Commutation relations angular momentum operators

This operator can now be shown to be identical with the operator for an infinitesimal rotation of the vector field multiplied by i, i.e. J = — M. The components of the angular momentum operator satisfy the commutation relations... 

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 ... 

Such triples of operators are often called angular momentum operators or generators of angular momentum. Sometimes they are indeed related to actual mechanical angular momentum more often, the label angular momentum is the physicists way of saying that the operators satisfy the commutation relations given above. 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an AT-electron wave function the operator a s) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

Using the anticommutation relations (13.15) we can readily verify that these operators obey the conventional commutation relations (14.2) for the irreducible components of the angular momentum operator. Further, from the definition... 

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 ... 

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 commutation relations of the orbital angular momentum operators can be derived from those between the components of r and p. If we denote the Cartesian components by the subindices i, k, and /, we can use the short-hand notation... 

We present a detailed description of angular momentum theory in chapter 5, and the reader may wish to examine the results given there at this stage. It emerges that the angular momentum operator / commutes with L and S in this axis system and its molecule-fixed components obey the usual commutation relations for angular momentum operators provided that the anomalous sign of i is used,... 

The angular-momentum operators obey the following commutation relations ... 

To avoid abstract definitions we shall give a simple example. As is well known, the angular momentum operators J+, Jo, J- satisfy the commutation relations... 

It should be noted that the simplified angular momentum operators (48) satisfy the commutation relations (49) but are not tensor operators in that sense. However, one can use them to construct angular momentum operators that are first-rank tensors (i.e., so, (3) vectors) in the following way ... 

In the following the commutation relations and the nonvanishing matrix elements of the direction cosines and angular momentum operators are summarized. For an excellent discussion of the theory of angular momentum operators the reader is referred to Ref. Derivations of the matrix elements may also be found in many textbooks on quantum mechanics. 

The operators Si satisfy the standard commutation relation for the angular momentum operators ... 

A major difficulty for molecular as opposed to atomic systems arises from the fact that two different reference axis systems are important, the molecule-fixed and the space-fixed system. Many perturbation related quantities require calculation of matrix elements of molecule-fixed components of angular momentum operators. Particular care is required with molecule-fixed matrix elements of operators that include an angular momentum operator associated with rotation of the molecule-fixed axis system relative to the space-fixed system. The molecule-fixed components of such operators have a physical meaning that is not intuitively obvious, as reflected by anomalous angular momentum commutation rules. 

Since the commutation relations determine which physical quantities can be simultaneously assigned definite values, we investigate these relations for angular momentum. Operating on some function/(jt, y, z) with Ly, we have... 

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. 

In computing the rotation Hamiltonian matrix in eqn (14.25), we should note that Hj is the projection of the angular momentum operator H along the molecular axis. Thus the angular momentum operator H,- satisfies the anomalous commutation relation ... 

The observed spin components behave like components of a vector, on rotating the coordinate axes, and from the general postulates of quantum mechanics it is inferred that the spin operators satisfy commutation relations exactly like the orbital angular-momentum operators (see... 

Instead of Cartesian coordinates it is convenient to use spherical coordinates. Properties of physical operators can be characterized according to the way they behave under rotation of the axes. These properties can be cast into a simple mathematical form by giving the commutation relations with the angular momentum. It is convenient to introduce the linear combinations... 

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... 

In any quantum-mechanical problem, we have a set of operators F, <7,... which obey certain relations. (An example is the set of angular-momentum commutation relations.) What we want to show is that the matrices formed from these operators obey the same relations as the operators, so that we can, if we like, work with the matrices instead of with the operators. There are three ways of combining operators we can add two operators we can multiply an operator by a constant we can multiply two operators. We shall examine each process in turn. 

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