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10 The Coulomb Problem In The Basis |H đť’śz Lz >
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Published:November 1991
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Abstract
The discussion presented below illustrates the application of operator methods in a coordinate system that is not Cartesian. The analysis is based on shift operators for n, λ, and μ. The shift operators for λ and μ are easily constructed (Section 10.1) and their properties determine in a simple manner the possible values of n, λ, and μ (Section 10.2). In Section 10.3 operators are expressed in terms of parabolic coordinates, and shift operators for n are derived in Section 10.4. The hypervirial and Hellmann-Feynman theorems are used to obtain expectation values in Section 10.5. The shift operators generate all excited bound states from the ground state (Section 10.6) and this property is used to construct normalized coordinate-space wavefunctions in parabolic coordinates (Section 10.7).
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