Commutation relations operators

The following commutation relations are readily derived from the definitions of the operators ... 

According to Eqs.(6) and (7), this becomes — (QL — LQ) = TihT 1, which when compared with the original commutation relation yields T T 1 — i. Therefore the time reversal operator is anti-linear. It can also be shown that the time reversal operator T is anti-unitary. 

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... 

Particles whose creation and annihilation operators satisfy these relationships are called fermions. It is found that [119] these commutation relations lead to wave functions in space that are antisymmetric. 

Indeed, these operators satisfy the usual commutation relations at equal times... 

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 ... 

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... 

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... 

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate, r2,... and momentum p, p2,..., boson creation and annihilation operators, b ia, bia. The index i runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i j,... 

The expectation value of H in the coherent state (7.17) can be evaluated explicitly for any Hamiltonian. However, an even simpler construction of Hd (valid to leading order in N) can be done (Cooper and Levine, 1989) by introducing intensive boson operators (Gilmore, 1981). In view of its simplicity, we report here this construction. If one divides the individual creation and annihilation operators by the square root of the total number of bosons, the relevant commutation relations become... 

Some algebras have identical commutation relations. They are therefore called isomorphic algebras. A list of isomorphic algebras (of low order) is shown in Table A.3. In this table, the sign denotes direct sums of the algebra, that is, addition of the corresponding operators. There is also the trivial case U(l) SO(2). 

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

Here a and are the usual oscillator creation and annihilation operators with bosonic commutation relations (73), and 0i,..., 1 ,..., 0Af) denotes a harmonic-oscillator eigenstate with a single quantum excitation in the mode n. According to Eq. (80a), the bosonic representation of the Hamiltonian (79) is given by... 

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. ... 

The operator a i) in the Heisenberg algebra, of course, corresponds to the operator constructed in Chapter 8. But our commutator relation (8.14) differs from the standard one, we need to modify operators. In fact, it is more natural to change also the sign of the bilinear form. Hence we dehne... 

These operators satisfy remarkable commutation relations (see Refs. 10 and 15) ... 

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

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 = ... 

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. 

The potential is treated as usual [6] as an operator subject to the commutator relation of quantum mechanics. This procedure gives the positive definite Hamiltonian (521) and vacuum energy (524) self-consistently. The scalar potential , is Fourier expanded as... 

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