Commutation relations systems

The second axiom, which is reminiscent of Mach s principle, also contains the seeds of Leibniz s Monads [reschQl]. All is process. That is to say, there is no thing in the universe. Things, objects, entities, are abstractions of what is relatively constant from a process of movement and transformation. They are like the shapes that children like to see in the clouds. The Einstein-Podolsky-Rosen correlations (see section 12.7.1) remind us that what we empirically accept as fundamental particles - electrons, atoms, molecules, etc. - actually never exist in total isolation. Moreover, recalling von Neumann s uniqueness theorem for canonical commutation relations (which asserts that for locally compact phase spaces all Hilbert-space representations of the canonical commutation relations are physically equivalent), we note that for systems with non-locally-compact phase spaces, the uniqueness theorem fails, and therefore there must be infinitely many physically inequivalent and... 

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

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 ... 

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... 

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 ... 

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 = ... 

Finally, we make a few additional remarks. First, note that a pure number state is a3j= state whose phase 0k is evenly distributed between 0 and 2n. This is a consequence of the commutation relation [3] between Nk and e,0 <. Nevertheless, dipole mafKi w elements calculated between number states are (as all quantum mechanical amplitudes) well-defined complex numbers, and as such they have well-defined phajje j S Thus, the phases of the dipole matrix elements in conjunction with the mode ph f i f/)k [Eq. (12.15)] yield well-defined matter + radiation phases that determine the outcome of the photodissociation process. As in the weak-field domain, if only gJ one incident radiation mode exists then the phase cancels out in the rate expres4<3 [Eq. (12.35)], provided that the RWA [Eqs. (12.44) and (12.45)] is adoptedf However, in complete analogy with the treatment of weak-field control, if we irradh ate the material system with two or more radiation modes then the relative pb between them may have a pronounced effect on the fully interacting state, phase control is possible. 

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,... 

Its primary use in our applications is for expressing various commutation relations in compact form and performing calculations involving them. The Levi-Civita symbol also arises naturally in vector analysis. Thus, if et, i = 1, 2, 3, are three mutually perpendicular unit vectors defining a right-handed coordinate system, then... 

Quantum mechanics involves two distinct sets of hypotheses—the general mathematical scheme of linear operators and state vectors with its associated probability interpretation and the commutation relations and equations of motion for specific dynamical systems. It is the latter aspect that we wish to develop, by substituting a single quantum dynamical principle for the conventional array of assumptions based on classical Hamiltonian dynamics and the correspondence principle. 

It is noted that the complete Schrodinger equation is a second-order differential equation in the spatial coordinates and a first-order differential equation in the variable time. Therefore, it is not rigorously a wave equation (which would require a second derivative with respect to time). On the other hand, the variable time does not enter the equation as an observable but as a parameter to which well-defined values are attributed. Thus, there are no commutation relations involving a time operator. Nevertheless, it is possible to establish an indeterminacy relation involving energy and time, similar to those previously found for position and momentum. If At is the lifetime of a given state of the system, there will be an indeterminacy in the energy of such a state ... 

The angular momenta of Eq. (3.60) behave quite differently from those referring to a molecular coordinate system with three rotational degrees of freedom. In particular this relates to the commutation relations of the corresponding quantum mechanical operators. This is briefly discussed below in connection with the evaluation of the quantum kinetic energy. 

In order to rigorously describe the nonlinear interaction between the weak pulsed fields, we now turn to the fully quantum treatment of the system. The traveling-wave electric fields can be expressed through single mode operators as j(z, t) = dj(t) Cqz (j = 1, 2), where uj is the annihilation operator for the field mode with the wavevector kp + q. The singlemode operators a and aq possess the standard bosonic commutation relations... 

The first two tenns on the right describe the system and the bath , respectively, and the last tenn is the system-bath interaction. This interaction consists of terms that annihilate a phonon in one subsystem and simultaneously create a phonon in the other. The creation and annihilation operators in Eq. (9.44) satisfy the commutation relations ... 

In classical mechanics, positions and momenta are treated on an equal footing in the Hamiltonian picture. In quantum mechanics, they become operators, but it is true that the position r and momentum p of a particle are appropriate conjugate variables that can entirely equivalently describe a state of a system under the commutation relation [r, p] = i (Dirac, 1958). This equivalence is usually demonstrated by the example of the onedimensional harmonic oscillator. The choice of the most appropriate representation depends on convenient description of the phenomenon considered. Generally, the position representation is useful for most bound-state problems such as atomic and molecular electronic structures as well as for many scattering problems. The momentum-space treatment... 

Previous work has not investigated if commutation relations are conserved upon transformation to effective operators. Many important consequences emerge from particular commutation relations, for example, the equivalence between the dipole length and dipole velocity forms for transition moments follows from the commutation relation between the position and Hamiltonian operators. Hence, it is of interest to determine if these consequences also apply to effective operators. In particular, commutation relations involving constants of the motion are of central importance since these operators are associated with fundamental symmetries of the system. Effective operator definitions are especially useful... 

The commutation relation between two arbitrary operators is not conserved upon transformation to effective operators by any of the definitions. Many state-independent effective operator definitions preserve the commutation relations involving // or a constant of the motion, as well as those involving operators which are related to P in a special way, for example, A with [P, 4] = 0. Many state-dependent definitions also conserve these special commutation relations. However, state-dependent definitions are not as convenient for formal and possibly computational reasons. The most important preserved commutation relations are those involving observables, since, as discussed in Section VII, they ensure that the basic symmetries of the system are conserved in effective Hamiltonian calculations. 

In the usual procedure a classical Hamiltonian function for the model is formulated. Attention should be given to the choice of the molecular coordinate system. As the frame and top are rigid, a convenient choice is the principal axis system of the whole molecule.8 As a consequence of the symmetry of the top, the orientation of the coordinate system within the molecule is independent of the torsion angle. Another choice, which is called the internal axis system, is defined in such a way that the angular momenta produced by internal rotation of the top and frame compensate each other.10 The Hamiltonian functions in both coordinate systems are related by a contact transformation, which guarantees the invariance of Poisson brackets11 and, subsequently, of the commutation relations. 

Here, we used the Heisenberg commutator relation between the conjugate position and momentum operators x, px = ih. The magnetic moment matrix element of the intra-ligand transition with respect to the common origin of the coordinate system is given by ... 

Notice the x, y, z cyclic relation.) These commutation relations do not depend on choice of coordinate system. Use of the r,6,4> coordinate system would give the same results. Evidently, these operators do not commute with each other since their commutators are unequal to zero. 

---