Harish Srinivasan

It is with the principle of Least action that the law of refraction works - which implies that light has to minimize the time taken to travel in the medium. To my knowledge I think this effectively done by altering the wavelength rather than frequency since the frequency of light would have to be the inverse of time taken for light to travel a distance of wavelength. About the conservation of energy I am not sure, the conservation of energy (E = hv) - its not just v(freq) can remain constant; it can also be given by E=hc/λ but since c changes the value of λ also has to change to keep E constant. But again you might have to reason why c changes.

Classically electric fields are not physical entities although as mentioned by swansont they add up by the principle of superposition. However, quantum mechanically electric fields can be looked upon as virtual photons (from QED). But to me its quite unclear about the math and ideas in QED.

How am I supposed to use ladder operators in solving them likewise, as in separating the variables ? Also how am I supposed to calculate energy eigenvalues as well ?

An anharmonic oscillator with the potential V(x,y) = 1/2(mw2x2) + 2mw2y2 , How am I supposed to solve this to get position eigenfunctions and energy eigenvalues ?