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Closed-Form Spherical Harmonics: Explicit Polynomial Expression for the Associated Legendre Functions
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Abstract
Although the general solution of the nonrelativistic Schrödinger equation for the one-electron atom is well known, it does not figure prominently in textbooks. Instead, a few particular solutions are given for low values of the quantum numbers n, l, and m. The reason for this is most likely the assumed very complicated appearance of the general solution. This in turn is due to the necessary participation of the associated Legendre function, which does not have a simple polynomial form.
In this communication a polynomial form is given for the associated Legendre function by means of the simultaneous use of sin q and cos q as variables. This means that one now has the complete expression as a function of the polar coordinates r, q, and j, in a form that shows the relation between the number and type of nodes and the degree in q and r, as well as the connection between the three quantum numbers and the three variables, in a direct and transparent way. Any particular solution is simply and directly found by plugging in the values of n, l, and m. This should find its way to the textbooks.
Keywords (Audience):
Upper-Division UndergraduateKeywords (Domain):
Physical ChemistryKeywords (Pedagogy):
Textbooks / Reference BooksKeywords (Subject):
Atomic Properties / StructureTools
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- Received: August 03, 2009
ACS
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