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OrbitalEgnfns

# OrbitalEgnfns - Orbital Angular Momentum Eigenfunctions...

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Orbital Angular Momentum Eigenfunctions Michael Fowler 1/11/08 Introduction In the last lecture , we established that the operators 2 , z J J G have a common set of eigenkets , j m , ( ) 2 2 , 1 , , , z J j m j j j m J j m m j m = + = G = , = where j , m are integers or half odd integers, and we found the matrix elements of , J J + (and hence those of J x , J y ) between these eigenkets. This purely formal structure, therefore, nails down the allowed values of total angular momentum and of any measured component. But there are other things we need to know: for example, how is an electron in a particular angular momentum state in an atom affected by an external field? To compute that, we need to know the wave function . ψ If a system has spherical symmetry, such as an electron in the Coulomb field of a hydrogen nucleus, then the Hamiltonian H and the operators 2 , z J J G have a common set of eigenkets , , E j m . The spherically symmetric Hamiltonian is unchanged by rotation, so must commute with any rotation operator, [ ] 2 , 0 and , 0 z H J H J ⎤ = = G . Recall that commuting Hermitian operators can be diagonalized simultaneously—and therefore have a common set of eigenkets. Fortunately, many systems of interest do have spherical symmetry, at least to a good approximation, the basic example of course being the hydrogen atom, so the natural set of basis states is the common eigenkets of energy and angular momentum. It turns out that even when the spherical symmetry is broken, the angular momentum eigenkets may still be a useful starting point, with the symmetry breaking treated using perturbation theory. Two-Dimensional Models As a warm-up exercise for the complications of the three-dimensional spherically symmetric model, it is worth analyzing a two-dimensional circularly symmetric model, that is, ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 , , , 2 H x y x y V x y x y E x y M x y ψ ψ ψ = − + + + = = , . ψ (In this section, we’ll denote the particle mass by M , to avoid confusion with the angular momentum quantum number m – but be warned you are often going to find m used for both in the same discussion!) The two-dimensional angular momentum operator is .

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OrbitalEgnfns - Orbital Angular Momentum Eigenfunctions...

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