# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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the same interpretation to the motion of an electron and regard the spin as the angular momentum associated with the electron revolving on its axis. However, as Dirac\u2019s relativistic quantum theory shows, the spin angular momentum is an intrinsic property of the electron, not a property arising from any kind of motion. The electron is a structureless point particle, incapable of \u2018spinning\u2019 on an axis. In this regard, the term \u2018spin\u2019 in quantum mechanics can be misleading, but its use is well-established and universal. Prior to Dirac\u2019s relativistic quantum theory, W. Pauli (1927) showed how spin could be incorporated into non-relativistic quantum mechanics. Since the subject of relativistic quantum mechanics is beyond the scope of this book, we present in this chapter Pauli\u2019s modification of the wave-function description so 7.1 Electron spin 195 as to include spin. His treatment is equivalent to Dirac\u2019s relativistic theory in the limit of small electron velocities (v=c! 0). 7.2 Spin angular momentum The postulates of quantum mechanics discussed in Section 3.7 are incomplete. In order to explain certain experimental observations, Uhlenbeck and Goudsmit introduced the concept of spin angular momentum for the electron. This concept is not contained in our previous set of postulates; an additional postulate is needed. Further, there is no reason why the property of spin should be confined to the electron. As it turns out, other particles possess an intrinsic angular momentum as well. Accordingly, we now add a sixth postulate to the previous list of quantum principles. 6. A particle possesses an intrinsic angular momentum S and an associated magnetic moment Ms. This spin angular momentum is represented by a hermitian operator S^ which obeys the relation S^ 3 S^ i"S^. Each type of particle has a fixed spin quantum number or spin s from the set of values s 0, 1 2 , 1, 3 2 , 2, . . . The spin s for the electron, the proton, or the neutron has a value 1 2 . The spin magnetic moment for the electron is given by Ms ÿeS=me. As noted in the previous section, spin is a purely quantum-mechanical concept; there is no classical-mechanical analog. The spin magnetic moment Ms of an electron is proportional to the spin angular momentum S, Ms ÿ gse 2me S ÿ gsìB " S (7:1) where gs is the electron spin gyromagnetic ratio and the Bohr magneton ìB is defined in equation (5.82). The experimental value of gs is 2.002 319 304 and the value predicted by Dirac\u2019s relativistic quantum theory is exactly 2. The discrepancy is removed when the theory of quantum electrodynamics is applied. We adopt the value gs 2 here. A comparison of equations (5.81) and (7.1) shows that the proportionality constant between magnetic moment and angular momentum is twice as large in the case of spin. Thus, the spin gyromagnetic ratio for the electron is twice the orbital gyromagnetic ratio. The spin gyromagnetic ratios for the proton and the neutron differ from that of the electron. The hermitian spin operator S^ associated with the spin angular momentum S has components S^x, S^ y, S^z, so that 196 Spin S^ iS^x jS^ y kS^z S^2 S^2x S^2y S^2z These components obey the commutation relations [S^x, S^ y] i"S^z, [S^ y, S^z] i"S^x, [S^z, S^x] i"S^ y (7:2) or, equivalently S^ 3 S^ i"S^ (7:3) Thus, the quantum-mechanical treatment of generalized angular momentum presented in Section 5.2 may be applied to spin angular momentum. The spin operator S^ is identified with the operator J^ and its components S^x, S^ y, S^z with J^x, J^ y, J^z. Equations (5.26) when applied to spin angular momentum are S^2jsmsi s(s 1)"2jsmsi, s 0, 12, 1, 32, 2, . . . (7:4) S^zjsmsi ms"jsmsi, ms ÿsÿ s 1, . . . , sÿ 1, s (7:5) where the quantum numbers j and m are now denoted by s and ms. The simultaneous eigenfunctions jsmsi of the hermitian operators S^2 and S^z are orthonormal hs9m9sjsmsi äss9äms m9s (7:6) The raising and lowering operators for spin angular momentum as defined by equations (5.18) are S^ \ufffd S^x iS^ y (7:7a) S^ÿ \ufffd S^x ÿ iS^ y (7:7b) and equations (5.27) take the form S^jsmsi (sÿ ms)(s ms 1) p "js, ms 1i (7:8a) S^ÿjsmsi (s ms)(sÿ ms 1) p "js, ms ÿ 1i (7:8b) In general, the spin quantum numbers s and ms can have integer and half- integer values. Although the corresponding orbital angular-momentum quan- tum numbers l and m are restricted to integer values, there is no reason for such a restriction on s and ms. Every type of particle has a specific unique value of s, which is called the spin of that particle. The particle may be elementary, such as an electron, or composite but behaving as an elementary particle, such as an atomic nucleus. All 4He nuclei, for example, have spin 0; all electrons, protons, and neutrons have spin 1 2 ; all photons and deuterons (2H nuclei) have spin 1; etc. Particles with spins 0, 1, 2, . . . are called bosons and those with spins 1 2 , 3 2 , . . . are fermions. A many particle system of bosons behaves differently from a many 7.2 Spin angular momentum 197 particle system of fermions. This quantum phenomenon is discussed in Chap- ter 8. The state of a particle with zero spin (s 0) may be represented by a state function Ø(r, t) of the spatial coordinates r and the time t. However, the state of a particle having spin s (s 6 0) must also depend on some spin variable. We select for this spin variable the component of the spin angular momentum along the z-axis and use the quantum number ms to designate the state. Thus, for a particle in a specific spin state, the state function is denoted by Ø(r, ms, t), where ms has only the (2s 1) possible values ÿs", (ÿs 1)", . . . , (sÿ 1)", s". While the variables r and t have a continuous range of values, the spin variable ms has a finite number of discrete values. For a particle that is not in a specific spin state, we denote the spin variable by ó. A general state function Ø(r, ó , t) for a particle with spin s may be expanded in terms of the spin eigenfunctions jsmsi, Ø(r, ó , t) Xs msÿs Ø(r, ms, t)jsmsi (7:9) If Ø(r, ó , t) is normalized, then we have hØjØi Xs msÿs jØ(r, ms, t)j2 dr 1 where the orthonormal relations (7.6) have been used. The quantity jØ(r, ms, t)j2 is the probability density for finding the particle at r at time t with the z-component of its spin equal to ms". The integral jØ(r, ms, t)j2 dr is the probability that at time t the particle has the value ms" for the z- component of its spin angular momentum. 7.3 Spin one-half Since electrons, protons, and neutrons are the fundamental constituents of atoms and molecules and all three elementary particles have spin one-half, the case s 1 2 is the most important for studying chemical systems. For s 1 2 there are only two eigenfunctions, j1 2 , 1 2 i and j1 2 , ÿ1 2 i. For convenience, the state s 1 2 , ms 12 is often called spin up and the ket j12, 12i is written as j"i or as jÆi. Likewise, the state s 1 2 , ms ÿ12 is called spin down with the ket j12, ÿ12i often expressed as j#i or jâi. Equation (7.6) gives hÆjÆi hâjâi 1, hÆjâi 0 (7:10) The most general spin state j÷i for a particle with s 1 2 is a linear com- bination of jÆi and jâi 198 Spin j÷i cÆjÆi câjâi (7:11) where cÆ and câ are complex constants. If the ket j÷i is normalized, then equation (7.10) gives jcÆj2 jcâj2 1 The ket j÷i may also be expressed as a column matrix, known as a spinor j÷i cÆ câ \ufffd \ufffd cÆ 10 \ufffd \ufffd câ 01 \ufffd \ufffd (7:12) where the eigenfunctions jÆi and jâi in spinor