Half-integers

Suppose now that the total wave function (state) is nondegenerate, and F is half-integer. From Eq. (29), it then follows... 

Finally, let us consider molecules with identical nuclei that are subject to C (n > 2) rotations. For C2v molecules in which the C2 rotation exchanges two nuclei of half-integer spin, the nuclear statistical weights of the symmetric and antisymmetric rotational levels will be one and three, respectively. For molecules where C2 exchanges two spinless nuclei, one-half of the rotational levels (odd or even J values, depending on the vibrational and electronic states)... 

We now compare the results calculated for the fundamental frequency of the symmetric stretching mode with the only available experimental datum [78] of 326 cm . The theoretical result is seen to exceed experiment by only 8.3%. It should be recalled that the Li3 and Li3 tiimers have for lowest J the values 0 and respectively. Thus, the istopic species Li3 cannot contribute to the nuclear spin weight in Eq. (64), since the calculations for half-integer J should employ different nuclear spin weights. Note that atomic masses have been used... 

These two facts imply that in case of an odd dimension the quantization is characterized by (a series of) integers only [as in Eq. (62)] but in case of an even dimension it is characterized either by (a series oO integers or by (a series of) half-integers [as in Eqs. (72)]. 

Only integer and half-interger values ean range from j to -j in steps of unity. Speeies with integer spin are known as Bosons and those with half-integer spin are ealled Fermions. 

There are two possible cases for the wavefunction of a system of identical fundamental particles such as electrons and photons. These are the symmetric and the antisymmetric cases. Experimental evidence shows that for fermions such as electrons and other particles of half integer spin the wavefunction must be anti-symmetric with respect to the interchange of particle labels. This... 

The maximum value of a, our A, is usually called j the eigenvalues of Z, our a, are called m. We have seen that j is either an integer or a half-integer, and that m ranges from — j to +j in unit steps. The eigenvalue of f2, the quantity b2, is related to j by Eq. (7-31), which, when solved for 62, gives... 

The integer n labels the wavefunctions and is called a quantum number. In general, a quantum number is an integer (or, in some cases, Section 1.10, a half-integer) that labels a wavefunction, specifies a state, and can be used to calculate the value of a property of the system. For example, we can use n to find an expression for the energy corresponding to each wavefunction. 

This agrees with experimental findings on the decomposition of acetaldehyde. The appearance of the three-halves power is a wondrous result of the quasisteady hypothesis. Half-integer kinetics are typical of free-radical systems. Example 2.6 describes a free-radical reaction with an apparent order of one-half, one, or three-halves depending on the termination mechanism. 

The presence of low-lying excited levels can greatly increase the efficiency of the relaxation processes, especially in the case of paramagnetic centers with half-integer spins. 

It seems worthwhile giving one possible example of nonconservative schemes. A cross scheme was very popular and much applicable in recent decades. Any scheme of this structure can be written on a chess grid by regarding e, p, rj to half-integer nodes (sj+1/2.fj+1/2) 3-nd v, x to integer nodes of the grid at hand. 

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