Quantum Physics II, Lecture Notes 9 - MIT OpenCourseWare

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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 quantum mechanics the classical vectors lr, pl and Ll. become operators. More precisely, they give us triplets of operators: lr → (ˆx, y,ˆ zˆ), lp → ( ˆpx ,pˆy ,pˆz ), (1.3) Ll → (L. ˆ. x ,Lˆy ,Lˆz ). When we want more uniform notation, instead of x, y, and z labels we use 1, 2 and 3 labels:

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