Quantum numbers half-integral

V l5ini7iand S = I, respectively.. STmist be positive and can assume either integral or half-integral values, and the quantum numbers lie in the mterval... 

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

Beeause Pij obeys Pij Pij = 1, the eigenvalues of the Pij operators must be +1 or -1. Eleetrons are Fermions (i.e., they have half-integral spin), and they have wavefunetions whieh are odd under permutation of any pair Pij P = - P. Bosons sueh as photons or deuterium nuelei (i.e., speeies with integral spin quantum numbers) have wavefunetions whieh obey Pij P = + P. 

Fermions are particles that have the properties of antisymmetry and a half-integral spin quantum number, among others. 

The experimental figures, with one exception, were obtained from oscillation-rotation spectra with the use of integral rotational quantum numbers by Kratzer, Z. f. Physik, vol. 3, p. 289 (1920). The second figure for hydrogen chloride was calculated by Colby, Astrophys. Journ., vol. 58, p. 303 (1923), from the same data, with the use of half quantum numbers, and by Czerny,... 

Schrodinger s equation has solutions characterized by three quantum numbers only, whereas electron spin appears naturally as a solution of Dirac s relativistic equation. As a consequence it is often stated that spin is a relativistic effect. However, the fact that half-integral angular momentum states, predicted by the ladder-operator method, are compatible with non-relativistic systems, refutes this conclusion. The non-appearance of electron... 

Note the possibility of half-integral (as well as integral) values for the quantum number j. For orbital angular momentum, only integral quantum numbers occur, but for spin angular momentum, we can have half-integral values. 

It was shown in Section 1.7 that when the operators Px, PY, Pz °t>ey general angular-momentum commutation relations, as in (5.41), then the eigenvalues of P2 and Pz are J(J+ )h2 and Mh, respectively, where M ranges from — J to J, and J is integral or half-integral. However, we exclude the half-integral values of the rotational quantum number, since these occur only when spin is involved. 

NUCLEAR SPIN. The intrinsic angular momentum of the atomic nucleus due to rotation about its own axis, It is usually designated I and has the magnitude, JI 1 + 1 )h/2iz /(/z/2jt), where I is the nuclear spin quantum number which has different (integral or half-integral) values... 

Vector representations correspond to integral values of the angular momentum quantum numberj and therefore to systems with an even number of electrons. Spinor representations correspond to systems with half-integral j and therefore to systems with an odd number of electrons. Note that T is the complex conjugate of T. 

We may now use Table 8.7.1 to determine the representations, under O symmetry, spanned by a state with half-integer quantum number J. These results are summarized in Table 8.7.2. Those for states with integral J are included for convenience and completeness. In other words, this table contains all the results listed in Table 8.4.2, in addition to those for half-integer J states. 

An important postulate in connection with the spin of the electron is called the Pauli principle. It states that if a system consists of identical particles with half-integral spins, then all acceptable wave functions must be antisymmetric with respect to the exchange of the coordinates of any two particles. In our case, the particles are electrons, and the Pauli principle is formulated accordingly No two electrons in an atom can have the same set of values for all four quantum numbers. 

We know from basic quantum mechanics that angular momentum is always quantized in half-integral or integral multiples of h, where h is Planck s constant divided by 2tt. For the electron spin, the multiple (or spin quantum number) is Y2, but the value for nuclear spin differs from one nuclide to another as a result of interactions among the protons and neutrons in the nucleus. If we use the symbol I to denote this nuclear spin quantum number (or, more commonly, just nuclear spin), we can write for the maximum observable component of angular momentum... 

TTje nuclei of certain isotopes possess what can be thought of as a mechanical spin, or angular momentum. In the quantum mechanical description this is characterized by the spin number, l, which has integral or half-integral values. The spin number is related to the mass and atomic number, as summarized in Table 7-1. 

In a free multielectron atom or ion, the spin and orbital angular moments of the electrons couple to give a total angular momentum represented in the Russell-Saunders scheme by the quantum number J. Since J arises from vectorial addition of L (the total orbital quantum number) and 5 (total spin quantum number), it may take integral (or half-integral... 

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