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Light emission from the dipole's matrix elements

Why a certain seemingly abstract formula is rudimentary and essential

A participant at a popular talk about light that I gave one month ago has found a difficult comment somewhere in my slides that has scared her and she used it as an example to argue that my talk was too hard. (Perhaps a good reason not to send such documents to random people.) It was something like:

For most photons emitted around us, the probability that light is emitted by the transition from the atomic level \(\ket {E_1}\) to the level \(\ket{E_0}\) is proportional to the squared matrix element of the electric dipole \(\vec{d}_{1,0} = \bra {E_1} \vec p \ket {E_0}\).
I didn't really discuss this issue verbally – because I did make an estimate that the expectation number of the number of people who would get totally lost was above 99.5% so most likely, no one would benefit – but I do believe that a scientifically educated yet cultural human beings should be aware of the claim above.

It really describes the "bulk" of the reason why there is light around us and how matter emits it.




The category of the "scientifically educated yet cultural" people is fuzzy and there's no way to sharply define how much they should exactly know and how they should formulate it, either. All these things are fuzzy. But the value of quantum mechanics hides in a network of insights like that – this one is one of the most important ones, a "skeleton" for many quantitative predictions of this kind.




The question "Why is there any light around us?" is a rather natural question that a curious kid or another human being should ultimately ask. And when she understands that the only truly satisfactory answer is one that may be in principle made quantitative, she must be interested in the correct formulae and this is the correct formula that actually settles the question.

My quote above hides at least the following essential wisdoms. Well, the quote hides them if you really understand the quote. Before you do, you must make certain conclusions or hear additional more elementary insights to acquire the prerequisites:
  1. Light is an electromagnetic wave.
  2. Electromagnetic waves are emitted (and absorbed) when charges (and magnets) move or oscillate.
  3. Atoms are small antennas in which charges move.
  4. Their energy is discrete, not continuously adjustable.
  5. To determine something about the transition that emits the wave, one needs to specify both the initial and final state of the atom.
  6. The complex matrix elements of the evolution (unitary) operator between the initial and the final state determine the probability amplitudes. The square of their absolute value yields the probability of the \(i\to f\) transition.
  7. Quantum mechanics prohibits answers that are not probabilistic, and answers to questions that are not completely formulated (e.g. if the information about the final state is missing).
  8. We could also add where the formula comes from: the dipole is provably the relevant operator that has to be sandwiched because the interaction term of the electomagnetic field and the atoms is given by the \(j_\mu A^\mu\) term in Quantum Electrodynamics.
You know, quantum mechanics is not the set of fairy-tales and idiocies that you read in the media for the stupid people every other day. It is not spooky. It is not weird. It is not allowing any instantaneous or superluminal transmission of signals. Quantum mechanics is a "90-year-old and firmly established" or "new and currently tested" (depending on whether you consider 90-year-old grandmothers young, fresh, and sexually attractive) framework that answers all the truly physical questions, such as "Why is there light around us", and calculates all the observable quantities such as the intensity of all spectral lines of gases under specified conditions.

Some of the sentences above hold true in classical physics. Some of them say that the objects we're interested in, like atoms, may be viewed as smaller counterparts of some large classical objects, namely antennas, but because the size is so much smaller, the specific properties of quantum mechanics become very important. One of these quantum mechanical properties is that the energy spectrum (of the atoms) is not continuous but discrete – this is the observation that gave quantum mechanics (and previously the quantum theory of light) its name.



Oscillating electric currents in the antenna create electromagnetic waves. When you reduce the size of this antenna to the atomic radius, and the antenna becomes an atom, the quantum aspects of all the observables – their nonzero commutators, discrete spectra of some of them, and the probabilistic character of the predictions etc. – become essential and demonstrable by all measurements.

And most of the rest may only be obtained once you take the quantum mechanical framework totally seriously. Some items in my list are a template of the way how quantum mechanics answers all physically meaningful questions. There are some relevant physical quantities that may be in principle measured – the observables – and the universal postulates of quantum mechanics dictate that each of them is represented by a Hermitian linear operator on the Hilbert space.

Whenever we're talking about a particular transition between two states, \(\ket i\to \ket f\), the matrix elements of some operators are important to extract all the meaningful answers.\[

P_{i\to f} = \abs{\bra{f} U \ket {i}}^2

\] where \(U\) is the unitary evolution operator is the probability of the transition. And because the evolution operator \(U\) is constructed as some exponential of the Hamiltonian \(H\) (energy operator) and \(H\) is constructed out of field values and/or positions and velocities of particles, the "building blocks", the probability \(P_{i\to f}\) above may be reduced to the matrix elements of some "building blocks" operators such as the electric dipole moment.

People generally know and understand that the atoms only admit certain discrete energy levels. But the previous sentence may still be visualized and (mis)interpreted in many ways, including ways that are totally classical. People may imagine – and most of them probably do imagine – that just like a TV set where you can switch between channels, every atom objectively exists either in the state \(E_1\) or the state \(E_0\) or another one at every moment of time, but no complicated superpositions are possible. But they are always possible and whenever you measure something else than the energy \(H_a\) of the atom, you bring the atom into a state that is a general superposition of the energy eigenstates.

When you ask how much light is actually emitted, you are pretty much forced to consider the matrix elements of operators – the objects that "quantum mechanics is all about". Without an external electromagnetic field, i.e. if the Hamiltonian were just \(H_a\) for the atomic part, an atom in an energy eigenstate \(\ket{E_i}\) would be totally stationary. Nothing would ever change about it even if it were an excited level. But the full Hamiltonian is\[

H = H_a + \int d^3 x\,\,j^\mu A_\mu

\] where \(j_\mu\) is the 4-current (its temporal component is the charge density) and \(A_\mu\) is the electromagnetic potential. In quantum field theory, \(A_\mu(x,y,z,t)\) is a linear combination of operators that may create and/or destroy a photon of (almost) any frequency, direction, and polarization. So if you add the \(j\cdot A\) interaction term, the total Hamiltonian \(H\) suddenly becomes able to create and destroy photons.

How many photons are created or destroyed in given conditions is dictated by the coefficients in the decomposition of \(A_\mu(x,y,z,t)\) to the creation and annihilation operators; and by the coefficient in front of \(A_\mu\) in the Hamiltonian. This coefficient is an operator field, \(j^\mu(x,y,z,t)\), too. So the probability amplitude for the emission or absorption of a photon will unavoidably be given by this operator of the current. We want one probability amplitude, so we need to take some matrix elements and not the whole operator. Clearly, we want the matrix element between the chosen initial and final state and we must make this choice to calculate the probability of a particular transition. In classical physics, the final state was determined by the initial state as well; classical physics was deterministic. But quantum mechanics is not. We must specify both states and only when we do, the transition probability becomes calculable.

Now, the operator of the current \(j^\mu(x,y,z,t)\) is dependent on the location. But the visible light is an electromagnetic wave whose wavelength is half a micron or so. It is much longer than the atom (than the typical distance between the electron[s] and the nucleus); the situation looks like a surfer dude on top of a very long (a kilometer) and slow water wave on the ocean. So it is an extremely good approximation to assume that the atom basically feels that it's in a uniform electromagnetic field – the wave is too long and the sine locally looks like a straight line. So if we take the modes in the decomposition of \(A_\mu(x,y,z,t)\) with these low frequencies, one such relevant term is linear in \((x,y,z)\). The same function of \((x,y,z)\) "picks" a combination of the coefficients \(j^\mu\) and such a combination of the current ends up being nothing else than the electric dipole moment \(\int d^3x \,\rho\vec r\). At least the strongest or most likely transitions reduce to the electric dipole moment. The "higher" magnetic and electric moments are proportional to higher powers of \(v/c\) and the speed of electrons in the atom is much lower than the speed of light.

Students of quantum mechanics and quantum field theory learn lots of complicated enough calculations whose purpose is something seemingly "trivial", to reduce the probability amplitude which was a matrix element \(U_{fi}\) to matrix elements of the dipole moments and other things we can measure, like the intensity of light around the atom, and so on. Quantum mechanics seemingly requires "difficult enough mathematics" even to achieve things that were "almost trivial" in classical physics. But that's how Nature works.

(In classical physics, the transition probability between two microstates – points in the phase space – is always \(P=0\) or \(P=1\) depending on whether a classical solution of the time-dependent differential equations connecting the two states exists or not. The previous Yes/No classical logic is replaced by probabilities in quantum mechanics and one needs all the operators, their products, and their matrix elements to correctly calculate those probabilities. The "heart" of our thinking about the transition has to change.)

But my point is that if someone wants to know what quantum mechanics actually is and what is at least the "flavor" of the way how it explains the phenomena in Nature, it's essential to be curious at least about one truly physical and nontrivial enough question such as
Why does matter emit light? And perhaps, why do gases emit spectral lines and can one determine which of them are the most intense ones?
It's a curiosity about rudimentary observable facts about Nature – such as the existence of light – but if someone never wants to understand this thing or similar thing, he will never learn any actual physics. In particular, quantum mechanics is definitely not a discipline of science explaining things like
How did some Dutchmen proved a spooky action at a distance that Einstein claimed to be impossible?
They didn't. There is no action at a distance and what they found has been known at least for 90 years from other, equivalent experiments. These folks aren't doing physics – in the sense of the enterprise to scientifically explain phenomena that are actually observed in Nature. They're "inventing" new phenomena that may be easily misinterpreted, hyped, and sold to the stupid people as some kind of magic. But Nature actually isn't supernatural – Nature is natural – and every magician who tells you that he's doing something supernatural is fooling you.

Quantum mechanics is a pillar of science and science exists to answer questions about Nature and its inner workings. The existence of light emitted by matter is an example of a feature of the world around us that a scientifically curious mind actually wants to be clarified and quantum mechanics does so.

During the talk, I didn't say the word "matrix element" or anything like that. On the other hand, I do feel that it's shameful that popular lectures in 2015 about "the physical explanation of light" can't get to this point which has been known to be true and relevant exactly for 90 years.

Light is arising due to transitions between energy eigenstates of the atoms. But the transition is mainly linked to the electric dipole (and, less importantly, other multipoles) because an atom is a "tiny antenna", after all! However, it's such a small antenna that the electric dipole has to be treated carefully and quantum mechanically. And when we do so, we always have to discuss some linear operators associated with observable physical quantities. And the (complex) probability amplitudes are always proportional to some matrix elements between the initial state and the final state.

Assume that we talk about a "scientifically and mathematically educated and cultural" person who can learn some basic things in mathematics and wants to know what quantum mechanics has to do with the phenomena around us. One can perhaps find it's OK she won't understand the logic of the calculation of the emission of light by the atoms above.

But if she doesn't know the sketch of any similar quantum mechanical explanation of a phenomenon that would force her to see that all the physics is stored in the observables which are linear operators and their matrix elements determine probabilities, then I think that it's right to say that she doesn't have a clue what quantum mechanics actually is. All the popular stories about waves and interference and stationary states and entanglement are counterproductive to the extent to which they are compatible with the idea that physics fundamentally doesn't need any new kind of thinking and calculations – such as the matrix elements.

The very thinking about physics without non-commuting operators and/or their matrix elements is a qualitatively flawed and for 90 years obsolete way of thinking. The right way of thinking about modern physics is nowhere near such an operator-free picture.

If someone is led to believe that she understands the basic structure of quantum mechanics but she has no clue why matrix elements of anything should play any role, she's been brainwashed and gained some self-confidence that is not backed by her actual knowledge at all. And that's arguably worse than her knowing that she knows basically nothing about quantum mechanics. The illusion of understanding of science – when the people actually understand just ludicrous pop-science caricatures of science – is dangerous. The bigger is the gap between the pop-science caricatures and the actual science, the more dangerous it is (the power law actually depends on a higher exponent but I don't want to go into these details LOL).

And the gap between "what quantum mechanics actually does and how" and the "popular caricature of quantum mechanics in the media" is about as deep as the gap between Darwin's theory and creationism. You just can't and shouldn't sell elaborate stories rooted in the latter as explanations of the former. They're completely different things.

And by the way, I think that people at the level of the self-selected participants of popular talks on physics simply have to know much more about physics than the average citizens of their countries – or the average readers and viewers of the mass media. This statement seems as obvious to me as the statement that the (numerically much less selective) fans attending the ice-hockey or soccer matches know substantially more about the league and the players than the average readers of the newspapers. The apparent fact that they don't is a sign of some toxic egalitarianism – a sign that stuff that goes beyond the mediocre caricatures is largely absent everywhere. The people who want to know more or should naturally know more about science are simply not offered the same thing as the sports fans. Only stuff that "average person understands" is often considered legitimate in the public. This populist egalitarianism makes the abyss separating the lay public from the experts so deep that it's dangerous for the co-existence of these two groups.

It's pathetic that in 2015, science is getting so much worse a treatment than unimportant things like sports and "the life of socialites". It's shocking that science going beyond the ideas of Joe Sixpack is still considered politically incorrect in the public – while no one is ever discouraged from saying or writing things about relatively irrelevant things such as "celebrities" and athletes. We don't live in a scientific world yet and it's questionable whether we ever will.

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