A few years back,I stumbled upon a nice idea which I am reporting below:-(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Path integral formulation of wave-optics

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