Exploring Spontaneous Decay in QM with Griffiths

[Silviu]

Hello! I am reading about Time Dependent Perturbation Theory from Griffiths book on QM. He derives the fact that for a system with 2 levels with energies $E_b > E_a$, if you send light to it (or a sinusoidal perturbation) the probability of an electron going from state a to state b is the same as going from state b to state a. I can't say it is very intuitive but the math is clear. Then he talks about spontaneous emission, when the electron goes from b to a without shining any light due to the fact that the ground state is never zero in QED. He then says that stimulated and spontaneous emissions are basically both stimulated emissions, just that in the first case it is a filed put there on purpose (not existing there without any external intervention). Based this explanation I am a bit confused. If spontaneous emission is the same as stimulated emission, why don't we have spontaneous absorption, too (i.e. an electron to go from a to b with no external intervention)? If for an external photon with the energy ($E_b-E_a$) the electron is equally likely to go from a to b and from b to a, why when the photon comes from the ground state of the vacuum, it doesn't behave the same way? Does this have to do with QED arguments, or am I missing something here?

 

[PeterDonis]

for a system with 2 levels with energies $E_b > E_a$, if you send light to it (or a sinusoidal perturbation) the probability of an electron going from state a to state b is the same as going from state b to state a.

It's important to be clear about exactly what this is saying. Here is a more elaborate statement of the result: if the external EM field is in a state containing photons of frequency $\omega_{ab} = (E_b - E_a) / \hbar$ the probabilities of the following two events are equal:

(1) The system in the higher energy level $E_b$ emitting a photon and dropping to the lower energy level $E_a$; this also adds a photon of frequency $\omega_{ab}$ to the external EM field.

(2) The system in the lower energy level $E_a$ absorbing a photon and raising to the higher energy level $E_b$; this also removes a photon of frequency $\omega_{ab}$ from the external EM field.

Obviously these two events can't both happen at the same time, because they require different initial conditions. But also, they both require the external EM field to be in a particular state.

Then he talks about spontaneous emission, when the electron goes from b to a without shining any light due to the fact that the ground state is never zero in QED.

Again, let me make a more elaborate statement of this result:

(3) If the external EM field is in the ground (i.e., vacuum) state, and the system is in the higher energy level $E_b$, there is a nonzero probability for the system to emit a photon and drop to the lower energy level $E_a$. This process also takes the external EM field from the ground (vacuum) state to the (non-vacuum) state containing one photon of frequency $\omega_{ab}$.

Notice that the initial condition for this is different from either of the initial conditions for #1 or #2 above.

He then says that stimulated and spontaneous emissions are basically both stimulated emissions, just that in the first case it is a filed put there on purpose (not existing there without any external intervention).

The logic behind this is that process #3 above can be viewed as a virtual photon of frequency $\omega_{ab}$ stimulating the emission of a real photon of the same frequency, i.e., the same as process #1 except that the photons already "contained" by the external EM field are virtual instead of real. However, it's important to recognize that this is just a heuristic description, and you have to be careful drawing inferences from it.

If for an external photon with the energy ($E_b-E_a$) the electron is equally likely to go from a to b and from b to a, why when the photon comes from the ground state of the vacuum, it doesn't behave the same way?

Because the process of absorbing a photon, process #2 above, requires there to be real photons in the external EM field to be absorbed. Or, to put it another way, process #2 above absorbs energy from the EM field, so the EM field after the absorption is in a lower energy state than it was before. But if the EM field is in its ground (vacuum) state, there is no lower energy state for it to go to, so there is no way for the two-level quantum system to absorb a photon from it. (By similar logic, the two-level system in its lower energy state cannot emit a photon, because there is no lower energy state for it to drop to after the emission.)

 

[bhobba]

Just to add a bit to Peters excellent post, note that you don't have to view spontaneous emission as heuristically stimulated by a virtual photon.

Its actually caused by the electron not being in a stationary state because its coupled to Quantum EM Field that exists everywhere. This is the precice reason you need Quantum Field Theory (from which the heuristic of a virtual photon comes from - why is it a heuristic - well as an actual particle they don't exist - but that is too much to go into now). It all explained in this interesting paper that since you are doing/have done Griffiths you likely can understand the detail of:
http://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf

Its your first step into the beautiful world of Quantum Field Theory. If you want to go further I suggest the following books in no particular order:
https://www.amazon.com/dp/0984513957/?tag=pfamazon01-20
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

If you want to go further still write back for how to progress to THE book on QFT - Weinberg:
https://www.amazon.com/dp/0521670535/?tag=pfamazon01-20

Thanks
Bill

 

[vanhees71]

It is really important to get the idea about what a photon is straight from the very beginning. Usually atomic physics in the introductory textbooks is treated in the semiclassical approximation (in the terminology of quantum optics), i.e., the electrons in the atom are quantized (moving in a static (Coulomb) field, describing the much heavier nucleus), while light (electromagnetic waves) are treated classically. From this you only get absorption and induced emission. If you also want spontaneous emission you also have to treat the ligth (em. fields) as a quantum system, and only then you can talk about photons, which are special states of (free) electromagnetic fields, the socalled one-particle Fock states.

Einstein has found the necessity for spontaneous emission before relativistic QFT and QED has been worked out completely, which happened in 1926 in the famous "Dreimännerarbeit", where Jordan already introduced the complete picture with the quantized em. field, but at this time most physicists considered this to go too far. So Jordan's contribution got forgotten, and usually Dirac is mentioned as the inventor of the quantized radiation field, where he explained spontaneous emission in a consistent way for the first time. Indeed, spontaneous emission is the most simple proof for the necessity of field quantization and thus, in the case of the em. field, photons. The claims in many introductory textbooks that the photoelectric effect or Compton scattering prove the existence of photons are flawed, because in modern QT the photoelectric effect at the level of Einstein's famous 1905 paper is easily derived in the semiclassical approximation (electrons quantized, em. field classical) using 1st-order time-dependent perturbation theory. No photons in sight, only classical electromagnetic fields! The same holds true for Compton scattering which has been derived by Klein and Nishina also in the semiclassical approximation first. In modern QED with the em field quantized you get this result in leading order perturbation theory by evaluating the tree-level Feynman diagrams, i.e., diagrams without closed loops. The closed loops provide the quantum effects of field quantization.

Einstein's ingenious discovery of the necessity of spontaneous emission came from re-analyzing the problem of black-body radiation in terms of kinetic theory, considering the emission and absorption process of "photons" (in the "particle sense" of old quantum theory, which is completely wrong from the modern point of view of modern Q(F)T) at the walls of the cavity. To obtain the correct Planck law for the spectrum you have to take into account not only absorption and induced emission (which both is possible in the semiclassical approximation, i.e., with classical em. fields) but also spontaneous emission (which is possible only due to the quantum fluctuations of the electromagnetic field, i.e., from a modern point of view via its quantization). As I said, the final breakthrough to explain spontaneous emission came from Dirac's 1927 paper, introducing annihilation and creation operators for photons.