Now back to our program. We've looked at introducing quantum mechanics from the photon point of view, mainly as described in the first chapter of Gordon Baym's book Lectures on Quantum Mechanics. Baym makes the point that once photons are introduced, things get weird. Namely, in a linearly polarized beam of light, all the photons (we'd think) would be identical, so how come some get through a polarizer oriented at an angle different to that of the polarization plane of the beam, and some don't get through?
The reason is quantum superposition, which, even more than Heisenberg's uncertainty principle, is what makes quantum theory so different and inexplicable in terms of the classical ideas of continuity and unique identity.
Baym's discussion focuses on the concepts of the photon and quantum (or coherent) superposition. He doesn't really emphasize the importance of how these concepts together lead to the fundamental formalism of quantum mechanics, which requires a complex vector space, as opposed to the "real" vector space with which classical mechanics can be fully described.
One of the other books I discussed earlier is Modern Quantum Mechanics, by J. J. Sakurai. Whereas Baym takes the quantization of the electromagnetic field--the existence of the photon--as his starting point, Sakurai introduces quantum superposition as a way, the only way, of explaining certain experiments on the spin of the electron, and then points out how beams of light passing thru polarizers can be explained the same way without even invoking the concept of photons.
Sakurai opens his book by discussing an experiment involving the orientation of individual silver atoms after they pass through an inhomogeneous magnetic field. (Silver atoms ejected from an oven in gaseous form and collimated into a beam which passes thru the poles of a magnet is what we're talking about, y'all.) This famous experiment was first done in 1922 by Otto Stern and Walter Gerlach. The inhomogeneous B field is used in order to exert a net force up-or-down on the atoms, due to the atoms having one unpaired outer electron. The first non-classical aspect of this arrangement is that it shows the outer electrons of the silver atoms have only two orientations in space instead of a continuum of values. Thus the demonstration of two discrete beams of silver atoms emerging from the Stern-Gerlach (SG) setup shows experimentally the quantized "spin" of the electron.
But the second non-classical effect that can be demonstrated with this experiment is the really strange one. In a sequential setup that 1) blocks one of the two up-or-down (z-direction) beams (say the downward oriented beam) emerging from the standard SG setup, then sends the one remaining beam thru 2) a B field in the x direction (horizontal; standard SG setup tilted 90 degrees), resulting in two horizontally deflected discrete beams, then 3) blocks one of these beams and sends the remaining single beam thru a final B field in the z direction (same as original SG setup), the result is that there are again two beams with z direction orientations, up and down. The downward oriented beam that was blocked in 1) has come back in 3) without a fresh new group of downward-z-oriented atoms being used! Yeh, I had to study on it myself before I could wrap my mind around it for any length of time.
Then Sakurai shows how, in order to describe all possible orientations of the beams in relation to each other--the z, the x and the y orientations--complex numbers (a "complex vector space") must be used. His introduction to the use of complex vector space is by way of polarized light. The third orientation (the y direction) of the spinning electron is analogous to circularly polarized light, he says. He also describes how th the 3-polarizer setup for light beams, is just like the three-SG setups in that coherent superposition is the only way to describe how a polarized beam can get thru the three consecutive, differently oriented polariods (polarizers). Sakurai then mentions that he did not have to invoke the concept of photons in order to accurately describe the 3-polarizer effect. Coherent superposition and complex vector space together are sufficient.
In a footnote, Sakurai says even though he didn't make use of the idea of photons, the reader can see Baym's book for an "extremely illuminating" discussion from the photon point of view (no pun intended, probably). One reason Baym's description is illuminating is that he introduces the essential mystery of the 3-polarizer effect by commenting on the weirdness of one polarizer's effect on supposedly identical photons--some get thru and some don't.
The reason is quantum superposition, which, even more than Heisenberg's uncertainty principle, is what makes quantum theory so different and inexplicable in terms of the classical ideas of continuity and unique identity.
Baym's discussion focuses on the concepts of the photon and quantum (or coherent) superposition. He doesn't really emphasize the importance of how these concepts together lead to the fundamental formalism of quantum mechanics, which requires a complex vector space, as opposed to the "real" vector space with which classical mechanics can be fully described.
One of the other books I discussed earlier is Modern Quantum Mechanics, by J. J. Sakurai. Whereas Baym takes the quantization of the electromagnetic field--the existence of the photon--as his starting point, Sakurai introduces quantum superposition as a way, the only way, of explaining certain experiments on the spin of the electron, and then points out how beams of light passing thru polarizers can be explained the same way without even invoking the concept of photons.
Sakurai opens his book by discussing an experiment involving the orientation of individual silver atoms after they pass through an inhomogeneous magnetic field. (Silver atoms ejected from an oven in gaseous form and collimated into a beam which passes thru the poles of a magnet is what we're talking about, y'all.) This famous experiment was first done in 1922 by Otto Stern and Walter Gerlach. The inhomogeneous B field is used in order to exert a net force up-or-down on the atoms, due to the atoms having one unpaired outer electron. The first non-classical aspect of this arrangement is that it shows the outer electrons of the silver atoms have only two orientations in space instead of a continuum of values. Thus the demonstration of two discrete beams of silver atoms emerging from the Stern-Gerlach (SG) setup shows experimentally the quantized "spin" of the electron.
But the second non-classical effect that can be demonstrated with this experiment is the really strange one. In a sequential setup that 1) blocks one of the two up-or-down (z-direction) beams (say the downward oriented beam) emerging from the standard SG setup, then sends the one remaining beam thru 2) a B field in the x direction (horizontal; standard SG setup tilted 90 degrees), resulting in two horizontally deflected discrete beams, then 3) blocks one of these beams and sends the remaining single beam thru a final B field in the z direction (same as original SG setup), the result is that there are again two beams with z direction orientations, up and down. The downward oriented beam that was blocked in 1) has come back in 3) without a fresh new group of downward-z-oriented atoms being used! Yeh, I had to study on it myself before I could wrap my mind around it for any length of time.
Then Sakurai shows how, in order to describe all possible orientations of the beams in relation to each other--the z, the x and the y orientations--complex numbers (a "complex vector space") must be used. His introduction to the use of complex vector space is by way of polarized light. The third orientation (the y direction) of the spinning electron is analogous to circularly polarized light, he says. He also describes how th the 3-polarizer setup for light beams, is just like the three-SG setups in that coherent superposition is the only way to describe how a polarized beam can get thru the three consecutive, differently oriented polariods (polarizers). Sakurai then mentions that he did not have to invoke the concept of photons in order to accurately describe the 3-polarizer effect. Coherent superposition and complex vector space together are sufficient.
In a footnote, Sakurai says even though he didn't make use of the idea of photons, the reader can see Baym's book for an "extremely illuminating" discussion from the photon point of view (no pun intended, probably). One reason Baym's description is illuminating is that he introduces the essential mystery of the 3-polarizer effect by commenting on the weirdness of one polarizer's effect on supposedly identical photons--some get thru and some don't.
