Those delicate phase relationships

Okay, to get back to the predicament of Schroedinger's cat, the reason we can't apply normal logic and say the cat is either alive or dead and we don't know which until we look is because quantum logic defies normal logic.  The cat is in a superposition of states of being alive and being dead.  In the state vector notation of quantum mechanics, this can be written

|Φ> = α|Ψ_alive> + β|Ψ_dead>,

where in this equally probable two-state example, α = β = 1⁄√2.  In Lectures on Quantum Mechanics, page 26, Gordon Baym discusses the difference between the classical either/or case and the quantum "unique superposition" case (which Gary Zukav noted the strangeness of by calling it a "thing in itself").  Instead of a cat Baym is talking about a single photon and how it's a superposition of polarization states:  

“It is very important to realize that the statement ‘the photon is either in state |Ψ₁> or |Ψ₂>, but we don’t know which,’ is not the same statement as ‘the photon is in a superposition of |Ψ₁> and |Ψ₂>.’ … The point is that when we say that a photon is in a state |Φ> that is a linear combination of states |Ψ₁> and |Ψ₂>,

|Φ> = α|Ψ₁> + β|Ψ₂>,

then we are implying that the relative phase of the coefficients α and β is certain.  For example, if |Ψ₁> = |x> and  |Ψ₂> = |y>, then if α = 1⁄√2 and β = i ⁄ √2, clearly |Φ> = |R>, but if if α = 1⁄√2 and  β = – i ⁄ √2, then |Φ> = |L>, which is a state completely different from |R>.  The relative phase must be fixed to specify the linear combination uniquely.  On the other hand, when we say that a photon is either in state |Ψ₁> or |Ψ₂>, but we don’t know which, then we are implying that there is no connection between the phases of these states, and hence no possibility of interference effects, which depend critically on delicate phase relations…”

The state |R> represents right circular polarization, |L> represents left circular polarization, and, of course, i is the square root of negative one.  Who would have thought this could be so much fun?!

Father's Day '77 and a 2-D complex vector space

I found an example from electrical engineering of a complex two-dimensional vector space.  If you already dislike math, you won't be too happy with these two pages from Circuits, Devices and Systems, © 1976, by Ralph J. Smith, used for a beginning EE class I took in the first summer term of 1977 at the University of Arkansas at Little Rock.

 

I'm not sure if engineers still prefer to use the letter j instead of the usual letter i in writing a complex number, but they preferred it back in the seventies, so that's what the j is doing here (below), representing the imaginary part of a complex number.  Vectors may be written in different equivalent ways, and one of those ways is to use complex numbers, which have a real (x-axis) part and an imaginary (y-axis) part.  For example, in the number 8.66 - j5  (see below) the real part is 8.66 and the imaginary part is 5. 

What this author says about electrical engineering is also true of quantum mechanics.  We have electronic devices (especially iPod, iPhone, iPad, etc) that depend on recently-discovered quantum effects, but in studying how these devices work, and in making the discoveries they are based on, we need abstract ideas.  (Very abstract, not to mention formal, ideas!  That's what I'm trying to make more sense of in these writings of mine.  Quantum theory is too abstract to understand as it stands.)

Yeh, those are my comments in the margin from 36 summers ago concerning TV ownership.

Our vector in this example, below, is called a phasor, a term used in physics as well as engineering, and also in Star Trek, but with a different meaning.  Electrical voltages and currents in alternating current (AC) circuits are generally not "in phase" with each other.  A phasor has a voltage or current as its "amplitude," and it has a phase angle ("phase" for short).  The variation in time of amplitudes and phases of different currents and voltages can be kept track of by the use of phasors.

The "Imaginary" and "Real" axes make this graph (above)  a complex two-dimensional vector space.  The variation in time of  60 Hz, by the way, is the standard frequency of household AC voltage.  Whatever the frequency is, Kenneth, it must be the same for both vectors in order for them to be represented in this simple way.

So, the amplitudes and the phase angles of the two vectors (above) are what are important. And here's the importance of my choosing this example:  Amplitude and phase are what are important in quantum mechanics. They take on strange new significance:  In quantum measurements, an "amplitude" becomes a "probability amplitude," and phase angles between vectors become those "delicate phase relationships" that are responsible for the weirdness of quantum superposition.

But how does this relate to Father's Day of 1977?   Well, I was enrolled in the EE class and reading and working problems in this textbook then.  Also I'd recently found a job as the 4-pm-to-midnight night watchman at the Old State House in Little Rock.  So that's a couple of possible reasons I'm remembering this Father's Day so well.

My family had lunch at The Embers (actually the Plantation Embers), a white-table-cloth restaurant in a better-than-average highway motel called the Pine Bluff Motel.  I usually got roast lamb with mint jelly when we ate there, so that's probably what I ate for lunch.  And my gift for Dad on that Father's Day was a copy of the New York Times Book Review.

 

This may be another reason I remember that day better than other Father's Days.  I had just discovered that the Book Review by itself was available at Publisher's Bookshop in Little Rock.  Back then, it was impossible in Little Rock to get a copy of the New York Times on the day it came out. It took until Tuesday or Wednesday for the Sunday NYT to arrive by truck and be available for sale in LR.

When I handed a copy of the Book Review to Dad at The Embers, I said it was not an expensive gift but also not an easy one to get.  He said something like "Is this last week's?"  I had the pleasure of saying no, it was the current issue.  Whether he knew about the early availability of the Book Review, I don't know.  But he sure didn't let on.  Dad and I had been to New York together (on our way to Rome) the previous October.  Not that he even read the NYT, but he was bookish.

He also got a kick out of the big headline for the leading book review in that June 19, 1977, issue:  "Newton, Kant, Ruskin, Virgins All."  It was a review of "The Book of Lists". The memorable title of that review could be another reason I recall that day and giving him that as a gift.

P.S.  And let's not forget that one scene in A Serious Man takes place at a restaurant called Embers.

 

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.

More about delicate phase relations (quotes)

In 1918 Hermann Weyl tried to reproduce electromagnetism by adding the notion of an arbitrary scale or gauge to the metric of general relativity—and noted the “gauge invariance” of his theory under simultaneous transformation of the electromagnetic potentials and multiplication of the metric by a position-dependent factor.  Following the introduction of the Schrödinger equation in quantum mechanics in 1926 it was almost immediately noticed that the equations for a charged particle in an electromagnetic field were invariant under gauge transformations in which the wave function was multiplied by a position-dependent phase factor.  The idea then arose that perhaps some kind of gauge invariance could also be used as the basis for formulating theories of forces other than the electromagnetism.  After a few earlier attempts, Yang-Mills theories were introduced in 1954 by extending the notion of a phase factor to an element of an arbitrary non-Abelian group.  In the 1970s the Standard Model then emerged, based entirely on such theories.  In mathematical terms, gauge theories can be viewed as describing fiber bundles in which connections between values of group elements in fibers at neighboring spacetime points are specified by gauge potentials—and curvatures correspond to gauge fields.  (General relativity is in effect a special case in which the group elements are themselves related to spacetime coordinates.)

—Stephen Wolfram, A New Kind of Science, Notes for Chapter 9, p. 1045.

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. . .  The difference between a neutron and a proton is then a purely arbitrary process.  As usually conceived, however, this arbitrariness is subject to the following limitation:  once one chooses what to call a proton, what a neutron, at one space-time point, one is then not free to make any choices at other space-time points.

          It seems that this is not consistent with the localized field concept that underlies the usual physical theories.  In the present paper we wish to explore the possibility of requiring all interactions to be invariant under independent rotations of the isotopic spin at all space-time points. …

—Yang and Mills (1954).  Used as epigraph to Part III (Non-Abelian Gauge Theory and QCD) in Gauge Theories in Particle Physics:  A Practical Introduction, second edition, by Ian J. R. Aitchison and Anthony J. G. Hey.   

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Isotopic spin (isospin; isobaric spin)  A quantum number applied to hadrons (see elementary particles) to distinguish between members of a set of particles that differ in their electromagnetic properties but are otherwise apparently identical.  For example if electromagnetic interactions and weak interactions are ignored, the proton cannot be distinguished from the neutron in their strong interactions:  isotopic spin was introduced to make a distinction between them.  The use of the word “spin” implies only an analogy to angular momentum, to which isotopic spin has a formal resemblance.

—A Dictionary of Physics, third edition, Oxford University Press, 1996.

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     The theory of the addition of quantum mechanical angular momenta has found an interesting application in the study of strongly interacting particles.  Basically the proton and neutron are very similar particles.  Their masses are nearly equal, they each have spin ½, and when interacting with themselves or other particles via strong interactions  they behave very similarly  Of course, the one striking difference between the neutron and proton is that the proton has a charge, whereas the neutron has none.  This means that they have very different electromagnetic interactions;  e.g., an electron will be attracted to a proton but will feel rather indifferent about a neutron.  However, the strong forces are several orders of magnitude stronger, at small distances, than electromagnetic forces, and in studying such forces, one can to a first approximation neglect the electromagnetic forces.  Thus from the point of view of strong interactions, the neutron and proton are practically identical.

     In fact, one can say that the neutron and proton are the same particle—the nucleon, N—which has an internal degree of freedom which can take on two possible values—protonliness or neutronliness.  This is analogous to our thinking of a spin-up electron and a spin-down electron as being the same kind of particle—the electron—in different states of the internal degree of freedom—spin.  By analogy, one calls the internal degree of freedom of the nucleon isotopic spin, or isospin for short.  The “isospin up” state of the nucleon is the proton and the “isospin down” state is the neutron.

   —Gordon Baym, Lectures on Quantum Mechanics, first two paragraphs of Chapter 16: Isotopic Spin.

The Φ in eiΦ

Well, sure, of course, the Φ in e^iΦ is some kind of angle, and if Φ = 0, then e^iΦ = 1, and we have the usual state vector for Schrödinger’s Cat.  So in my previous post, I was only wondering what the variable Φ could represent in this particular coherent superposition.  Φ could certainly be time-dependent, resulting in the "delicate phase relation" between the live and dead components being time-dependent.  This could be just a more general expression for the Shroe cat state, for all I know.  I just haven't seen it before and thus am prompted by the movie to check into it.  It could just be another foolin' around on the part of the prone-to-fool-around Coen brothers.  Except they'd have to rely on a knowledgeable physicist to have given them the opportunity.

If Φ = π = 180 degrees, then e^iΦ = -1.  Live and dead components are "out of phase."  Of course, just as an abstract entity, e^iπ = -1 is interesting in its own right.  Right?

 

Polymeric molecules, pure and mixed states

While it's still August, I'll quote again from friendly Frank J. Blatt's physics textbook, this time concerning one particular type of polarizer.  I bought the textbook at a yard sale at 4110 Avenue F in Austin, Texas in August 1997.  Or maybe it was 4112 Ave F.  Anyway, I like Blatt's writing:

The most common polarizer in use today is a synthetic dichroic material invented by E. H. Land. Polaroid is produced by stretching a sheet of polyvinyl alcohol, thereby aligning the long hydrocarbon chains of the polymeric molecules. The sheet is then impregnated with iodine, which attaches itself to the polymers and results in good electrical conduction along the chains, but allows little conduction perpendicular to them. Electromagnetic waves whose E vector is parallel to the polymers are then strongly absorbed by this material.

The E vector represents the electric field's direction and strength. 

Now a bit more from Gordon Baym on the mysteries of quantum mechanics:

There are really two levels of probability in quantum mechanics. The first, which is called the pure case, is when we have a system that is in a definite state (often called a pure state). Then the behavior of this system in a given experiment is governed by the probability amplitude rules we have been discussing. The second case is when the system can be in any of several states, with certain probabilities. This case is called the mixed case, and we say that the photon is in a mixed state. Then one must calculate the results one expects in a given experiment for each of the separate states that can be present, and take the weighted average of the result over these, as we have done ...   In averaging over the various states that can be present, one uses the ordinary classical probability rules; there is no possibility of interference occurring between these states.

Now, I'm not saying it makes any sense, yet.  But I learn stuff from copying it down, so by typing the quote, I will I hope remember its content.  I'll have more to say on pure and mixed states later.  To begin with, however, you can see that purity involves those delicate phase relationships and the possibility of interference, and mixed-ity involves only ordinary classical probabilities, with no chance of quantum mechanical interference.  Quantum superposition and interference are thus closely related.

And speaking of closely related:  what kicked off this whole polarization discussion was the mention by Emil Wolf of the close relationship of polarization and coherence (of light), and the fact that I had been thinking something similar myself before Wolfie wrote about it.

And now back to our program

Symmetry and invariance in physics and their relation to conservation laws, both classical and quantum: that's the program, for the most part.  Oh, yeh, and Schrödinger's Cat, also.  And A Serious Man, peripherally, as a sort of springboard.  Remember in the movie there is a very unusual form of the Schrödinger's Cat quantum superposition equation on the board in the non-dream college classroom scene?  An unusually delicate phase relationship is involved in that particular equation.

  

How did we get to where we are now in physics?  Where are we now, anyway?  We're deeply into quantum field theory (QFT), and looking to make it work also for gravity.  I don't like quantum field theory and only barely understand it, so my plan now is to temporarily, in the next few posts, describe it as if I do like it!  I'll be a gauge-invariance/QFT cheerleader for the next month or so and see where that leads. Let's hear it for those hardworking virtual particles and perturbation expansions!