

    [tex] \int{D[r,t]e^{iS(r,t)/\hbar} [/tex]
     
  4. Apr 12, 2006 #3
    The beauty about the form that I suggest is that for a photon using E=h\nu,we arrive naturally at the particle action(starting with the optical lagrangian/action).
    My quick impression is that your S(phase I believe) is nothing but my S(modulo T_0)/T_0*2\pi---don't know where you get the \hbar from.
     
    Last edited: Apr 12, 2006
  5. Apr 12, 2006 #4
    Eikonal equation should follow easily from my optical action S .[tex] \delta S=0[/tex] i.e. Fermat's principle should easily lead to the eikonal equation.
    The nice thing would be to show that the wave equation follows from this sum over paths approach.
     
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