Condon and Shortley Phase

By standard procedures the tx states may be diagonalized with respect to the z operators, yielding / mr> kets (with / = 1). If one adopts a Condon and Shortley phase convention [9] these eigenkets read ... 

Brown and Howard (1976) have suggested that all molecule-fixed matrix elements be evaluated in terms of space-fixed operator components. The reasons for this are the space-fixed components of all operators obey normal commutation rules it is natural to adopt the Condon and Shortley phase convention for... 

This phase convention is similar to what is often called the Condon and Shortley phase convention, which specifies that ... 

In recent years a more sophisticated approach using irreducible spherical tensor operators has been found advantageous [13,16-18]. Symmetric top matrix elements can be obtained by using the following extension of the Wigner-Eckart theorem to axially symmetric systems (within the Condon and Shortley phase convention)[16]... 

The angular dependence of the coefficients C R, < a. < b) can be expressed in a closed form. The relevant formulae are obtained by asymptotic expansion of the polarization series truncated at some finite order. In practice such an asymptotic expansion is best performed by evaluating the polarization energies (as given by equations 9, 18, and 21) using the multipole expansion of the electrostatic potential l/ ri — r2. The latter expansion can be written in terms of either the Cartesian or the spherical tensors. The spherical formulation appears to be more popular because it leads much more easily to closed formulae and only this formulation will be considered in this article. Denoting by (r) the regular solid harmonic r Cim(0,), where Cim 0,) is the spherical harmonic in the Racah normalization and with the Condon and Shortley phase, one can write ... 

The spherical harmonics with Condon and Shortley phase [16, p. 52] are given by ... 

It is customary to follow the Condon and Shortley (1967) (CS) choice of phase for the eigenfunctions j m) by setting 7 = 0, thus taking the phase factor exp(i7) as unity for all values of the quantum numbers j and m. The derivation of the corresponding relation... 

It is important to note that different authors use different phase conventions. Those of Condon and Shortley [CSh35] will be employed here, requiring [STa63]... 

Phase conventions have been chosen to be consistent with those of Condon and Shortley.13 In terms of tensor operators, the square modulus of f becomes... 

There is a phase convention implicit in these two equations, the so-called Condon and Shortley convention [9], which is universally adopted. 

Cji(Q) is a Racah spherical harmonic, in the phase of Condon and Shortley, which can also be written as a special (K = 0) rotation matrix ... 

We choose, as an archetype, the one-electron function of Schrodinger, ft. Using Condon and Shortley (1967) phase, this is expressed in somewhat modified form as,... 

The choice of a phase convention is a matter of taste. However, the convention adopted must be internally consistent. See Brown and Howard (1976) for a discussion of the Condon and Shortley (1953) phase convention and molecule-versus space-fixed angular momentum components see Larsson (1981) for a brief but comprehensive summary of all of the most frequently encountered phase conventions. Throughout this book, an attempt has been made to use the phase conventions of Eq. (3.2.82), Eq. (3.2.85a), Eq. (3.2.86), and Eq. (3.2.87), and... 

For reasons of notation we have included a phase factor i, and the spherical harmonics Y (r) have the phase defined by Condon and Shortley [5.3]. Inside the muffin-tin well the radial part p (E,r) must be regular at the... 

The spherical harmonics are defined with a specific phase factor that may be different in different presentations. Here, we followed Edmonds [70], which is the convention by Condon and Shortley [71] commonly used, where the (—1) prefactor of the spherical harmonics, Eq. (4.121), multiplied by the (—1) prefactor of the Legendre polynomials, Eq. (4.124), yields a total prefactor of (+1) for Y/m in the case of I = m. The definition of Y in Eq. (4.121) differs from the one of Bethe [72] by a factor of (—1) . Compared to Schiff [73] the Y/ are equal for negative values of m, while for positive values they differ by the factor (—1) . 

Now consider the commutator of J3 with the isotopic spin raising operator T+ = Ti -I- iT2 which, in the Condon-Shortley phase convention (Condon and Shortley, 1963) has the following effect ... 

The associated Legendre polynomuds [11,12] may be obtained from the Rodrigues expression (in the phase convention of Condon and Shortley)... 

The spherical harmonics are complex functions difficult to visualise and also their handling is impractical. A simple unitary transformation exists, yielding the real, normalised and orthogonal functions—angular wave functions Yfa for m positive only. They are collected in Table 1.9, using the Condon-Shortley phase convention. 

This transformation is in accordance with the Condon-Shortley phase conventions for the spherical basis functions [7]. In fact, our initial Hamiltonian matrix in Eq. (7.21) was constructed in this way. The resulting vector corresponds to the triplet spin functions, which we used in Sect. 6.4. The total spinor product space has dimension 4. The remainder after extraction of the three triplet functions corresponds to the spin singlet, which is invariant and transforms as a scalar. Spinors are thus the fundamental building blocks of 3D space. Their transformation properties were known to Rodrigues as early as 1840. It was some ninety years before Pauli realized that elementary particles, such as electrons, had properties that could be described... 

Note, that this definition includes the Condon-Shortley phase (-1) , which is sometimes left for the sake of convenience (e.g. Bohren and Huffman, p. 90 Spiegel 1995, pp. 243-244), but has then to be included at other places of the spherical harmonics. 

The phase choice (sign) is referred to the Condon-Shortley convention the factor —1 occurs only for positive odd values of m. Another phase choice, according to the Fano-Racah convention, can be met in literature and hence... 

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