Quantum MeasurementMeasurement
Spin, Polarization, and Mixed States |
Bell's Inequality
Stern-Gerlach ExperimentWe have already seen how the spin of the electron is quantized, once the electron is placed in an external magnetic field. In their experiment, German physicists Stern and Gerlach created a region of non-uniform magnetic field and passed through it a beam of atoms (for our purposes we can think of this beam as a beam of electrons, instead). What they expected to observe was a sort of fanning of the beam. This is because, depending on the orientation of the magnetic moment vector, each individual atom (or electron) would experience a force related to the orientation of its magnetic moment resulting in a vertical deflection of the beam. They also expected this deflection to be continuous for their "un-polarized" beam. Instead they observed that the beam separated into only two vertical directions (two beams: an "up" and a "down" beam) once it traversed through the non-uniform magnetic field. This "up" and "down" beams suggest that the atom's magnetic moment vector must have been either "up" or "down" and in no other orientation; i.e. quantized in the vertical direction. But what if the magnet were to be rotated to instead create a non-uniform magnetic field in the horizontal direction? They found that the beam separates into still two beams but these split horizontally; a "right" and a "left" beam! In fact, any orientation of this non-uniform magnetic field is shown to create just two beams along the direction of the magnetic field; i.e. electron's spin is quantized in the direction of the external magnetic field. Try the JAVA applet developed by the Oregon State: Do the following:
What happens if two or three of such magnets were to be used to check the "spin" of electrons? For our discussion purposes let us call a magnet of the type used in the Stern & Gerlach's experiment a SG-magnet. Now, it turns out that a vertically oriented SG-magnet creates an Up and a Down beam. A second SG-magnet oriented in any direction other than vertically will create two beams out of either the Up or the Down electrons. (A second SG-magnet analyzing the Up beam will find that all of the electrons are indeed Up; i.e. this beam will no longer separate into two beams.) A third SG-magnet in an orientation other than the one of the second magnet also creates two beams, even if it is in the vertical orientation! These experiments seem to suggest that when an electron is placed in an external magnetic field its spin can align parallel or anti-parallel to the field direction with a 50-50 probability. But these two are the only possible spin directions. This property (attribute) of the electron - its spin direction - doesn't stay with the electron, as does, say, its charge or its mass. We can call a spin "up" electron a spin-up electron only after we've just checked for vertical spin; a horizontal spin measurement "erases" the vertical spin property. To get more familiarity with this type of measurement try playing with the simulation of Stern-Gerlach experiments, courtesy of David McIntyre of Oregon State University: Spins Java Applet at: http://www.physics.orst.edu/paradigm5/spins/ Simple Math RelationshipEvidently, we cannot say that an electron that is going through a horizontal SG-magnet is a spin up (or a spin down) electron. But could we say that it is either a spin up or a spin down? Clearly, our experiment checking for vertical spin has no other possibility! Well, it turns out that we cannot make such assumptions! To see how such a supposition can be tested, let us consider the experimental arrangement that includes a source of correlated electrons pairs, four SG magnets, and four detectors. The source creates separate pairs of electrons that have a total spin of zero. That is to say, if one electron in the pair has a spin Up, its other pair has a spin Down. These electrons travel in opposite directions so that the total momentum of the pair is also zero. Each of these electrons is then send through one of two SG magnet randomly, and in this way its spin is measured. Let us label these SG magnets A, B, C, and D and arrange them so that one of the electrons in the pair goes through, say, A and B while the other goes through C and D. Now, if the electron that goes through A has its spin Up, then we record a value of +1 for this measurement; but if its spin turns out to be Down, then we record a value of -1. The same way we record values of +1 and -1 for when an electron goes through the other three SG magnets (this is all independent of the SG's orientation, but for the sake of a simpler discussion let us assume that all magnets are oriented vertically). This means then that for any electron pair we record a +1 for one electron and a -1 for the other, independent of which SG magnet any electron goes through. Is this correct? To check this assumption (why an assumption?) let us consider the following algebraic expression: X = A·C + B·C + A·D - B·D where A, B, C, and D represent the SG magnets' assigned values of +1 and -1when a magnet records an Up versus a Down spin. So, for example, in a case that one electron goes through A and is found to be Up and the other goes through D and is found to be Down, then A = +1 and D = -1. Of course, the other two variables, B and C, have not been assigned a value for this electron pair, but their values are either +1 or -1 each. What's significant here is that independent of which magnet any electron of any pair goes through and independent of what spin it has, the value of X, above, is either +2 or -2. This is of course because the algebraic expression for X was cooked up to give us this result! What value do we expect to obtain for X after a large number of trials? If we assume that the electron pair are generated randomly and each is send through one of its two possible SG magnets randomly, then we expect as many +1 as -1 values for any of the four SG magnets A, B, C, and D. That is to say, A must have as many +1 value occurrences as for its -1 value occurrences; the same of the rest of the three SG magnet measurements. As a result, the summed average of X must be zero. But when these measurements are performed, it is found that X takes on an average value that is almost as large as 3. (This happens despite of the fact that electron pairs are verified to be generated randomly and each is send to one of its two SG magnets randomly, but for a special choice of angles of the SG magnets.) So, this result tells us that we cannot assume that electron's spin is Up or Down without performing a measurement despite the fact that when we make the measurement we find that it is Up or Down!
Consequence of EPR ExperimentsThe above mathematical relation is just one way that we could test the reality as prescribed by nature. To-date many different versions of EPR experiments have been performed and have verified that our notion, based on our every day experience in our macroscopic world, that physical reality has "pre-ordained" measurable values does not apply, at least in the microscopic world. This rather odd result is not only in accordance with, but it is also predicted by the theory of quantum mechanics. According to quantum mechanics a mixed state evolves in time as a mixed state. But when we make a measurement the mixed state "collapses" into one of its possible definite values. Every one seems to agree with these two facts; i.e. that 1) nature violates Bell's inequality, and 2) quantum mechanics is the theory that correctly describes the microscopic world. But there is no clear agreement on what this really means. Some seem to believe that these results tell us that in the absence of a measurement there is no reality. Others go further by saying that it is the measurement that create the result. In the context of the classical statement: "When a tree falls in a forest will it make a sound, if no one is there to hear it?" the first group will say that there will be no sound and the second group believe that it is the presence of the observer that causes the fall of the tree to make a sound. No matter which of these descriptions one believes (or not) there are two important consequences of EPR experiments namely that two previously held doctrines of our physical reality are violated: causality and local reality. We have already discussed one quantum eraser type experiment; other disturbing experiments that demonstrate the above mentioned violations include Interaction-free Measurements, Quantum Cryptography, and Quantum Teleportation. In the context of these new types of measurements we see that despite the oddness that the quantum mechanical world has created for us, it has clearly expanded our scope of reality. Some of these, seemingly X-file physics, have already produced applications. See, for example: quantum applications of correlated photon pairs in Dr. Alan Migdall's Lab at NIST and Professor B. Saleh's Center at Boston University (go to Research). For an online tutorial and a rather cool site see QuBIT the publication of Center for Quantum Computation at Oxford University. |