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Energy of a photon is $E=nhf$ where $n$ is an integer number, $f$ is a frequency and $h$ is Planck's constant. When a moving light source radiates blue light the wave will appear at two difference frequencies when observed toward the point the source is moving and at the point from which the source is getting further away right? Does it mean that the photons going into two different directions will have different energy levels? The same source different energy photons. I do not understand something here....

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First of all the energy of a photon is $=hν$ where $ν$ is the frequency of light that will become manifest if there is an accumulation of n photons building up a classical electromagnetic beam. Photon energy itself covers all the realm of positive real numbers, as all energies.

The classical beam is built up by a confluence of zillions of photons, explained quantum mechanically here, because photons are quantum mechanical entities, not classical. The Doppler shift happens at the spectra , a shift to light frequency dependent on the relative velocities of source and observer.

doppler

The accumulation of photons shows a shifted atomic spectrum.

Energy is conserved only within a fixed inertial frame. The frame of the observer of the spectra above is not the same as the frame of the moving source from where the light beam with its photons emanated. The same is true for the shift in frequencies of sound. Energy will be conserved in the individual interaction center of mass frame of "atom emitting photon".

Here is an archive paper that goes into the mathematics of the Doppler shift in the cosmological models , and energy conservation .

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