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Abstract
The azimuthal and magnetic quantum numbers of spherical harmonicsYlm(θ,ϕ)describe quantization corresponding to the magnitude andz-component of angular momentum operator in the framework of realization ofsu(2)Lie algebra symmetry. The azimuthal quantum numberlallocates to itself an additional ladder symmetry by the operators which are written in terms ofl. Here, it is shown that simultaneous realization of both symmetries inherits the positive and negative(l-m)- and(l+m)-integer discrete irreducible representations forsu(1,1)Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation ofsu(2)compact Lie algebra via theYlm(θ,ϕ)’s for a givenl, we can also representsu(1,1)noncompact Lie algebra by spherical harmonics for given values ofl-mandl+m.
Highlights
The set of principal, azimuthal, magnetic, and spin quantum numbers describe the unique quantum state of a single electron for any system in which the potential depends only on the radial coordinate
In order to provide the necessary background and to attribute a quantization relation for azimuthal quantum number l, here, we present some basic facts about the spherical harmonics [2, 3]
Our proposed ladder operators for simultaneous shift of l and m are of first-order differential type, contrary to [2]. They lead to a new perspective on the two quantum numbers l and m in connection with realization of u(1, 1) (su(1, 1)) Lie algebra which in turn is accomplished by all spherical harmonics Ylm(θ, φ) with constant values for l − m and l + m, separately
Summary
The azimuthal and magnetic quantum numbers of spherical harmonics Ylm(θ, φ) describe quantization corresponding to the magnitude and z-component of angular momentum operator in the framework of realization of su(2) Lie algebra symmetry. The azimuthal quantum number l allocates to itself an additional ladder symmetry by the operators which are written in terms of l. It is shown that simultaneous realization of both symmetries inherits the positive and negative (l − m)- and (l + m)-integer discrete irreducible representations for su(1, 1) Lie algebra via the spherical harmonics on the sphere as a compact manifold. In addition to realizing the unitary irreducible representation of su(2) compact Lie algebra via the Ylm(θ, φ)’s for a given l, we can represent su(1, 1) noncompact Lie algebra by spherical harmonics for given values of l − m and l + m
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