Hermite operators

Evolution equation of the Hermitic operator of the quantum system... 

With these, one may consider the further case in which there are two wave functions, both as eigen-values of the same hermitic operator, yet producing two different eigen-values ... 

THEOREM Two non-commutative hermitic operators cannot provide simultaneous observable measurements with the same precision on a given (prepared) eigen-state. 

THEOREM Quantum hermitic operators of coordinate and momentum fulfill the Newtonian laws of motion... 

The proof is based on evaluation of the time-evolution for the expectation value for a given hermitic operator... 

However, worth mentioning that the above transfer from classical to quantum picture has to follow also the rule according which the result of the Poison equation to be written in such manner so that when quantized to provide a Hermitic operator. An example is here given for particular functions and p of conjugated variables q and p their commutator looks like ... 

VII. Any linear operator F may be decomposed on one hermitic and one anti-hermitic or on only hermitic operators as the context demands ... 

Here, Qa and Fa are Hermite operators in the system and the stochastic bath subspaces, respectively. The deterministic system Liouvillian operators C t) and s are defined as... 

There are infinitely many solutions and we assume that they are labelled according to their energies, Eq being the lowest. Since the H operator is Hermitic, the solutions form a complete basis. We may furthermore chose the solutions to be orthogonal and normalized. 

The vanishing of this matrix element is, in fact, independent of the assumption of current conservation, and can be proved using the transformation properties of the current operator and one-partic e states under space and time inversion, together with the hermiticity of jn(0). By actually generating the states q,<>, from the states in which the particle is at rest, by a Lorentz transformation along the 3 axis, and the use of the transformation properties of the current operator, essentially the entire kinematical structure of the matrix element of on q, can be obtained.15 We shall, however, not do so here. Bather, we note that the right-hand side of Eq. (11-529) implies that... 

The hermitian character of an operator depends not only on the operator itself, but also on the functions on which it acts and on the range of integration. An operator may be hermitian with respect to one set of functions, but not with respect to another set. It may be hermitian with respect to a set of functions defined over one range of variables, but not with respect to the same set over a different range. For example, the hermiticity of the momentum operator p is dependent on the vanishing of the functions ipi at infinity. 

In some of the derivations presented in this section, operators need not be hermitian. However, we are only interested in the properties of hermitian operators because quantum mechanics requires them. Therefore, we have implicitly assumed that all the operators are hermitian and we have not bothered to comment on the parts where hermiticity is not required. 

The operator n is linear and hermitian. In the one-dimensional case, the hermiticity of TI is demonstrated as follows... 

Hermite polynomials 104-107 integrals 99-100 matrix methods 172-175 operators 151-153 particle in a box 96-100,122, 309-311... 

Some of the Hermite polynomials and the corresponding harmonic-oscillator wave functions are presented in Thble 1. The importance of the parity of these functions under the inversion operation, cannot be overemphasized. 

There is no proper perturbative basis for the mnemonic diagram in Fig. 3.58, because the non-orthogonal unperturbed orbitals cannot correspond to any physical (Hermitian) unperturbed Hamiltonian operator,79 as illustrated in Examples 3.17 and 3.18 below. The PMO interpretation of Fig. 3.58 therefore rests on an nnphysical starting point. Removal of orbital overlap (to restore Hermiticity) eliminates the supposed effect. 80... 

On account of the hermiticity of the operators, only even values of L are allowed. Since the D operators are tensor operators of rank 1, the only allowed values are L = 0,2. The L = 0 contribution has been treated in type (1). Hence, only the L = 2 contribution must be considered here. The matrix elements of the operators (4.122) with L = 2 are difficult to evaluate. Nonetheless, by making use of the angular momentum algebra, they can be evaluated in explicit... 

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