March 7 notes-part 2: Laser physics

I think in class on March 7, we discussed that, in the presence of the interaction term, $e E_x x cos (2 \pi f t)$ associated with a "photon",  the coefficient for the excited state state is proportional to $\gamma t$, where
$\gamma = e E_x  \int \psi_o (x) x \psi_1 (x) dx/\hbar$.
(Note that  $e E_x x$ has units of energy (an interaction energy) and thus $e E_x x/\hbar$ has units of $sec^{-1}$.)
This indicates that the probability of an induced transition, which is proportional to $(\gamma t)^2$, is proportional to $(E_x)^2$, i.e, the energy in the electric field.

When considering an induced transition from the ground state to the first excited state, it makes intuitive sense that the probability of a transition is related to energy in the photon field (the number of photons). How could it be otherwise?

The key insight to understanding a laser is to realize that these equations are symmetric in the sense that they don't care whether the transition is up or down. The equations are unchanged if you interchange the initial and final states.  Thus for an atom starting in the excited state, the probability of a downward transition to the ground state, in which a photon is emitted, is also proportional to  $(E_x)^2$ (to the energy in the electric field). Thus, for a gas of atoms in an excited state, a passing resonant photon will tend to induce downward transitions, leading to photon emission, and then greater likelihood of emission from another atom. The strength of the electric field grows with each emission, that stimulates more emissions, etc. This leads to in-phase emissions of photons all traveling in the same direction. This is critical to how a laser works.