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PHYS 30101 Quantum Mechanics PHYS 30101 Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10) j.billowes@manchester.ac.uk These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101 Lecture 22

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Syllabus 1.Basics of quantum mechanics (QM) Postulate, operators, eigenvalues & eigenfunctions, orthogonality & completeness, time-dependent Schrödinger equation, probabilistic interpretation, compatibility of observables, the uncertainty principle. 2.1-D QM Bound states, potential barriers, tunnelling phenomena. 3.Orbital angular momentum Commutation relations, eigenvalues of L z and L 2, explicit forms of L z and L 2 in spherical polar coordinates, spherical harmonics Y l,m. 4.Spin Noncommutativity of spin operators, ladder operators, Dirac notation, Pauli spin matrices, the Stern-Gerlach experiment. 5.Addition of angular momentum Total angular momentum operators, eigenvalues and eigenfunctions of J z and J 2. 6.The hydrogen atom revisited Spin-orbit coupling, fine structure, Zeeman effect. 7.Perturbation theory First-order perturbation theory for energy levels. 8.Conceptual problems The EPR paradox, Bell’s inequalities.

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Einstein, Podolski, Rosen (EPR) problem and Bell’s theorem What is the correct interpretation of quantum mechanics? System itself does not know position of particle or direction of spin until it is forced to decide by our measurement (our present QM opinion) System has “hidden variables” which we do not know but which gives the system an underlying deterministic structure. We hide our ignorance by describing the “most probable” outcomes of measurement

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John Bell proposed a theorem “it is impossible for any local hidden variable theory to reproduce all predictions of QM” which could be tested experimentally by comparing the outcomes of spin-polarization measurements of pairs of “entangled” particles. Systems exist which emit pairs of particles in an overall spin = 0 state although each particle has non-zero spin Examples: Low-energy proton s-wave scattering from hydrogen π0π0 Back-to-back emission of photons (e - e + ) Electron-positron annihilation from 1 S 0 state

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Consider a source that emits two photons in mutually perpendicular polarization states. What happens when the polarization of one photon is measured source AB QM: Particles do not decide on polarization direction until one (A) is measured. This instantaneously (faster than light) forces B into the orthogonal state. This can be confirmed by a polarization measurement. Local hidden variables: Assumes mutually orthogonal polarization directions are determined as photons are emitted. QM prediction confirmed by series of experiments by Alain Aspect (France) testing polarizations against Bell’s Inequalities relations

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So, either a problem with quantum mechanics or relativity (instantaneous collapse of wavefunction is a problem) But so far NO QM prediction has been experimentally falsified. (ditto relativity)

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