Condon and Shortley phase convention

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]... 

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. 

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... 

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... 

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 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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