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I am trying to understand the physical meaning behind transition amplitudes and transition probabilities. By my current understanding if $A$ is the transition amplitude then the corresponding transition probability is $A^*A$ (here $A^*$ is the complex conjugate of the transition amplitude). Is that correct?

For the Physics part, I understand that the transition probability is the probability that a certain transition between two atomic level is going to happen when the atom is excited. But as I understand, the difference of energy between two atomic levels is fixed (if we ignore the broadening effects due to uncertainty principle) then if the required difference of energy is provided then the transition should happen so why the probability? Or is the uncertainty principle at work behind inducing the probabilistic nature of atomic transitions?

So, if I excite an atomic level with a tuned laser, what exactly is happening physically in terms of transition amplitude? Even if the laser is tuned to the right frequency to cause the transition it will still be probabilistic if the transition is going to happen or not?

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Even if the laser is tuned to the right frequency to cause the transition it will still be probabilistic if the transition is going to happen or not?

That is roughly correct. A more precise statement is that the end result will be a superposition of the atom in an excited state (and one photon subtracted from the field) and the atom in its initial state with an untouched field. Thus far the interaction is deterministic, but if you then decide to perform a projective measurement to see what state the atom is in, then yes, the outcome will be probabilistic.

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