If an electron oscillates about a mean position, it will create a time varying electric filed which in turn will create a time varying magnetic field and so on to create an electromagnetic wave. How does this wave carry energy , in which amount and how can this be quantised?
I know that energy is quantised when we see it through the particle nature of EM waves but how can it be defined in terms of time varying electric and magnetic fields?
It is a difficult thing to visualize and connected with the wave-particle duality of photons. I think what you are interested in is the Second Quantization. This is where an electromagnetic wave is decomposed into its Fourier modes and each Fourier mode can be interpreted as simple harmonic oscillator. The energy levels of such oscillators corresponds to $E = nh\nu$, where each electromagnetic mode with that energy is a state that has $n$ photons with energy $h\nu$.
As Einstein said and proved, with his photo-electric effect, light is quantized, photon's, 'packets' of light.
The 'sum' of one, two, three or more, simple harmonic oscillations, in space-time.
Unless I'm making a mistake (somebody correct me if I'm wrong), that electromagnetic wave is identical to the wave of probability amplitude. (The word "amplitude" is the key. Think of it as a complex number spinning around in a circle.)
The way you get a particle out of it is by being uncertain of it's frequency.
If its frequency is totally certain, then its position is totally uncertain, and it's just infinitely "spread out".
If its frequency has a distribution, like a gaussian distribution about a mean, that's equivalent to an infinite sum of probability amplitude waves of different frequencies and powers. When they are added together, the only place they are "in sync" is one place, and either side of that, the power tapers off, because the waves all cancel out. The location of where they are in sync moves at what's called "group velocity".
That's the wave packet, whose power represents the probability of a photon appearing there.
