Product of two operators

Quantum mechanically the combination of two angular momenta is more complicated since the angular momenta are operators. (They are tensor operators of rank one.) The law of combination of angular momenta can be expressed, in general, by the so-called tensor product of two operators, indicated by the symbol x,... 

And, the product of two operators, which is equal to another single permutation operator is written, for example,... 

The condition that is a positive semidefinite operator ensures that belongs to the Kummer cone. Unfortunately, the Grassmann wedge product of two operators is not explained explicitly in Ref. [1] but the section, p.79 may be sufficient. Here are two other references [4, 5] that may be helpful. Be sure that you understand Exercise 4 on p.73 of Ref. [1]. 

Problem 3-1. Verify that the set of covering operations of the water molecule is a group, with the definition of the product of two operations as the compound operation resulting from applying them in succession. 

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

Let us first specify what we mean by a complete set of symmetry operations for a particular molecule. A complete set is one in which every possible product of two operations in the set is also an operation in the set. Let us consider as an example the set of operations which may be performed on a planar AB3 molecule. These are E, C3, Cjj, C2, C2, CJ, symmetry operations are possible. If we number the B atoms as indicated, we can systematically work through all binary products for example ... 

The submatrix element of the tensorial product of two operators, acting on one and the same coordinate, may be calculated applying the formula... 

In other words, in a term which is the product of two operators, each operator evolves under the effect of chemical shift and scalar coupling independently. A further example is... 

In products of two operators, each one evolves independently also under the effect of pulses. For instance ... 

Performing the partial Wigner transform in eq.(l) and applying the rule for the partial Wigner transform of the product of two operators [10]... 

Hence, for the scalar product of two operators which both act on the jx part of a coupled scheme, we have... 

The second (antiphase) term in eq. A6-14 contains the product of two operators, 4(/)4(7 ), for the first time. Coupling thus can be considered to create a new operator through the process whereby one operator is transformed into the product of... 

The time evolution of a quantum operator C = AB, which is the product of two operators, can be written in the partial Wigner representation as... 

The new feature which arises when considering two spins is the effect of coupling between them. The Hamiltonian representing this coupling is itself a product of two operators ... 

If the product of two operators is equivalent to the unit operator, each is said to be the inverse of the other so here both operators are said to be self-inverse . 

There are various possible strategies for a formulation of this transformation [13], but for our purpose the FW transformation as a sequence of two transformations [49], is particularly recommended. Let us write the transformation operator Wpw as a product of two operators, such that the first one removes the non-diagonal blocks of D, while the second one reestablishes the normalization, such that the final transformation is unitary. 

As a specific example, consider the matrix elements of J-S, which is a product of two operators, one having anomalous, the other regular molecule-fixed commutation rules. The value of a specific matrix element, nJQM J S n J Q M ), cannot depend on whether J-S is written in terms of space-fixed components,... 

The product of two operators is defined as the successive operation of the operators, with the one on the right operating first. If... 

Since we have defined the product of two operators, we have a definition for the powers of an operator. An operator raised to the nth power is the operator for n successive applications of the original operator ... 

To find [z, d/dz], we apply this operator to an arbitrary function g(z). Using the commutator definition (3.7) and the definitions of the difference and product of two operators, we have... 

By repeated application of the definition of the product of two operators show that... 

A further property of rotations about an axis is that they commute. The following relations are easily verified by inspection of Figure 1.6 (the product of two operations commands—first apply the right-hand then the left-hand transformation). 

A product of two operators is executed consecutively, and hence the one closest to the ket acts first. In detail. 

Thus if we define matrix multiplication by (1.16), then the matrix representation of the product of two operators is just the product of their matrix representations. 

A product of two operators A and B in the LiouviUe space is defined as follows ... 

Let us now use this table to see if the requirements for a group are satisfied by these symmetry elements. Since every product of two operations is equivalent to one of the operations of the set (i.e., since every spot in the table is occupied by one of our six symbols) the set exhibits closure. Also, the identity element is present in the set and occurs once in each column and row (Problem 13-2), and so every element has an inverse in the set. The associative law is satisfied, as one can establish by trying various examples, e.g., D(CB) = DF= E, DC)B = BB = E, so that D CB) = (DC)B. (In general, symmetry operations satisfy the associativity law.) Therefore, our set of six symmetry operations constitutes a symmetry point group of order 6. 

This choice sets the location of each mirror plane in space so, to work out the product of two operations, it is necessary to hold the planes in place during the whole manoeuvre. For example, the product is illustrated in Figure 2.4. This involves a rotation around the principal axis which brings H3 into the plane. Reflection by the plane then swaps Hi and H2. Comparing the final configuration with the start point, it can be seen that H2 has returned to its original position but Hi and H3 have been interchanged, a result that can be achieved by [Pg.29]

All the operations in a cyclic group commute with one another, because a product of two operations can easily be written as the root operation applied multiple times. For example ... 

For the matrices to be proper representations of the operations their products should reproduce the group multiplication table (Table 4.4a). The product of two operations, such as C2<7v, means take the original vectors, transform by the vertical reflection and then apply a C2 rotation to this intermediate result. We could carry this out in two steps ... 

There is a model for each of the operations in the point group of H2O. To perform a product of two operations, start by picking up the model which represents the first part of the product and hold it with the labelled side toward you. To carry out the second operation, do one of the following ... 

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