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Appendix 16 SOME TECHNICAL DETAILS CONCERNING BELL’S THEOREM quantization of Angular Momenta VECTORS Among the dynamic physical attributes of things are those that are fully characterized by merely specifying an amount, or a magnitude , such as for example the energy or the mass of a system, and there are those that need both a magnitude and a direction to be fully defined, such as for example a force acting on a mass. In dealing with forces it is not only important to know how strong a given force is, but also the direction in which it acts. In physics, directed quantities that carry both a magnitude and a direction are called vector quantities. ANGULAR MOMENTUM In a closed system of moving objects, one of the constants of motion is linear momentum, the product mv of mass, m, and velocity, v. Since mv is a constant of motion, we say that in every mechanical process linear momentum is conserved. Similarly, when a particle with mass m is moving in a circle with radius r and its velocity is v, its motion is characterized by a quantity called angular momentum. Angular momentum is also a vector quantity and a constant of motion; that is, in every mechanical process angular momentum is conserved. For the particle moving in a circle the magnitude of angular momentum is given by the product mvr, mass times velocity times radius, and its direction is along the axis of rotation. It is seen from this formula that, the faster a greater mass moves in a larger circle, the greater its orbital angular momentum. Similarly, the faster a bigger spherical mass spins about an axis—like the earth is spinning about an axis through its poles—the greater its spin angular momentum.  1SCHÄFER_PAGES:SCHÄFER PAGES 4/29/10 11:14 AM Page 198 QUANTIZATION OF ANGULAR MOMENTUM It was one of the unexpected discoveries of this century that masses cannot revolve in an orbit or spin about an axis with arbitrary velocities; they can do so only at certain speeds. More precisely, angular momentum is quantized. In addition, common elementary particles, like electrons, protons, or neutrons, were unexpectedly found to carry an intrinsic spin angular momentum, as though they were spheres spinning about an axis like a top. This intrinsic momentum is also quantized, being allowed a single, fixed value for each type of particle. Furthermore, when its direction is probed with respect to an axis in the laboratory, only a limited number of orientations with respect to that axis is found to exist. We say that orientation is quantized and often refer to it as space quantization. For a classical particle, the amount of spin can be measured in terms of speed, but for the point-like quantum particles speed of spin has no meaning and magnitude of spin can only be measured in terms of angular momentum. If we use the magnitude of the spin of photons as a unit, assigning it the value of , then most particles either have , ½, or  unit of angular momentum. Electrons and protons are spin-½ particles. Invariance of spin means that the amount cannot be changed; that is, it cannot be accelerated nor stopped. Spin is an intrinsic property that elementary particles own, regardless of observation. SPACE QUANTIZATION Space quantization is an amazing phenomenon. We can take an electron as a specific example. Each electron has intrinsic angular momentum with fixed magnitude (√)(h/π), where h is Planck’s constant . In addition, when the components of this spin are measured along a direction in space, only one of two possible values can be found, either +h/π or -h/π, showing that only two orientations are allowed for this vector property with respect to a given axis, one up and one down, respectively. Whenever an axis is chosen in the laboratory, no matter in what direction it is pointing, the spin angular momentum (SAM) vector of the electron must orient in space in such a way that its projections along this axis assumes one of the two allowed values, one pointing up, ↑, the other one down, ↓. A question that immediately comes to mind concerns the meaning of the phrase “given axis in the laboratory.” What is that given direction or axis, often denoted as the z-axis or z-direction, who gives it, and who  1SCHÄFER_PAGES:SCHÄFER PAGES 4/29/10 11:14 AM Page 199 [3.143.17.128] Project MUSE (2024-04-20 02:13 GMT...

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