Angular Momentum Theory for Diatomic Molecules by Brain Judd (Auth.)

By Brain Judd (Auth.)

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The quantum-mechanical form of a was first introduced by Pauli [ 4 0 ] . The Lenz vector shares many properties with the generator A. In analogy with Eqs. 3), we can define two vectors ki and k 2 by writing ki = i ( ' + a ) , k2 = J ( ' - a ) . 38) It is at once found that ki and k 2 are angular momentum vectors, and also that [kx, k 2 ] = 0. 39) This means that ki and k 2, like Ki and K 2, can be taken as the generators of an Λ (4) group. 6). The eigen­ functions corresponding to a given energy level form basis functions for an irreducible representation of the new i2(4) group.

This follows because the Hamiltonian Η for the hydrogen atom is given by 2 2 Η = fi p /2m 2 2 Ze /r, 2 and the eigenvalue Ε is — ft p0 /2m. Thus we have a - [ ( ! x Ρ) - (P x I) + 2Zr/roo]/2p 0. 37) Runge- 59 Problems Lenz) vector. Classically it defines the direction of the major axis of an elliptical orbit. 5). In classical mechanics, as well as in quantum mechanics, /-a = 0. Both I and a commute with the Hamiltonian and are thus constants of the motion. The quantum-mechanical form of a was first introduced by Pauli [ 4 0 ] .

T h e n a t u r a l choice for our polynomials is t h e four-dimen­ sional spherical harmonics. If we specialize t o t h e hydrogen a t o m , we can set R = 0, corresponding t o a charge Ze a t t h e origin. T o expand t h e 2 kernel esc κ / 2 , we can use E q . 44), getting *(0) = — Σ " ί OoPo n ml η J , r Λ ί *m( Ω ' ) F n ί m ( Ω ) v ? ( Ω ) cW. W e now develop φ(Ω) as a series of spherical harmonics: p(Q) = Σ α > λ Μ Κ , λ μ ( Ω ) . T h e integration over Ω' can be performed a t once, a n d we get Ζ = Σ**βΥ*βΦ) ,λμ 1 -OnlmYnlmiQ).

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