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25) where the powers of have been inserted in order to make F and G dimensionless, and the nature and possible values of the quantum numbers j and m appearing in the eigenvalues F and G, respectively, remain to be determined. 26) so that jˆx = (jˆ+ + jˆ− )/2 and jˆy = (jˆ+ − jˆ− )/2i. 5) jˆx† = jˆx and jˆy† = jˆy , (jˆ+ )† = jˆ− and (jˆ− ) † = jˆ+ . we have Thus, jˆ+ and jˆ− are mutually adjoint operators. i jˆx = − jˆ+ . Similarly, [jˆ− , jˆz ] = jˆ− , [jˆ+ , jˆ− ] = 2 jˆz , [jˆ+ , jˆ2 ] = [jˆ− , jˆ2 ] = 0.

For example, ∂ ∂φ100 ∂x ∂x = −∞. 47) is singular at r = 0 (it tends to −∞ at r = 0), so that, for a given fixed total energy E, the kinetic energy tends to +∞ at r = 0. 53) has a natural interpretation. Functions with l = 0, 1, 2, 3, 4, . . are called s, p, d, f, g, . . functions, respectively. Since for l = 0 we must have m = 0 always, all s functions contain the factor √ Y00 = 1/ 4π , and so are independent of θ and ϕ, that is, are spherically symmetric functions. For any given value of the principal quantum number n (except n = 1) there are three p functions (l = 1, m = 1, 0, –1).

Thus, wave functions with even l are even under inversion, and wave functions with odd l are odd under inversion. 3, have a cusp (a discontinuous gradient) at r = 0. For example, ∂ ∂φ100 ∂x ∂x = −∞. 47) is singular at r = 0 (it tends to −∞ at r = 0), so that, for a given fixed total energy E, the kinetic energy tends to +∞ at r = 0. 53) has a natural interpretation. Functions with l = 0, 1, 2, 3, 4, . . are called s, p, d, f, g, . . functions, respectively. Since for l = 0 we must have m = 0 always, all s functions contain the factor √ Y00 = 1/ 4π , and so are independent of θ and ϕ, that is, are spherically symmetric functions.

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Acoustic Pressure Promoted by Cooperating Quill Tubes without Pinholes by Barus C.


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