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:
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:
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.
(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:
(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.)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.
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.
