Commutator Equation

Let us therefore determine the rate of change of the position projection operator for the coordinate qi, which in the Heisenberg picture is given by the following commutator equation ... 

When the idempotent density operator p is constructed from orbital solutions of the Hartree-Fock equations, (Ti - ) = 0, it satisfies the commutator equation... 

The order of operation is very important the operation is not commutative (equation 5.15). 

T = d>)( for the operator for the reference state. Starting from a given basis B in the operator space, we will further assume that we have constructed an approximate eigenoperator D = Bd by properly solving the double-commutator equations (1.49) and (1.50). The approximate unnormalized wave function for the final state is then given by the relation ... 

It turns out that one may present an exact integration of the equations for geodesics of the metrics (pahD on the group SL(m,C). Metrics of the type (pabD appeared for the first time in the course of construction of nonlinear differential equations integrable by the inverse scattering method. FVom the paper [38] it readily follows that the Euler equation X = [X,ipahD ) on a classical Lie algebra of series Afn-i serves as a commutativity equation for a pair of operators. 

The operators act on the space of smooth vector functions, and the Lie algebra Afn i is realized by square zero-trace matrices. The matrices a and b are diagonal with distrinct diagonal elements. According to the finite-zoned integration theory (see [77]), the commutativity equations [Lai Aa] = 0 are integrated by means of the theta-functions of the Riemann surface of the algebraic curve Q W A) = det(lV — X - Aa) = 0. 

From the commutation equation (7), it may now be seen first of all, that each diagonal block must commute with the corresponding furthermore, the off-diagonal blocks, Fw, satisfy conditions of the type ... 

The subscript minus sign in [7f, /3(/)] indicates that this qirantity is a commutator. Equation (225) is the quantum analogue of the classical Liouville equation. Thus, we call it the quantum Liouville equatioa... 

It is more convenient to re-express this equation in Liouville space [8, 9 and 10], in which the density matrix becomes a vector, and the commutator with the Hamiltonian becomes the Liouville superoperator. In tliis fomuilation, the lines in the spectrum are some of the elements of the density matrix vector, and what happens to them is described by the superoperator matrix, equation (B2.4.25) becomes (B2.4.26). 

We are dealing with real numbers that commute hence, it is evident that the right side of Equation set (3-59) is... 

However, equation 2.(i-2 IS valid because A, B are commuting variables that lead to equation 2.6-3. Rearranging, results in one of the usual forms of the Bayes equation (equation 2.6-4). PiA E) is the prior probability of A given E. P(B A E is probability that is... 

For a more complicated [B] matrix that has, say, n columns whereas [A] has m rows (remember [A] must have p columns and [B] must have p rows), the [C] matrix will have m rows and n columns. That is, the multiplication in Equations (A.21) and (A.22) is repeated as many times as there are columns in [B]. Note that, although the product [A][B] can be found as in Equation (A.21), the product [B][A] is not simultaneously defined unless [B] and [A] have the same number of rows and columns. Thus, [A] cannot be premultiplied by [B] if [A][B] is defined unless [B] and [A] are square. Moreover, even if both [A][B] and [B][A] are defined, there is no guarantee that [A][B] = [B][A]. That is, matrix multiplication is not necessarily commutative. 

In spite of its simplicity and the visual similarity of this equation to Eq. (7), we would like to note that Eq. (11) leads to a nontrivial thermodynamics of a partially quenched system in terms of correlation functions, see, e.g.. Ref. 25 for detailed discussion. Evidently, the principal route for and to the virial theorem is to exploit the thermodynamics of the replicated system. However, special care must be taken then, because the V and s derivatives do not commute. Moreover, the presence of two different temperatures, Pq and P, requires attention in taking temperature derivatives, setting those temperatures equal, if appropriate, only at the end of the calculations. 

While each term in the resulting series has the form required by equation 12,29, the caveat is that as each commutator connects sites x and x that are a distance d apart, the terms of order n in Ai and Aj yield contributions to H x) that obey the third condition in equation 12.29 only for d = nd. In other words, we also need to require that the Baker-Campbell-Hausdorff series has a sufficient degree of convergence, t... 

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

Since div ( ) mid div 3 (x) commute with 8(x ) and 3 t (x ) for x0 —x, they have vanishing commutators with the hamiltonian and hence, they are time-independent operators. In fact, their constancy in tame implies that they commute with 3 (x) and S(x) at all times and hence they must be c-number multiples of the unit operator. If these c-numbers are set equal to zero initially, they will remain zero for all times. With this initial choice for div 8(x) and div 3tf(x), the operators S and satisfy all of the Maxwell equations (these now are operator equations ) ... 

A particular representation of the commutation rules (9-636) and (9-637) characterized by the existence of a no-particle state 0> is exhibited by the equations... 

Summarizing, we have noted that the Heisenberg operators Q+(t) obey field free equations i.e., that their time derivatives are given by the commutator of the operator with Ha+(t) = Ho+(0) and that this operator H0+(t) is equal to H(t) = H(0). The eigenstates of H0+ are, therefore, just the eigenstates of H. We can, therefore, identify the states Tn>+ with the previously defined >ln and the operator [Pg.602]

Hence, by virtue of the equation of motion (10-1), which >fi(x) obeys, and the equal time commutation rules (10-8), the Green function GA obeys the following equation... 

We shall again postulate commutation rules which have the property that the equations of motion of the matter field and of the electromagnetic field are consequences of the Heisenberg equation of motion ... 

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