Thursday March 7 notes. Quantum transitions and mixed states.

Here is something new.  This may help us understand spectroscopy. It also lays the foundation for understanding lasers. In particular, the concept of stimulated emission, which is critical to a laser, comes from the result, embedded in this calculation, that the probability of an induced transition is proportional to the square of the electric field amplitude at the frequency of the photon that would be emitted or absorbed in the transition. The direction of the transition, up or down, makes no difference. All transitions are stimulated. There is no such thing as a spontaneous transition.

The theoretical result that the probability of emitting a photon is proportional to the oscillating electric field (squared) at the "resonant" frequency, means that (slipping into photon language) a photon going by an excited atom makes it more likely for that atom to emit another identical photon. Laser physics is a confirmation of this theoretical concept and prediction.

In absorption spectroscopy, if an incoming photon has an energy that coincides pretty exactly with the difference in energy between two quantum states, then it can stimulate a transition between those states. Actually one does not represent the photon in a quantum manner, but rather as a (classical) electric field oscillating at a frequency f. When the oscillation frequency is "tuned" so that $\hbar 2 \pi f = E_1 - E_o$, the oscillating electric field can stimulate a transition between a state of energy $E_o$ and a state of energy $E_1$. In that case the state may be of the unusual time dependent form
$\Psi(x,t) = (1-\gamma^2 t^2)^{1/2} \Psi_o(x,t) + \gamma t \Psi_1(x,t)$, where, because of the time dependence of the interaction potential one gets time dependent coefficients in front of the "before"state and "after" state.

Perhaps one can think of it this way. At t=0, the electron is purely in the ground state. And I should have mentioned before, let's also suppose that before t=0 the electron is just in the ground state, and then at t=0 the time dependent interaction term appears. Then the state will begin to evolve from a pure state into mixed state with an increasing amount of 1st excited state mixed in. Really this approach is only good for a fairly short range of time, from t=0 to about $\gamma t = 0.2$.

Let's see, what can we say about this and what does it seem to show us?
1) Well first maybe we should mention again that the first excited state was selected to be the 2nd energy eigenstate in the mixed state by the "tuning" of the frequency of the oscillating electric field to just the right frequency $2 \pi f = (E_1 - E_o)/\hbar = \omega_1 - \omega_o$.
2) Second, what one calls a quantum jump (from the $E_o$ to $E_1$ state) is not really sudden at all, in our formulation, but rather a continuous evolution of the state due to the influence of the E field.
3) Third, there presumably is a jump at some point. How does that happen? How should we think about that?...  Let's take the example of an Mg atom in a photosynthetic molecule. Suppose it starts out in its ground state and an oscillating electric field puts the Mg+ valence electron into a mixed state, as outlined above. The resolution of the energy of the mixed state, in this case and in all cases, is associated with some interaction with the environment outside the atom. In the case of a photosynthetic molecule, the Mg atom is connected to a pretty complex molecular environment via two nitrogen atoms. Once its excited valence electron travels across one of those nitrogen bridges, to begin its photosynthesis journey, the mixed state is over. The probability for the electron to start that journey is related to the coefficient of the 1st excited state in the above mixed state. In that sense one can say that the electron was induced to jump to the excited state and that that was essential to its beginning its travels. Also, the energy lost by the electromagetic field, hf, is equal to the energy gained by the electron.

The resolution of a mixed state, into one energy or the other, tends to involve an interaction between our simple quantum system, which we can model and describe pretty well, and a quite complex environment, which is difficult (or impossible) to model with any precision.

Ideas like wave-function collapse do not have any actual theory behind them. Most physicists in the current era do not believe in anything like that in my experience.