Transcription of Quantum Physics II, Lecture Notes 9 - MIT OpenCourseWare
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Quantum Physics II, Lecture Notes 9 - MIT OpenCourseWare
angular momentum B. Zwiebach December 16, 2013 Contents 1 Orbital angular momentum and central potentials 1 Quantum mechanical vector identities.. 2 Properties of angular momentum .. 6 The central potential Hamiltonian.. 9 2 Algebraic theory of angular momentum 11 3 Comments on spherical harmonics 18 4 The radial equation 20 5 The free particle and the infinite spherical well 24 Free particle .. 24 The infinite spherical well .. 25 6 The three-dimensional isotropic oscillator 28 7 Hydrogen atom and Runge-Lenz vector 33 1 Orbital angular momentum and central potentials Classically the angular momentum vector Llis defined as the cross-product of the position vector lr and the momentum vector lp: Ll= lr lp . ( ) In cartesian components, this equation reads Lx = ypz zpy , Ly = zpx xpz , ( ) Lz = xpy ypx . 1 In Quantum mechanics the classical vectors lr, lp and Llbecome operators.
The classical angular momentum operator is orthogonal to both lr and p as it is built from the cross product of these two vectors. Happily, these properties also hold for the quantum angular momentum. Take for example the dot product of r with L to get . r · L = xˆ ˆ. i Li ...
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