Commutation relations spin operators

In Chapter 1 we have discussed the familiar realization of quantum mechanics in terms of differential operators acting on the space of functions (the Schrodinger wave function formulation, also called wave mechanics ). A different realization can be obtained by means of creation and annihilation operators, leading to an algebraic formulation of quantum mechanics, sometimes called matrix mechanics. For problems with no spin, the formulation is done in terms of boson creation, b (, and annihilation, ba, operators, satisfying the commutation relations... 

There are a variety of formalisms that allow for a mapping of a discrete quantum system onto a continuous analog (for reviews see Refs. 218 and 219). The most prominent examples are Schwinger s theory of angular momentum [98] and the Holstein-Primakoff transformation [97], both of which allow a continuous representation of spin degrees of freedom. To discuss these two theories, we consider a spin DoF that is described by the spin operators Si,S2,Si with commutation relations... 

Consider, furthermore, a (2i- - 1)-dimensional subspace of the Hilbert space with fixed 5. Then, according to Schwinger s theory of angular momentum [98], this discrete spin DoF can be represented by two bosonic oscillators described by creation and annihilation operators with commutation relations... 

The mapping preserves the commutation relations (70) of the spin operators. As can be seen from (75d), the image of the 2s + 1)-dimensional spin Hilbert space is the subspace of the two-oscillator Hilbert space with 2s quanmm of excitation— the so-called physical subspace [218, 220], This subspace is invariant under the action of any operator which results by the mapping (75a)-(75d) from an arbitrary spin operator A(5 i, S2, S3). Thus, starting in this subspace the system will always remain in it. As a consequence, the mapping yields the following identity for the matrix elements of an operator A ... 

The Holstein-Primakoff transformation also preserves the commutation relations (70). Due to the square-root operators in Eqs. (78a)-(78d), however, the mutual adjointness of S+ and 5 as well as the self-adjointness of S3 is only guaranteed in the physical subspace 0),..., i- -m) of the transformation [219]. This flaw of the Holstein-Primakoff transformation outside the physical subspace does not present a problem on the quantum-mechanical level of description. This is because the physical subspace again is invariant under the action of any operator which results from the mapping (78) of an arbitrary spin operator A(5i, 2, 3). As has been discussed in Ref. 100, however, the square-root operators may cause serious problems in the semiclassical evaluation of the Holstein-Primakoff transformation. 

It follows from the fermion commutation relations that the entries of a fe-matrix are related by a system of linear equalities. For example, consider the pair transport operator = 2 b a a b + h ala b ), which moves a spin-up, spin-down pair of electrons between sites p,v of A. If we define w o = ... 

These commutation relations are taken to be the basic property of all angular momentum operators, including the spin operators which cannot be expressed in terms of the position coordinates xyz. In addition to the components of A/, we must deal with the operator M2 ... 

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. 

The indices i and j in (3 7) now refer to the n molecular orbitals without the spin factor.These operators fulfill the same commutator relation as the generators of the unitary group of dimension n, and are often referred to as generators. The commutator relation has the following form ... 

A vectorial product will be defined below by (5.14), and V as a tensor of first rank is defined by (2.12). Operator L may be defined also in a more general way by the commutation relations of its components. Such a definition is applicable to electron spin s, as well. Therefore, we can write the following commutation relations between components of arbitrary angular momentum j ... 

A detailed analysis (Chapter 11) shows that this result depends upon the commutation relations for the L operators, and, since the spin and the total angular momentum operators obey the same commutation relations (CRs), this formula holds also for S and for J ... 

L, S, J AAA L, S, J L, S, J A J j =jl +j2 orbital, spin, and total angular momenta quantum mechanical operators corresponding to L, S, and J quantum numbers that quantize L2, S2, and J2 operator that obeys the angular momentum commutation relations total (j) and individual (ji, j2, ) angular momenta, when angular momenta are coupled... 

The eigenvalues and eigenfunctions of the orbital angular momentum operators can also be derived solely on the basis their commutation relations. This derivability is particularly attractive because the spin operators and the total angular momentum obey the same commutation relations. 

The Hamiltonian (1) is spin free, commutative with the spin operator S2 and its z-component Sz for one-electron and many-electron systems. The total spin operator of the hydrogen molecule relates to the constituent one-electron spin operators as... 

When this operator is substituted into equation (7.246), it corresponds to a small rotation in rotational-spin-orbital space. The parameter. si governs the magnitude of this rotation. It is a variable parameter which can be chosen to eliminate terms from the transformed Hamiltonian. Using this form for F and the well known commutation relations between the molecule-fixed components of J, S and L, it is easy to show that... 

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

Note that commutation of cluster operators holds only when the occupied and virtual orbital spaces are disjoint, as is the case in spin-orbital or spin-restricted closed-shell theories. For spin-restricted open-shell approaches, where singly occupied orbitals contribute terms to both the occupied and virtual orbital subspaces, the commutation relations of cluster operators are significantly more complicated. See Ref. 36 for a discussion of this issue. 

With Ki a. spherical tensor of arbitrary rank, either spatial or spin, irreducible spherical tensors are defined by their commutation relations with linear Cartesian operators in the following manner... 

It is seen that (157) has the operator structure and algebraic properties similar to those of (136)—(137). At r 0, the set (157) exactly coincides with (136)— (137). Due to the commutation relations (155), the operators (157) have the same algebraic properties as do (136)—(137) at any given point. In particular, we can construct the local representation of the radiation phase operators in the same way as in Section IV, using the operator (158) instead of (63). By construction, this gives us the 57/(2) quantum phase of spin or polarization with the properties described in Section IV.C. 

We start again with the pointgroup D4. The eigenvalues under the C4 operation are now e m/4 and e l3n/4. The commutation relations are as for integer spin and operation with Q transforms an eigenstate X4) into another eigenstate X4 ). Again there is freedom in the choice of phases. Since now, however, we have ... 

When speaking of kinematic interaction, it should be noted that the problem of its separation in connection with the transition from Pauli operators to Bose operators is far from new. This problem arises, in particular, for the Heisenberg Hamiltonian, which corresponds, for example, to an isotropic ferromagnet with spin a = 1/2 when spin waves whose creation and annihilation operators obey Bose commutation relations are introduced. This problem was dealt with by many people, including Dyson (6), who obtained the low-temperature expansion for the magnetization. However, even before Dyson s paper, Van Kranendonk (7) proposed to take into account of the kinetic interaction by starting from a picture where one spin wave produces an obstacle for the passage of another spin wave, since two flipped spins cannot be located at the same site (for Frenkel excitons this means that two excitons cannot be localized simultaneously on one and the same molecule). 

Tensors (15.39)—(15.41) meet commutation relations (14.2) for irreducible components of the momentum operator, and, in addition, they commute with the operators of orbital (14.15) and spin (14.16) angular momenta for the lN configuration, since they are scalars in their respective spaces. Accordingly, the states of the lN configuration can be characterized by the eigenvalues of operators L2, Lz, S2, Sz, Q2, Qz. 

The corresponding Hamiltonian operator will still be given in terms of proper expansions over bilinear forms of (boson) creation and annihilation operators. (The more complex situations including half-spin particles can be addressed as well by using fermion operators [20].) The general rule is that one introduces a set of (n -I-1) boson operators b, and b (/, y = 1,. . . , n + 1) satisfying the commutation relations... 

We postulate that the spin angular-momentum operators obey the same commutation relations as the orbital angular-momentum operators. Analogous to [Lj, Ly = ihL [Ly, LJ = ihL [Lj, Lj,] = ihLy [Eqs. (5.46) and (5.48)], we have... 

The spin angular-momentum operators obey the general angular-momentum commutation relations of Section 5.4, and it is often helpful to use spin-angular-momentum ladder operators. 

It took a little while for the full power of the WE theorem to be appreciated. In its most obvious applications, it allows any vector operator to be replaced by another provided a proportionality constant is included. For example, we can write L+2S = gJ since L, S and J all satisfy similar commutation relations with respect to J. The spin-orbit equivalence. 

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