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A Representation of Angular Momentum Operators We would like to have matrix operators for the angular momentum operators L x; L y, and L z. In the form L x; L y, and L z, these are abstract operators in an inﬂnite dimensional Hilbert space. Remember from chapter 2 that a subspace is a speciﬂc subset of a general complex linear vector space. Tutorial: projection of the angular momentum operator along the z axis Spherical coordinates (r, ,Ï) rather than cartesian coordinates (x,y,z) will be used in this exercise. In addition, we assume that the wavefunction describing a given particle depends only on the angle Ï.Thus

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Quantum Mechanics Concepts and Applications Second Edition Nouredine Zettili Jacksonville State University, Jacksonville, USA A John Wiley and Sons, Ltd., Publication 6.1 Central potential, angular momentum The Schroedinger equation for the Hamiltonian with central potential, is (6.3) For the appropriate spherical coordinates , one has (6.4) for the metric elements defined in Eq.(5.1). Using Eq.(5.5) and Eq.(5.9), we get for the gradient and Laplacian operators, (6.5) The Uncertainty Principle, Hilbert Space, Operators and Observables, Dirac Notation • Time-Independent Schrödinger Equation: Stationary States, Simple Exactly Solvable Quantum Mechanical Systems • Quantum Mechanics in Three Dimensions: Spherical Coordinates, The Hydrogen Atom, Angular Momentum, Spin

Action of Operators ∇, n and Angular Momentum Operators. Sums of Tensor Spherical Harmonics. Orthogonality, Normalization and Completeness. Expansion in a Series of Tensor Spherical Harmonics. SPINOR SPHERICAL HARMONICS . Definition. Components of Spinor Spherical Harmonics. Complex Conjugation. Time Reversal. Transformation of Coordinate ... specific spherical harmonic solutions m (θ,φ) Yl. Formally the spherical harmonics )(θ,φ m Yl are the angular portion of the solution to Laplace's equation in spherical coordinates derived in the appendix. m (, ) Yl θφ are found in the solution of any PDE which contains no explicit angular dependence. The