Quantum mechanics wears a tuxedo

Classical electromagnetism and classical mechanics are rather intuitive and informal when compared to quantum mechanics.  In fact, so are  special and general relativity, which are updated versions of Newton's mechanics and Maxwell's electrodynamics (special) and of Newton's law of universal gravitation (general), and which are applicable to the large-scale structure of space and time. To put it differently, quantum mechanics, which applies to the sub-microscopic space and time scale, is strangely formal when compared to everything else in physics.

Why, you ask, is general relativity applicable only on the large-scale space and time stage?  Good question!  It would be great to think of gravity also acting on the teeny tiny scale, in particular if black hole theory could be invoked to explain some of the things now ruled over by quantum field theory!  This would be the reverse of what the quantum gravity gang is attempting to do.  It would remake the strong, weak, and electromagnetic forces in the image of the gravitational force, and the hail with virtual particles and complete reliance on gauge invariance!

In line with the opening pages of Sakurai, where he discusses quantum mechanical spin, we can start to discuss quantum mechanics in terms of a complex 2-D vector space.  Hoo boy. Why vector space?  Why 2-D?  Why complex?  Why why why?

On a side note, if you take the "i" off Sakurai, you have Sakura, which, according to my Level 2 Adult Piano book, translates as Cherry Blossoms, the name of a traditional and rather haunting little Japanese melody.

Looking at the emphasized words (see below) in the first chapter of Baym, you can get an idea of quantum theory's strict formality.  One underlying trait of QM is the separation of the observer and the observed.  A wave function (not too formal a word) or state vector (the preferred formal description) evolves in time deterministically as described by the Schrödinger equation--until (cymbal clash) a measurement occurs.

A measurement is a quantitative observation--that is true in all of physics.  But the quantum measurement problem brings statistics into play when a measurement is made.  An observer is not necessarily a person but could be any device capable of not only causing a measurement to be made but also recording the measurement. 

To give an overview of how QM is related to some other fields of theoretical physics, I'll quote what David J. Griffiths says in the preface of his live-cat-on-the-front-dead-cat-on-the-back textbook Introduction to Quantum Mechanics:

Unlike Newton's mechanics, or Maxwell's electrodynamics, or Einstein's relativity, quantum theory was not created--or even definitively packaged--by one individual, and it retains to this day some of the scars of its exhilarating but traumatic youth.  There is no general consensus as to what its fundamental principles are, how it should be taught, or what it really "means."  Every competent physicist can "do" quantum mechanics, but the stories we tell ourselves about what we are doing are as various as the tales of the Scheherazade, and almost as implausible.  Niels Bohr said, "If you are not confused by quantum physics then you haven't really understood it"; Richard Feynman remarked, "I think I can safely say that nobody understands quantum mechanics."

 

Next time, the quantum formalism of 2-D complex vector space...

Baym's words of emphasis

Emphasized words (those in italics) in Chapter 1 of Lectures on Quantum Mechanics, by Gordon Baym:

Polaroid filter correspondence principle exactly probability complex basis superposition principle orthonormal basis probability amplitude transformation matrix eigenvector eigenvalue angular momentum spin operator for the photon definitely definitely expectation value eigenvalue eigenstate operator completeness relation probability amplitude superposition interference never indistinguishable probability amplitudes or amplitude amplitude exactly interference completely unpolarized or not mixed state mixed case pure case extraordinary ray ordinary ray optic axis transition amplitude transition probability change unitary.

The movie cat equation described by Asher Peres

The "Live Cat Dead Cat" superposition equation that appears in A Serious Man (see my Back to the Cat and Back of the Cat blog entry from last year) also shows up in the November 1975 issue of the American Journal of Physics--the only other place I've seen it--in a short paper by a fellow named Asher Peres.  I don't know if he was (he died in 2005) related to the current president of Israel, Shimon Peres, but that is not our concern of the moment.  (However: he wasn't.  See Wikipedia article about him.)

Our concern is partly that I just stumbled upon this article and equation while looking over some old discarded issues of the AJP that I picked up when I worked as a lab assistant at Austin Community College in the 1990s.  Our other concern is for the physics involved, and whether these old 1975 statements would be legitimate today.  What statements?  Oh, just that dead cats might be made alive again (the question of whether this could happen exactly nine eight times is not addressed). 

So, first we have Asher publishing his 1974 AJP paper (which I don't have a copy of, yet) called “Quantum Measurements are Reversible,” then we have P. J. van Heerden of Polaroid Research Laboratories, Cambridge MA, in the November 1975 AJP, on pages 1014 and 1015, complaining about this particular statement in Peres’ 1974 paper:  “An ensemble of identical systems is in some state, but a single system has no state.”  P. J. v H.  retorts “I believe that this statement is incorrect,” and he goes on to give an example he claims shows that “single systems do have eigenstates,” which is the title of his paper. 

So then in response Asher pipes up in the following paper, “A single system has no state,” on pages 1015 and 1016, with, among other responses to P. J. v H.'s paper, this paragraph:  “Most puzzling is van Heerden’s statement, ‘one can even think of an experiment exhibiting the interference pattern between the cat alive and the cat dead.’  If such an experiment could indeed be performed, then the phase θ in the state

ψ = 2^-1/2[ |live> + exp(iθ)|dead>]

would be meaningful.  One could then resuscitate dead cats in the following way:  Take an ensemble of dead cats and measure on each one of them the projection operator on state ψ.  In 50% of the cases, the state of the cat will become ψ.  Now measure whether the resulting cat in state ψ is alive or dead. In 50% of the cases, it will turn out alive.  I did not say this is impossible, but only that I don’t know how to construct the ψ-measuring machine.”

Were the Coen Bros friends with Asher Peres?  Asher's homebase was Technion-Israel Institute of Technology, in Haifa, so it's unlikely that the C brothers and A. Perez crossed paths.  Ethan and-or Joel may have heard second hand through the Jewish physicists' grapevine about the outrageous idea of reviving dead cats using the superposition equation for the live and dead states with the extra ingredient of a phase factor multiplying the dead "ket." 

Sorry, that's a stupid pun, but an accurate one, because our old untalkative pal, PAM Dirac, invented a terminology that is now universally used in quantum mechanics where |a> is called a ket and <a| is called a bra.  Bras and kets, which are vectors, often get multiplied together, most often with the bra vector first.  Whadda ya got then?  Bra-ket, or bracket!  And also, you have "bras" appearing in a pointedly bra-like way in the equations of physics. 

Asher Peres spent some time, on sabbatical, at the University of Texas at Austin, in 1979 or 1980.  His Physical Review paper from 1980, called "Can We Undo Quantum Measurements?" appears in a book I have (Quantum Theory and Measurement, published in 1983 and edited by J. A. Wheeler and W. H. Zurek).  I haven't made much sense of this paper yet, due partly to my ignorance of the subject and partly to the fact that it isn't very well written.  The latter happens a lot in physics papers, and books.  Or does it just happen a lot in general?  Yeh, it may not just be your stupidity that keeps you from understanding something!  Maybe the article or book is just poorly written, or written in such a pedestrian manner as to be sleep inducing.  Stephen Hawking has confessed to not doing a very good job with the writing in A Brief History of Time, probably the most purchased yet unread physics book of all time.  Most others who don't write well don't bother with the confessions.  


(My profile lists science and math books that I think are good.)

Anyway, at the end of that 1980 paper Peres says: "I am very grateful to J. A. Wheeler for the warm hospitality of the University of Texas and to E.C.G. Sudarshan for many stimulating discussions.  The final version of this paper has benefitted from comments by J. S. Bell (CERN) and A. Ron (Technion)."

I'll be discussing John S. Bell and his famous theorem later.  A well-written article by a fellow named Travis Norsen concerning Bell's theorem just recently appeared in the American Journal of Physics.  The AJP is not for publication of professional original results, which appear in other journals--you know, those "professional" journals such as Physical Review Letters or Nature or countless others (and also of course online at open access sites)--but is instead at the level of "pedagogy."  But Norsen's article does break new ground nevertheless, I think.

Epigraph(s) from my physics MS thesis

"The investigations of the self-energy of the electron by men like Abraham, Lorentz and Poincare have long since ceased to be relevant. All that has remained from those early times is that we still do not understand the problem."
--Abraham Pais, 'Subtle is the Lord...': The Science and the Life of Albert Einstein (1982). The full names of the men mentioned by Pais are Max Abraham, Hendrik A. Lorentz, and Henri Poincare. See thesis for a discussion of their contributions.

"I do not believe that the problem of matter is to be solved by a mere field theory."

--Hermann Weyl, Gravitation and Electricity (1923). Weyl was a mathematical physicist who became an expert in general relativity.
Einstein wasn't very mathematically gifted himself, and had help developing the non-Euclidian geometry used to describe general relativity, mainly from his friend Marcel Grossmann: "Grossmann, you must help me or I will go crazy!" was Einstein's way of initially requesting assistance.

"...'matter' has lost its role as a fundamental concept."

--Albert Einstein, Relativity: The Special and the General Theory. This is a book Einstein wrote for the general public in the early 1920s. The comment here comes from a final short chapter added to the last edition of the book, published in 1952, three years before Einstein died.


"Physics is the study of the fundamental laws of nature, but what constitutes a law and which laws are taken to be fundamental are matters of evolving consensus among physicists."

--Richard A. Matzner and Lawrence C. Shepley, Classical Mechanics (1991). Matzner and Shepley are physicists at the University of Texas at Austin.

 

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These quotes are meant to touch upon a common theme. The theme is that I believe physics is off track in its reliance on virtual particles, quantum field theory and the attempt to quantize gravity.  I would like to see a return to a reliance on physical intuition as being important for understanding physics.  This is one reason I don't like Feynman diagrams.  They're a poor substitute for physical intuition, especially the simplest diagram, which is supposed to show emission of a photon by an electron.  The diagram is a ghastly schematic squiggly line plus two straight lines.

For a humorous mention of the physicist Hermann Weyl, see the 1929 interview with Dirac by the Wisconsin sports writer nicknamed "Roundy." (Somewhere else in this old blog I reference this interview, but it's worth a second link.) For a look at my thesis, see my "old writings" blog, or (maybe) Google books. 

In Dirac's book Principles of Quantum Mechanics, a copy of which can be seen on Larry Gopnik's desk when he discovers the money in the envelope, there's a famous statement about the superposition and interference of light. Normally, interference is something that occurs between two or more waves that overlap or superpose. But the representation of a photon as a coherent superposition (of left and right circular polarization states, for example) is described by Dirac in this book as a self-interference: "Each photon interferes only with itself. Interference between two different photons never occurs."

Radical!  But also maybe limited in it's applicability... 

Uncle Ilya weighs in

There were two Nobel prize winners among the faculty in the physics department at the University of Texas at Austin when I was attending classes, seminars and colloquia there during the landmark years 1987 through 1999. 

One of them was Steven Weinberg, whom UT-Austin managed to recruit from Harvard in 1980, just after Weinberg, Salam, and Glashow won the 1979 physics Nobel for their theory unifying the weak nuclear force with the electromagnetic force.  That theory goes by the name electro-weak, although now maybe the name should be electro-weak-strong, since the strong nuclear force has also been subsumed under the same field theoretical tent, leaving only our dear friend (or enemy if you fall down and get hurt) the gravitational field of force unaccounted for by quantum field theory. 

The other Nobel winner who resided in the physics department at UT-Austin during my years there was Ilya Prigogine, or Uncle Ilya as my friend Tom Mellet liked to call him.  There's some controversy about the value of Prigogine's contributions, as described in one opinionated biographical sketch, but I think he was onto something in his attempts to look at quantum theory and chaotic systems in new and often philosophically interesting ways.  In contrast, I see nothing philosophically interesting in the mathematical tour de force that is quantum field theory.

Weinberg is still living and still at UT-Austin.  Prigogine actually divided his time between the Free University of Brussels and UT-Austin, and died in Brussels in 2003, aged 86.  He was also director (appointed in 1959) of the International Solvay Institute, and it was due to his presence at UT-Austin that original black & white photos from the famous early Solvay Conferences (some with participants signatures on them)  were displayed outside the large ground floor lecture hall in the Physics-Math-Astronomy building (R. L. Moore Hall).  I once went by Prigogine's office to see if he had a copy of a book about the Solvay Conferences that I'd been unable to find elsewhere.  He wasn't there, but his secretary (first name Amy is all I remember) found the book and loaned it to me without asking who I was or when I'd return it. 

Prigogine's Nobel prize was in chemistry, awarded in 1977 for his "definition of dissipative structures and their role in thermodynamic systems far from equilibrium," as the Wikipedia article about him says.  I have a partial paperback copy (and just ordered a full hardback copy) of his 1980 book From Being to Becoming, and offer a quote from it now concering a subject I've discussed here before, the difference in pure states and mixed states in quantum mechanics:

As noted in Chapter 3, a fundamental distinction is made in quantum mechanics between pure states (wave functions) and mixtures represented by density matrices. Pure states occupy a privileged position in quantum mechanics, somewhat analogous to orbits in classical mechanics. As indicated by the Schrodinger equation (see equations 3.17 and 3.18), pure states are transformed into other pure states during the time evolution. Moreover, observables are defined as Hermitian operators mapping vectors of the Hilbert space into itself. These operators also preserve pure states. The basic laws of quantum mechanics can thus be formulated without ever invoking the density-matrix description of states corresponding to mixtures. The use of that description is considered to be only a matter of practical convenience or approximation. The situation is similar to that considered in classical dynamics in which the basic element corresponding to the pure state is the orbit or the trajectory of a dynamical system (see, in particular, Chapters 2 and 7).

In Chapter 3, the question was asked: Is quantum mechanics complete? We have seen that one of the reasons for asking this question in spite of the striking successes of quantum mechanics in the past fifty years is the difficulty of incorporating the measurement process (see the section titled The Measurement Problem in Chapter 3). We have seen that the measurement process transforms a pure state into a mixture and therefore cannot be described by the Schrodinger equation, which transforms a pure state into another pure state.

In spite of much discussion (see the beautiful account by d'Espagnat'), this problem is far from being solved. According to d'Espagnat (p. 161), "The problem [of measurement] is considered as non-existent or trivial by an impressive body of theoretical physicists and as presenting almost insurmountable difficulties by a somewhat lesser but steadily growing number of their colleagues."

I do not wish to take a position that is too strong in this controversy, because, for the present purpose, the measurement process is simply an illustration of the problem of irreversibility in quantum mechanics.

Whatever the position one takes, the fundamental distinction between pure states and mixtures and the privileged position of the pure states in the theory must be given up. Thus, the problem is to provide a fundamental justification for this loss of distinction. It is a remarkable fact that the introduction of the entropy operator M (see the section on irreversibility and the formalism of classical and quantum mechanics in Chapter 8) as a fundamental object of the theory entails just this loss of distinction betWeen pure states and mixtures.

Einstein-Schrödinger, Summer of 1935

Letters exchanged between Einstein and Schrödinger in the summer of 1935 had a lot to do with Schrödinger 's article published later that year introducing his quantum cat problem to the world. I've taken the quotes below from The Shaky Game: Einstein, Realism and the Quantum Theory, (C) 1986 by Arthur Fine, who, in this book, was the first to point out  how much Einstein influenced Schrödinger's invention of the simultaneously-alive -and-dead cat in the box.  

Google Books has at least some of Fine's book avaiable for viewing online. 

"During the summer of 1935," says Fine, "Schrödinger was in residence at Oxford, while Einstein was spending the summer at Old Lyme, Connecticut.  The Einstein, Rosen, Podolsky (EPR) paper. .. came out in the May 15, 1935, issue of The Physical Review. Schrödinger wrote about it to Einstein on June 7, and his part of the correspondence continued with letters to Einstein on July 13, August 19, and October 4. Before receiving the letter of June 7, Einstein had written to Schrödinger on June 17 and then wrote again, responsively, on June 19, August 8, and September 4."

Schrödinger to Einstein, June 7:

I am very pleased that in the work that just appeared in Physical Review you have publicly called the dogmatic quantum mechanics to account over those things that we used to discuss so much in Berlin. Can I say something about it? It appears at first as objections, but they are only points that I would like to have formulated yet more clearly.

Einstein, on June 17, before he'd received Schrödinger's June 7th letter, wrote to him and mentioned, among other things, the possibility of Schrödinger's appointment to the Institute for Advanced Study, in Princeton, New Jersey, where Einstein found employment soon after Hitler rose to power and Einstein gave up his professorship in Berlin. The Nazi's wanted to fire Einstein anyway (they labeled relativity "Jewish physics" and wouldn't allow it to be taught in Germany) but he pre-empted them by quitting. 

Einstein to Schrödinger, June 17:

From the point of view of principles, I absolutely do not believe in a statistical basis for physics in the sense of quantum mechanics, despite the singular success of the formalism of which I am well aware. I do not believe such a theory can be made general relativistic. Aside from that, I consider the renunciation of the spatio-temporal setting for real events to be idealistic-spiritualistic. This epistemology-soaked orgy ought to come to an end. No doubt, however, you smile at me and think that, after all, many a young whore turns into an old praying sister, and many a young revolutionary becomes an old reactionary.

"Einstein wrote again just two days later," Fine writes, "expressing his pleasure at  Schrödinger's 'detailed letter.' . . .  Recall that Einstein uses the analogy of finding a ball after opening one of two covered boxes in order to explain the idea of completeness and to motivate the intuitive concept of local causality (his 'separation principle'). It is in the context of this ball-in-the-box analogy, I believe, that Schrödinger's cat begins.  It has to do with the idea of completeness, concerning which Einstein writes this:"

Einstein to Schrödinger, June 17:

Now I describe a state of affairs as follows:  The probability is 1/2 that the ball is in the first box. Is this a complete description?

NO:  A complete statement is: The ball is (or is not) in the first box.  That is how the characterization of the state of affairs must appear in a complete description.

YES:  Before I open them, the ball is by no means in one of the two boxes. Being in a definite box only comes about when I lift the covers. This is what brings about the statistical character of the world of experience, or its empirical lawfulness. Before lifting the covers the state [of the two boxes] is completely characterized by the number 1/2, whose significance as statistical findings, to be sure, is only attested to when carrying out observations. Statistics only arise because observation involves insufficiently known factors, foreign to the system being described.

We face similar alternatives when we want to explain the relation of quantum mechanics to reality. With regard to the ball-system, naturally, the second "spiritualist" or Schrödinger interpretation is absurd, and the man on the street would only take the first, "Bornian" interpretation seriously. But the Talmudic philosopher dismisses "reality" as a frightening creature of the naive mind, and declares that the two conceptions differ only in their mode of expression.

Schrödinger to Einstein, July 13:

You have made me extremely happy with your two lovely letters of June 17 and 19, and the very detailed discussion of very personal things in the one and very impersonal things in the other. I am very grateful. But I am happiest of all about the Physical Review piece itself, because it works as well as pike in a goldfish pond and has stirred everyone up. . .

I am now having fun and taking your note to its source to provoke the most diverse, clever people: London, Teller, Born, Pauli, Szilard, Weyl. The best response so far is from Pauli who at least admits that the use of the word "state" ["Zustand"] for the psi-function is quite disreputable. What I have so far seen by way of published reactions is less witty. ... It is as if one person said, "It is bitter cold in Chicago"; and another answered, "That is a fallacy, it is very hot in Florida." . . .

My great difficulty in even understanding the orthodoxy over this matter has prompted me, in a lengthy piece, to make the attempt to analyze the current interpretation situation once and for all from scratch. I do not know yet what and whether I will publish on it, but this is always the best way for me to make matters really clear to myself. Besides, a few things in the present foundation strike me as very strange.

Einstein to Schrödinger, August 8:

The system is a substance in chemically unstable equilibrium, perhaps a charge of gunpowder that, by means of intrinsic forces, can spontaneously combust, and where the average lifespan of the whole setup is a year.  In principle this can quite easily be represented quantum-mechanically.  In the beginning the psi-function characterizes a reasonably well-defined macroscopic state.  But, according to your equation, after the course of a year this is no longer the case at all. Rather, the psi-function then describes a sort of blend of not-yet and of already-exploded systems.  Through no art of interpretation can this psi-function be turned into an adequate description of a real state of affairs; in reality there is just no intermediary between exploded and not-exploded.
. . .
My solution of the paradox presented in our work is this.  The  ψ function does not describe a state of one system, rather (statistically) an ensemble of systems. For a given

ψ₁ wavefunction a linear combination c₁ψ₁  +  c₂ψ₂  signifies an expansion of the totality of systems.  In our example of the system composed of two parts A, B, the change that the ψ function suffers if I make an observation on A signifies, conversely, the reduction to a subensemble from the whole ensemble; the reduction simply occurs in accord with a varying point of view, depending on the choice of the quantity that I measure on A.  The result is then an ensemble for B, that likewise depends on this choice.

 Schrödinger to Einstein, August 19:

Many thanks for your lovely letter of 8 August.  I believe it doesn't work [das geht nicht] that one relates the psi-function to an ensemble of systems and thereby solves the antinomy or paradox.  To be sure I do not like the idiom "das geht nicht" at all, for it expresses the prejudice of the people with blinders who take certain computational devices as permanently established because otherwise they could not advance their own [ideas].

By the way, The Hebrew University's Einstein website is putting or has put about 80,000 Einstein documents online, as announced last year on Einstein's birthday. 

Get on the wave train...

I still have almost all the math and physics textbooks I bought for classes during my undergrad and graduate school careers, and one of my favorite texts is simply called Optics.  My copy of the book is a first edition (1974) and third printing (December 1976), and was the chosen textbook for an undergraduate class I took in the spring of 1978 at the University of Arkansas at Little Rock called Optics and Wave Motion.

(Bill Clinton was elected governor of Arkansas for the first time in 1978.  In '77 and '78, I worked as a nightwatchman at the Old State House while attending UALR.  Clinton gave a fundraising party at the Old State House in early 1978, at night.  Making one of my hourly rounds through the building, I crossed paths with him.  He shook my hand and said, "I'm Bill Clinton.  I'm running for governor."  I said, "I'm David Trulock.  I work here."  That was it, as far as I recall.)

The authors of this particular text on optics are Eugene Hecht and Alfred Zajac, professors of physics at the time at Adelphi University.  I think they did a good job writing the book, which is not the general rule with textbooks. Here's a quote from their book, one of several I'll be discussing related to the concept of coherent superpostion and polarization. This is from the section titled "Natural Light" in the chapter on polarization:

An ordinary light source consists of  a very large number of  randomly oriented atomic emitters. Each excited atom radiates a polarized wave train for roughly 10^-8 seconds.  All of the emissions having the same frequency will combine to form a single resultant polarized wave which persists for no longer than 10^-8 sec.  New wave trains are constantly emitted and the overall polarization changes in a completely unpredictable fashion. If these changes take place at so rapid a rate as to render any single resultant polarization state indiscernable, the wave is referred to as natural light. It is also known as unpolarized light, but this is a bit of a misnomer since in actuality the light is composed of a rapidly varying succession of the different polarization states.

 We can mathematically represent natural light in terms of two arbitrary, incoherent, orthogonal, linearly polarized waves of equal amplitude (i.e. waves for which the relative phase difference varies rapidly and randomly).

Okay. So natural light, say sunlight such as that streaming in the library window right now (11:27 am CST) onto my keyboard, involves a bunch of little bitsy wave trains, which is what I meant when I said light is produced by random atomic electromagnetic expectorations.  The often-depicted sine wave (single frequency) light wave is quite different:

Keep in mind that an idealized monochromatic plane wave must be depicted as an infinite wave train. If this disturbance is resolved into two orthogonal components perpendicular to the direction of propagation, they, in turn, must have the same frequency and be infinite in extent, and therefore be mutually coherent (ε = constant).  In other words, a perfectly monochromatic plane wave is always polarized. …

What I really want to get to is Hecht & Zajac's discussion of representing a linearly polarized beam of light as consisting of identical photons which themselves must be considered individually as coherent superpositions of left and right circularly polarized states.  This relates to Gordon Baym's discussion of delicate phase relationships, and is the same idea of the superposition of live and dead cat states in the dilemma of Schroedinger's cat:

We have already seen that an electromagnetic wave can impart both energy and linear momentum. Moreover, if a plane wave incident upon some material is circularly polarized we expect electrons within the material to be set into circular motion in response to the force generated by the rotating E-field.  Alternatively we might picture the field to be composed of two orthogonal P-states which are 90° out of phase.  These simultaneously drive the electron in two perpendicular directions with a π/2 phase difference.  The resulting motion is again circular.  In effect the torque exerted by the B-field averages to zero over an orbit and the E-field drives the electron with an angular velocity ω equal to the frequency of the electromagnetic wave.  Angular momentum will thus be imparted by the wave to the substance in which the electrons are imbedded and to which they are bound. …

Thus far we’ve had no difficulty in describing purely right- and left-circularly polarized light in the photon picture; but what is linearly or elliptically polarized light? Classically, light in a linear polarized state can be synthesized by the coherent superposition of equal amounts of light in right and left circularly polarized states (with an appropriate phase difference).  Any single photon whose angular momentum is somehow measured will be found to have its spin totally either parallel or anti-parallel to its direction of propagation.  A beam of linearly polarized light will interact with matter as if it were composed, at that instant, of equal numbers of right- and left-handed photons.

There is a subtle point that has to be made here. We cannot say that the beam is actually made up of precisely equal amounts of well-defined right- and left-handed photons; the photons are all identical. Rather, each individual photon exists simultaneously in both possible spin states with equal likelihood. On measuring the angular momentum of the constituent photons, -ћ would result equally as often as +ћ.  This is all we can observe. We are not privy to what the photon is doing prior to the measurement (if indeed it exists prior  to the measurement).  As a whole, the beam will therefore impart no total angular momentum to the target.

In contrast, if each photon does not occupy both spin states with the same probability, one angular momentum, say  +ћ, will be measured to occur somewhat more often than the other, - ћ.  In this instance, a net positive angular momentum will therefore be imparted to the target.  The result en masse is elliptically polarized light, i.e. a superposition of unequal amounts of right- and left-handed light bearing a particular phase relationship.

(For the uninitiated, E(t) = Asin(ωt + ε) is an equation for a sine wave, or single frequency wave, with amplitude A, frequency ω, phase difference ε relative to some other sine wave, and t representing time; E and B are the standard letters designating the electric and magnetic fields, respectively;  P-states are linearly polarized states,  π/2 is a different way of saying 90°, and  ћ is Planck's constant divided by 2π.)

So I hope you all can see again that even for describing a single photon, the same existential language and quantum superposition model as in the Schroedinger cat problem are necessary.  But are they sufficient?  That's the big question in the debate on the EPR problem, which Einstein and Niels Bohr continued to disagree on until they died (E. in 1955, B. in 1962).  Next time: The Shaky Game: Einstein, Realism & Quantum Theory, by Arthur Fine.