6_Identical_Particles

6_Identical_Particles - VI IDENTICAL PARTICLES Griffiths...

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Unformatted text preview: VI. IDENTICAL PARTICLES Griffiths Chapter 5, Shankar Chapter 10 1 A. Definition Two particles are said to be identical if they are exact replicas of each other in every respect. There should be no experiment that can detect any intrinsic difference between them. This definition of identical particles is the same in classical and quantum physics. The implications are very different. 2 B. The Classical Case Imagine a pool table with four holes labelled 1, 2, 3 and 4. Place two identical billiard balls, label them 1 and 2, near holes 1 and 2, respectively. 3 4 Let’s shoot these two balls down the table at the same time. Since they are identical, the resulting configuration should appear exactly the same if we interchanged the two of them. 3 1 1 2 2 B. The Classical Case 3 4 Let’s shoot the billiard balls towards the diagonally facing pockets and have two physicists predict what will happen. Physicist 1 predicts that ball 1 will end up in pocket 4, and ball 2 in pocket 3. Physicist 2 predicts that ball 1 will end up in pocket 3 and ball 2 in pocket 4. 1 2 1 2 Upon shooting the balls simultaneously, suppose we find that physicist 2 is right: ball 1 lands in pocket 3 and ball 2 in pocket 4. Now if a judge walks in only at the end of the experiment, who will he say is right? 4 B. The Classical Case Now if a judge walks in only at the end of the experiment, who will he say is right? 3 4 He only sees the final configuration, which is identical upon exchanging the balls, and he doesn’t know which ball is which. He will conclude that both physicists are correct. What is it that we know, that this judge doesn’t, that allows us to see that physicist 2 is right? We have seen the trajectories of the balls. 5 1 2 B. The Classical Case In classical physics, we can distinguish between identical particles by following their non-identical trajectories. We can, in principle, measure these trajectories without disturbing them. Consequently, two configurations related by exchanging identical particles are physically nonequivalent. In quantum physics, we have no trajectories that we can measure without disturbing the system. To be true to our postulates and the empirical evidence that inspired them, we can’t allow ourselves to distinguish configurations based on any notions of different trajectories. Consequently, two configurations related by exchanging identical particles must be treated as one and the same configuration, described by the same quantum wave function. 6 C. The Quantum Implications 1. Symmetric and Antisymmetric States Suppose we have two distinguishable particles 1 and 2, and a position measurement shows particle 1 to be at x = a and particle 2 to be at x=b. We could write this composite state as ψ (x1 = a, x2 = b) or just |ab￿ position of p article #1 position of p article #2 See board VI.C.1.a Since the particles are distinguishable, the exchange state |ba> is a different state. 7 C. The Quantum Implications Now let’s repeat this experiment but with two identical particles. Suppose we again find one particle at a and the other at b. Which is the correct state, |ab> or |ba>? Answer: Neither! The are two possibilities for the correct state: |ab￿ + |ba￿ symmetric state vector or |ab￿ − |ba￿ See board VI.C.1.b antisymmetric state vector In either case, it doesn’t matter which particle is at a and which particle is at b (aside from a minus sign for the antisymmetric case). 8 C. The Quantum Implications 2. Bosons and Fermions Suppose an electron, for example, could exist in either the symmetric or the antisymmetric state. Then, by postulate #2, it could also exist in a superposition of those states. But this linear combination of symmetric and antisymmetric states is itself neither symmetric nor antisymmetric. (Check this yourself!) So, according to our thinking, this is not a possible state. Therefore, an electron must be either symmetric or antisymmetric permanently. 9 C. The Quantum Implications It happens that electrons are antisymmetric. So are protons and neutrons. These are all examples of FERMIONS. Photons turn out to be symmetric. Photons, gravitons and pions are all examples of BOSONS. Existing in symmetric or antisymmetric states under exchange of identical particles is then, like spin, an intrinsic property of each type of quantum particle. In fact, it turns out (using relativistic quantum mechanics) that all half integer spin particles are Fermions (they exist in antisymmetric states) and all integer spin particles are Bosons (they exist in symmetric states). 10 C. The Quantum Implications How can we be sure that nature behaves like we are predicting here, with particles being Fermions or Bosons? Experimentally, we find that Fermions obey the Pauli Exclusion Principle, whereas Bosons do not. This means that two identical Fermions cannot exist in the same quantum state. See board VI.C.2 As a result, Fermions and Bosons obey different statistics when doing statistical mechanics. Fermions obey Fermi-Dirac Statistics and Bosons obey Bose-Einstein Statistics. See Griffiths section 5.3: Solids One can therefore do experiments to show that electrons are Fermions, for example, and that nature does behave with respect to identical particles just as we think. ...or one can use quantum field theory... 11 C. The Quantum Implications 3. Other Consequences Griffiths goes into a lot of depth about many of the implications of the existence of Fermions and Bosons for quantum identical particles. Exchange Forces Atoms and the Periodic Table Solids and Band Structure Quantum Statistical Mechanics and the Blackbody Spectrum Note that none of his discussion deals with entangled states. See his footnote 2 in section 5.1.1. 12 ...
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