The commutation relations

When the symmetry operator R( f) n) acts on configuration space, a function/(r, 6, ip) is transformed by the function operator R( o n) into the new function f (r, 0, ip), where 

The differences in sign in the operators in eqs. (17) and (21) and between eqs. (19) and (22) have arisen from our rules for manipulating function operators. 

Equation (3) is derived using the MRs of the infinitesimal generators (symmetry operators) and therefore holds for the operators, so that 

The commutators, eqs. (4) and (5), are derived in three different ways, firstly from eq. (11.3.9) and then in Exercises (11.4-1) and (11.4-2) and Problem 11.1. Note that It, /2, and /3 are components of the symmetry operator (infinitesimal generator) I which acts on vectors in configuration space. Concurrently with the application of a symmetry operator to configuration space, all functions fir, 0, p) are transformed by the corresponding function operator. Therefore, the corresponding commutators for the function operators are 

R( p n) is a unitary operator. Using eq. (11.3.19), which holds for small angles fr, and retaining only terms of first order in d , 

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If wx and w2 are spinors corresponding to definite energy, momentum, and helicity, the matrices ww are explicitly given by Eqs. (9-344) or (9-345). Finally the resulting traces involving y-matrices can always be evaluated using the commutation relations [y ,yv]+ — 2gr"v. Thus, for example... 

Neither nor J is hermitian. Application of equation (3.33) shows that they are adjoints of each other. Using the definitions (5.18) and (5.14) and the commutation relations (5.13) and (5.15), we can readily prove the following relationships... 

Using the commutation relation (5.10b), find the expectation value of Lx for a system in state lm). 

The commutation relations involving operators are expressed by the so-called commutator, a quantity which is defined by... 

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... 

The solute-solvent system is coupled via solvent operators (b+bf)k so that the equation of motion for the solvent operator is to be solved first. Using the commutation relations one gets for the linear term components the equation ... 

Third, the metric tensor is determined by the variables 4>, //, A. On the other hand and v never appear in Eqs.(6)-(9) (reflecting the fact that x° and x5 constant dilatations are always possible without harming the commutator relations for the Killing motions), so these equations are of first order on 4>, / and A. However, the equations can be rearranged resulting in the following symbolic structure ... 

In quantum mechanics, 1 is an operator whose Cartesian components satisfy the commutation relations... 

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... 

One defines2 a spherical tensor of rank k by the commutation relations [/z,rf] = K7f ... 

They are constructed from powers of the operators Xs and can be linear, quadratic, cubic,. Quite often a subscript is attached to C in order to indicate the order. For example, C2 denotes a quadratic invariant. The number of independent Casimir invariants of an algebra is called the rank of the algebra. It is easy to see, by using the commutation relation (2.3) that the operator... 

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... 

In addition to energy eigenvalues it is of interest to calculate intensities of infrared and Raman transitions. Although a complete treatment of these quantities requires the solution of the full rotation-vibration problem in three dimensions (to be described), it is of interest to discuss transitions between the quantum states characterized by N, m >. As mentioned, the transition operator must be a function of the operators of the algebra (here Fx, Fy, F7). Since we want to go from one state to another, it is convenient to introduce the shift operators F+, F [Eq. (2.26)]. The action of these operators on the basis IN, m > is determined, using the commutation relations (2.27), to be... 

A tensor operator under the algebra G 3 G, T, is defined as that operator satisfying the 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. 

The commutator relations (21) show that the true eigenfunctions can be chosen such that they are simultaneously eigenfunctions of the operators H, S, eind Sz. ... 

They obey the commutation relation [x, p ] = ih. We consider the transformed position and momentum operators given by... 

Since an arbitrary fc-matrix is orthogonal to S, the matrix representation for a Hermitian operator is far from unique. Suppose that H is the matrix representation of some Hamiltonian, and that S is an arbitrary matrix in S. Then H + S is an equally valid representation since the identity (p,h) = (P, H)j = (P, H - - S) clearly holds for all fe-matrices P. The explanation for such a large number of representations is straightforward the operator s corresponding to a matrix S E S is equal to the zero operator. The operator s is constructed by taking the matrix elements to be coefficients in an expansion but can then be reduced to the zero operator using the commutation relations. 

A canonical transformation, which may be single particle or many particle in nature, is one that preserves the commutation relations of the particles involved. Strictly speaking, it need not be unitary (it need only be isometric e.g., see Ref. [37]), but this distinction is less important for calculational purposes and we shall henceforth consider only unitary canonical transformations where U satisfies u = i. 

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