Dear Thomas,

I did myself look into this some time ago, in connection with the possibility to appreciate long range correlation and order at "higher" temperatures, including the relevance for complex enough systems in biology and the origin and evolution of life.

I enclose two chapters from my contributions to a book edition from 1996. On page 226 labeled 7.5 Spontaneous and Stimulated Emission of Radiation in Masers you can check the writing as well as some references therein. Unfortunately it is not easy reading.

Best

erkki

Dear Thomas,

I am not so sure whether I understand what exactly you ask, but I will give you the physical explanation of the phenomenon, as is known to me.

Imagine that in some medium many atoms are raised to the same excited level. Now, an external field of a given frequency hν irradiates the atoms, where hν is exactly the frequency at which the atom would emit if undergoing de-excitation.

Thus, due to the external field all the atoms begin to oscillate in phase. They begin to emit, and the emitted radiation is in phase with the external field. Due to this equality of phase between the external and emitted field there is constructive superposition and the emission is enhanced.

Well, for keeping rigorous, the phase of the coherent field is not perfectly defined, there is some imprecision in it, as is some imprecision in the intensity (number of photons). This is due to the uncertainty principle, the imprecision in phase is proportional with the imprecision in time, and the imprecision in number of photons is proportional with the imprecision in energy. But these are just details, the physical process is as I described above.

Dear Thomas,

Your questioning is, to my opinion, related to a very simple fact that is never commented in QM textbooks.`The wave function associated to a photon is a real function which squared amplitude measures an intensity not a probability of presence.

Thus this wave function has a classical behaviour even if it belongs to the realm of QM. The relationship :  ∆ω.∆t≥1/2 is no uncertainty principle, it is only a Fourier relationship between frequency and time. It has been inspiring to Heisenberg for defining the uncertainty principles that apply to complex functions. In the presence of a real wave-function (em field) an excited atom relaxes emitting another wave in phase with the impinging one. This is the stimulated emission introduced by Einstein who did not use at all the concept of photon at this stage. To my understanding it is therefore no surprise that the laws of optics do apply to a set of excited atoms that can emit a coherent beam.

Now the question is : why are excited atoms not stable when left apart from any em field?

And the answer is: because the vacuum fluctuations make em fields to appear in the vicinity of the atoms for very short periods of time and these em fields have the atoms relaxing through stimulated emission.

Stimulated emission is at the core of atoms relaxation, spontaneously or not.

Hope it answers your question.

Charles,

The squared amplitude of the photon wavefunction is a probability density and there is no restriction to real wavefunctions. 

The relationship should be ∆E.∆t ≥1/2ℏ.

Hello,

Erkki: Section 7.5.1 - Einstein Relation of your book only cites the 1951 book "Quantum Theory" by David Bohm. Is the 1951 description of stimulated emission the one everyones agrees is complete and correct? I am travelling right now so I cannot consult this particular book, and it cannot be viewed on Google books.

Sofia: You say that, "Thus, due to the external field all the atoms begin to oscillate in phase. They begin to emit, and the emitted radiation is in phase with the external field." But even in a laser beam there is some spontaneous emission, which is not necessarily in phase with the stimulated emission even though it has the same frequency. My question is what triggers the atoms to undergo stimulated emission. And I know you can answer that question by saying the external field, but the external field is what (the agent that) causes stimulated emission not why (the mechanism by which) it happens. If billard ball B, which is initially at rest, is struck by billard ball A, then one can certainly say A causes (is the agent) B to move. But the why (the mechanism) is momentum transfer. I am asking about the mechanism of stimulated emission.

One other point. You say: "...there is some imprecision in it, as is some imprecision in the intensity (number of photons). This is due to the uncertainty principle, ...and the imprecision in number of photons is proportional with the imprecision in energy." But the imprecision in the energy would affect the frequency, i.e., line width of the radiation, not the intensity, i.e., the number of photons.  Or, at least, that is my understanding.  As far as I remember from the photoelectric effect, energy and intensity are independent.  Of course, from classical theory, they are related.

Charles: You say: "The wave function associated to a photon is a real function which squared amplitude measures an intensity not a probability of presence." When I took Quantum Mechanics and we did the standard solution for a particle in box, the square of the wave function was always interpreted as the probability of the location of the particle. Of course, I don't remember ever solving this same problem for the case where the mass was identically zero, as it would be for a photon. And I guess this is what you are saying. But is your assertion explicitly mentioned in any books or papers? The book by Jurgen R. Meyer-Arendt; Introduction to Classical and Modern Optics; Prentice–Hall; 1972 makes the following statement "Intensity is a term much overworked; to be correct, it should be used to mean only the square of the wave amplitude and power per unit solid angle." But, I believe the author is speaking about a classical wave from Electricity & Magnetism, not a quantum wave function.

I'm sorry, but I am still baffled by the mechanism of stimulated emission.

Regards,

Tom Cuff

Thank you to Erkki for having pointed out my typo.

The time frequency duality writes down as:

∆ω . ∆t ≥ 1/2

It does not contain the quantum of action one encounters in the uncertainty inequalities.

Writing ∆E . ∆t ≥ 1/2ℏ is misleading as both sides of the inequality contains the quantum of action as ∆E = ℏ ∆ω.

See:

"Femtosecond laser pulses", 2nd edition, Ed. C. Rullière, Springer, 2003, p. 31

For French readers: https://fr.wikipedia.org/wiki/Optique_impulsionnelle

Thomas,

The real nature of em waves is something that struck me when I was a student, and I never found it ever written down anywhere. I never made the point that it might be related to the nullity of the mass but in some ways it is true as you cannot think about an em wave as a travelling particle. The quantum of energy (the photon) has only to do with energy and momentum exchange, not with propagation.

As I wrote, stimulated emission is the standard light-matter interaction mechanism. So to enter your argument with Sofia, when the field emitted by some relaxing atom somewhere reaches another excited atom with the same energy (within the uncertainty on this energy=the width of the absorption line) there is a quantified exchange between the field and the atom. This exchange involves both energy and momentum. This momentum exchange being at the core of the cooling down of atoms using lasers. So that you are perfectly legitimate to think that the mechanism is also momentum transfer.

If I understand well your concern, the emission from an atom has to be isotropic and the em wave that triggers the stimulated emission of a very far away excited atom should be fully located on this atom. I think that this has to do with the wave-packet reduction and non-locality in QM. These aspects of QM still remain largely mysterious.

Photons, gluons and gravitons are "no-mass" particles and as such are fully relativistic particles. Why should QM apply to them as we know that Quantum Mechanics and General Relativity are non-compatible theories?

Dear Thomas,

I would say that Bohm’s treatment is correct, but possibly not complete in view of what we know today.

I would recommend the photon paper by Bialynicky-Birula working with Scully, see acknowledgements, in Garching, as rather up-todate regarding a photon wavefunction.

Charles,

Yes the time frequency – time duality is a basic property of the Fourier transformation, which you wrote down as: ∆ω ∆t ≥ ½, while the quantum of action appears in the uncertainty inequality ∆E ∆t ≥ 1/2ℏ containing the quantum of action ∆E = ℏ ∆ω.

This points at a deeper connection between classical and quantum mechanics as both subscribe to the important concept of conjugate variables (operators in QM).

I am not sure why you say that “the wave function" associated to a photon is a real function”?

To Erkki, simply because it is not complex.

Why is it "not complex"?

In any case, why do you want a fully quantum mechanical treatment? In the case of lasers, we are fully in the semiclassical case, since the number of photons involved is very large. So whatever you need can be obtained by thinking in semiclassical terms. Spontaneous emission does need a full quantum treatment, stimulated does not.

Erkki I really don't understand what you don't understand.

An electromagnetic wave is the propagation of a couple of fields (E and B). This is a physical object which representation stands on a real function with real variables. The imaginary part of this function is null. But the point is not on math but on the fact that it is a physical object.

A wave function is the propagation of a function which square is a probability. Its representation is a complex function which imaginary part is not zero. A probability is one quality that characterizes an object it is not a physical object.

These two concepts are deeply different in nature.

Light is the archetype of a relativistic object. Relativistic objects are outside QM description as these two theories, QM and GR, are not compatible one with the other. There cannot be any full QM description of light. Of course this will change as soon as QM and GR will be unified if this can happen.

Charles,

I am just saying that the electromagnetic wave is defined by physical boundary conditions commensurate with the physical situation..

And?

Charles,

For instance and outgoing electromagnetic wave is represented by a complex wavefunction.

Dear Erkki,

Yes, but this is only a way to simplify math calculation by having only one equation instead of 3 for space and one for time, at least.

For an electron, the complex nature of its wavefunction is no math simplification, it is an intrinsic property that has no interpretation in our real word. For coming back to our world one needs to square the modulus of this complex function in order to get a real function, real having here both meanings: math and our world.

When Maxwell designed the em theory he used neither vectors nor complex notations. As far as I remember ( I read it, I am not that old :-)) he wrote 24 equations or more instead of the 4 ones we are used to. This had to be simplified.

If you look at it, the Schrödinger equation is a transposition of the em propagation equation into the complex plane.

Dear Charles,

Your interpretation of the Schrödinger equation as "a transposition of the em propagation equation into the complex plane"is not meaningful.

First of all, simplifying a bit, you may "derive" the Schrödinger equation from the Klein-Gordon equation, i.e. approximating and reducing a relativistic equation to the non-relativistic domain. The essential difference between em waves and "particle waves" are related to the non-zero restmass of QM "particles" as compared to solutions to the Maxwell equations.

As I said earlier the detailed mathematical nature, i.e. complex or real, of a wave being a solution to a wave equation rests in the boundary conditions.

Ok Erkki,

I think we will not converge, but is that so important?

Let me ask you this question: Do you have a citation to a full quantum mechanical derivation of stimulated emission?

Charles,

I think you mix two separate things: 1. the natural laws as formulated succinctly in mathematics and interpreted and expressed in ordinary language and 2. our private and subjective mindset depending on our background, theoretician, experimentalist, etc. which reflects our intuition when we debate and communicate science.

The latter may or may not be realistic or relevant, but if it is too distorted, it may not be meaningful even if it is impossible to grade as truth or false.

Regarding your question, see my first postings where I gave some reference to a review article.

Hello,

Francois: I am guessing that you are invoking the correspondence principle because of your statement that "the number of photons involved is very large." But when Millikan did his famous experiments on the photoelectric effect, the number of photons was large,and still the overall effect could not be explained classically. In the case of blackbody radiation, the number of photons is likewise quite large, but their spectrum cannot be explained classically. And, yes, in both the photoelectric effect and blackbody radiation cases, the initial approach to explaining them was semi-classical because quantum mechanics had not yet been developed. I suppose that even today most treatments are still semi-classical because the potential functions in the Schroedinger wave equation are usually obtained from their classical counterparts.

You say that, "Spontaneous emission does need a full quantum treatment, stimulated does not." It was my understanding that spontaneous emission, a type of fluorescence (resonance radiation), if you will,still requires a quantum treatment if for no other reason than the energy levels in the atoms are quantised.

Charles: I see your point that complex waveforms in EM are only mathematical artifices used to make the manipulations easier. Quaternoins were a likewise similar artifice championed by Hamilton and Tait to represent rotations and other transformations. No experimentalists believe that such representations (complex waveforms and quaternions) exist in reality.

However, your assertion that "The wave function associated to a photon is a real function which squared amplitude measures an intensity not a probability of presence.", and that this idea is never mentioned in QM textbooks reminds me of Collins' idea of "tacit knowledge" (cf. Harry Collins; Gravity's Shadow; The University of Chicago Press;2004; pp. 476, 608-10,etc.). Collins was speaking of experimental knowledge or procedures that are passed on by word of mouth as opposed to being written down in books or papers. The problem with tacit knowledge is vetting it, since it is hard to trace its provenance.

Sofia: My demand for a mechanism to explain stimulated emission is, no doubt, wedded to a classical interpretation of the phenomenon. Since QM eschews cause and effect, asking for a mechanism is very retro of me. I apologize. In QM, stuff happens, but exactly how it happens and with what speed is, in the main, never discussed.

Erkki: I am at home, now, and I have not been able to find my copy of Bohm's "Quantum Mechanics", but I did look in my copy of W. Heitler's book "The Quantum Theory of Radiation". Heitler treats spontaneous emission (he calls it "induced emission") classically on pp. 36-38, and quantum mechanically on pp. 175-180. It is still not clear to me that his QM approach is anything but a semi-classical approach in QM clothing, but that is my problem.

Regards,

Tom Cuff