Why the emitted photon has exactly the same energy,phase and direction as the incident photon in stimulated emission? My textbook says:-

When an atom emits a photon due to its interaction with a photon incident on it,the process is called stimulated emission.the emitted photon has exactly the same energy,phase and direction as the incident photon. 

Why the emitted photon has exactly the same energy,phase and direction as the incident photon in stimulated emission?Do we have a mathematical proof?
 A: the emission is triggered by an incident photon. assumes emission photon has equal energy. Now when electrons fall back to low energy EM wave emitted if we relate this emission due to electron oscillation around its atom. then we can relate it to dipole resonance. Now if the resonance has any phase difference with a field that's driving the resonance then it means the driving field is working on it. But driving field or incident photon is not absorbed so it means the phase must be equal to the incident photon & also the wave vector.
A: Its my understanding that stimulated emission is a resonant process.   It only occurs if the emitting electron is an excited state in which is is ready to emit a photon which is identical to the stimulating photon. To produce a laser beam, electrons in the source are “pumped” to a selected excited state.  When one of those emits a photon in the right direction, a cascade begins which is enhanced and shaped by multiple reflections.
A: You are basically asking about direction and frequency.


*

*Direction


You need a background photon field, and population inversion. Basically spontaneous emission happens in a random direction, and stimulated emission happens in the direction of the background photon field.
The background field creates extra photons, helping the transition from higher energy levels to lower levels
Stimulated emission is the inverse process of absorption, both effect happen at the same time, it is just that because of the background photon field, overproduction happens, and  more are emitted in the same direction as absorbed.

Coming back to lasers specifically. It is essential that you have a population inversion. That means you must have a configuration such that the number of electrons Nk around a certain energy Ek is bigger than that of another mode Ni where Ei

https://en.wikipedia.org/wiki/Stimulated_emission


*Frequency


You need an optical resonator. This helps the production of a certain frequency of photons.
https://physics.stackexchange.com/a/445279/132371
