reduced%20Planck%20constant
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1: 14.30 Spherical and Spheroidal Harmonics
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14.30.11
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14.30.11_5
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►where is the reduced Planck’s constant.
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14.30.12
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14.30.13
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2: 18.39 Applications in the Physical Sciences
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►where is a spatial coordinate, the mass of the particle with potential energy , is the reduced Planck’s constant, and a finite or infinite interval.
►Here the term represents the quantum kinetic energy of a single particle of mass , and its potential energy.
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►and , has eigenfunctions
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►, , Mohr and Taylor (2005, Table XXX, p. 71), where the relationship of to SI units is spelled out.
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►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry.
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3: 20 Theta Functions
Chapter 20 Theta Functions
…4: 33.22 Particle Scattering and Atomic and Molecular Spectra
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§33.22(i) Schrödinger Equation
►With denoting here the elementary charge, the Coulomb potential between two point particles with charges and masses separated by a distance is , where are atomic numbers, is the electric constant, is the fine structure constant, and is the reduced Planck’s constant. The reduced mass is , and at energy of relative motion with relative orbital angular momentum , the Schrödinger equation for the radial wave function is given by … ►For and , the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, , and to a multiple of the Rydberg constant, ► . …5: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►Such a set is a reduced
residue system modulo .
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6: 15.13 Zeros
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►where .
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►If , , , , or , then is not defined, or reduces to a polynomial, or reduces to times a polynomial.
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7: 30.12 Generalized and Coulomb Spheroidal Functions
8: 12.19 Tables
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Miller (1955) includes , , and reduced derivatives for , , 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.
Murzewski and Sowa (1972) includes for , , 7S.
9: 36.5 Stokes Sets
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►where denotes a real critical point (36.4.1) or (36.4.2), and denotes a critical point with complex or , connected with by a steepest-descent path (that is, a path where ) in complex or space.
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36.5.4
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36.5.7
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10: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Zhang and Jin (1996, Table 3.8) tabulates for , to 8D or 8S.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.