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 ... ii = 3i so that we have the amusing three-dimensional identity . r · p = p · r + 3i . (1.16) For cross products we typically have a × b = −b × a. Indeed,

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