Commutator quantum operators

In the system of quantum dipoles, dipole and momentum variables have to be replaced by the quantum operators, and quantum statistical mechanics has to be applied. Now, the kinetic energy given by Eq. 9 does affect the thermal average of quantity that depends on dipole variables, due to non-commutivity of dipole and momentiun operators. According to the Pl-QMC method, a quantum system of N dipoles can be approximated by P coupled classical subsystems of N dipoles, where P is the Trotter munber and this approximation becomes exact in the limit P oo. Each quantiun dipole vector is replaced by a cychc chain of P classical dipole vectors, or beads , i.e., - fii -I-. .., iii p, = Hi,I. This classical system of N coupled chains... 

The partial recovery of the quantum phase coherence of nuclear dipoles originates from the non-commutative property of the Zeeman energy with the quantum operator which represents the residual interaction after rotating the spins. This rotation has no effect on the magnetisation dynamics when the residual interaction, hHR, is equal to zero. No... 

Fig. 11. Schematic representations of the commutator sequences starting from /, for (A) effective POO and lOO, (B) effective ILL, (C) effective PPP, and (D) effective III coupling topologies. The following rules and abbreviations are used. Terms in the density operator appear as nodes in the diagram Circles contain polarization operators which are represented by the symbols z,. Rounded boxes contain zero-quantum operators (ZQ), = which are represented by the symbols A,. Straight-edge boxes contain an-...

The norm of the density operator o- = (Tr o-V ) is always conserved under the unitary transformation of Eq. (71). However, in general, additional constants of motion exist. If the effective mixing Hamiltonian is composed exclusively of zero-quantum operators, it commutes with the z component of the total angular momentum operator ... 

Since these quantum operators can be considered as canonical relations for commutation [q. P = ih, we will have [a, a = 1. Thus, expressing the Hamiltonian of Equation 6.56 with the a+ and a ... 

The two components are said to commute. Quantum-mechanically the components of linear momentum are represented by differential operators, such as / -> —ihdjdx, to give operators... 

Recognizing the fact that representation of observable quantum operators, in various bases, is made by (hermitic) matrices, Heisenberg had generalized the commutation rules to operators and thus to matrix level, while this way constructing the so-called quantum matrix mechanics. It is basically founded by the commutation rules among the coordinate [2] momentum [- ] matrices. 

The first paper in which the separability of the total infrared correlation function into its vibrational and rotational components was questioned is the paper by Van Woerkom, de Bleyser et al, (27). Their theory is based on use of the generalized cumulant expansion theorem for non-commuting quantum mechanical operators and involves the following assumptions, (i) The Hamiltonian of the liquid sample is written in the form H = E + F + G where E is the Hamiltonian for vibrational degrees of freedom, F is the bath Hamiltonian for rotational and vibrational degrees of freedom whereas G is the interaction Hamiltonian, (ii) G is small with respect to E + F. An interaction representation is used in which G is considered as a perturbation, (iii) The averaged ordered exponential is developped into a truncated cumulant expansion series. 

In the second quantization (SQ) one introduces quantum operators instead of quantum states. Thus, for instance, for a harmonic oscillator we would introduce the following set (1, a, a+). If we take the commutator between any two of these operators we get either zero or the third operator. Thus the set is closed with respect to commutations. The hamiltonian for the harmonic oscillator can be expressed in terms of these operators Hq = Tiw a i). If this is the case, i.e., that the hamiltonian for the system can be expressed in terms of operators, it is possible to solve also the TDSE using operator algebra. The advantage of using this technique is that the number of operators is smaller than the number of states. For the harmonic oscillator the number of operators is three but the number of states is infinite. Thus a certain class of problems can be solved analytically, i.e., formally. One such hamiltonian is... 

Note that we have furthermore dropped the two-quantum operators and Ok dkt. Thus two quantum transitions will take place through successive operations with the one-quantum operators Ok and d. This approximation is introduced in order to obtain a solvable algebra for the operators. Such an algebra should be closed with respect to commutations. The above hamiltonian fulfills this condition. 

Starting with the quantum-mechanical postulate regarding a one-to-one correspondence between system properties and Hemiitian operators, and the mathematical result that only operators which conmuite have a connnon set of eigenfiinctions, a rather remarkable property of nature can be demonstrated. Suppose that one desires to detennine the values of the two quantities A and B, and that tire corresponding quantum-mechanical operators do not commute. In addition, the properties are to be measured simultaneously so that both reflect the same quantum-mechanical state of the system. If the wavefiinction is neither an eigenfiinction of dnor W, then there is necessarily some uncertainty associated with the measurement. To see this, simply expand the wavefiinction i in temis of the eigenfiinctions of the relevant operators... 

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

Suggested Extra Reading- Appendix C Quantum Meehanieal Operators and Commutation]... 

We use an example to illustrate the importance of two operators commuting to quantum mechanics interpretation of experiments. Assume that an experiment has been... 

The subset 0kv 0k2> 0k3,. . . formed from the complete set by means of the projection operator 0k is called /l-adapted or symmetry-adapted in the case when A is a symmetry operator. From Eqs. III.81 and III.86 it follows that the projection operators 0k commute with H and, using this property, the quantum-mechanical turn-over rule/ and Eq. III.91, we obtain... 

Any operator J, which satisfies the commutation rule Eq. (7-18), represents quantum mechanical angular momentum. Orbital angular momentum, L, with components explicitly given by Eq. (7-1), is a special example5 of J. 

Stated more abstractly, in quantum mechanics, a particle is characterized by a set of dynamical variables, p,q, which are represented by operators that obey the fundamental commutation rules... 

Quantization of the Electromagnetic Field.—Instead of proceeding as in the previous discussion of spin 0 and spin particles, we shall here adopt essentially the opposite point of view. Namely, instead of formulating the quantum theory of a system of many photons in terms of operators and showing the equivalence of this formalism to the imposition of quantum rules on classical electrodynamics, we shall take as our point of departure certain commutation rules which we assume the field operators to satisfy. We shall then show that a... 

Now in quantum theory the description of a physical system in the Heisenberg picture for a given observer O is by means of operators Q, which satisfy certain equations of motion and commutation rules with respect to O s frame of reference (coordinate system x). The above notion of an invariance principle can be stated alternatively as follows If, when we change this coordinate frame of reference (i.e., for observer O ) we are able to find a new set of operators that obeys the same equations of motion and the same commutation rules with respect to the new frame of reference (coordinate system x ) we then say that these observers are equivalent and the theory invariant under the transformation x - x. The observable consequences of theory in the new frame (for observer O ) will then clearly be the same as those in the old frame. 

The statement that quantum electrodynamics is invariant under such a spatial inversion (parity operation) can be taken as the statement that there exist new field operators >p (x ) and A x ) expressible in terms of tji(x) and Au(x) which satisfy the same commutation rules and equations of motion in terms of s as do ift(x) and A x) written in terms of x. In fact one readily verifies that the operators... 

As indicated at the beginning of the last section, to say that quantum electrodynamics is invariant under space inversion (x = ijX) means that we can find new field operators tfi (x ),A v x ) expressible in terms of fj(x) and A nix) which satisfy the same equations of motion and commutation rules with respect to the primed coordinate system (a = igx) as did tf/(x) and Av(x) in terms of x. Since the commutation rules are to be the same for both sets of operators and the set of realizable states must be invariant, there must exist a unitary (or anti-unitary) transformation connecting these two sets of operators if the theory is invariant. For the case of space inversions, such a unitary operator is... 

There are various approximations (7) to the above expression for the absorption rate Rj that offer further insight into the photon absorption process and form a basis for comparison to the non Bom-Oppenheimer rate expression. The most classical (and hence, least quantum) approximation is to ignore the fact that the kinetic energy operator T does not commute with the potentials Vj f and thus to write... 

Since y commutes with p and z commutes with py, there is no ambiguity regarding the order of y and Pz and of z and Py in constructing Lx. Similar remarks apply to Ly and Lz- The quantum-mechanical operator for L is... 

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