Quantum mechanics operators

Although not a unique prescription, the quantum-mechanical operators A can be obtained from their classical counterparts A by making the substitutions x x (coordinates) t t (time) p -Uid/dq (component of... 

Dynamical variable A Classical quantity Quantum-mechanical operator A... 

The relationship between tire abstract quantum-mechanical operators /4and the corresponding physical quantities A is the subject of the fourth postulate, which states ... 

If the system property is measured, the only values that can possibly be observed are those that correspond to eigenvalues of the quantum-mechanical operator 4. 

Wliat does this have to do with quantum mechanics To establish a coimection, it is necessary to first expand the wavefiinction in tenns of the eigenfiinctions of a quantum-mechanical operator A,... 

This provides a recipe for calculating the average value of the system property associated with the quantum-mechanical operator A, for a specific but arbitrary choice of the wavefiinction T, notably those choices which are not eigenfunctions of A. 

Suppose that the system property A is of interest, and that it corresponds to the quantum-mechanical operator A. The average value of A obtained m a series of measurements can be calculated by exploiting the corollary to the fifth postulate... 

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... 

When we wish to replace the quantum mechanical operators with the corresponding classical variables, the well-known expression for the kinetic energy in hyperspherical coordinates [73] is... 

Before concluding this sketch of optical phases and passing on to our next topic, the status of the phase in the representation of observables as quantum mechanical operators, we wish to call attention to the theoretical demonstration, provided in [129], that any (discrete, finite dimensional) operator can be constructed through use of optical devices only. 

The appropriate quantum mechanical operator fomi of the phase has been the subject of numerous efforts. At present, one can only speak of the best approximate operator, and this also is the subject of debate. A personal historical account by Nieto of various operator definitions for the phase (and of its probability distribution) is in [27] and in companion articles, for example, [130-132] and others, that have appeared in Volume 48 of Physica Scripta T (1993), which is devoted to this subject. (For an introduction to the unitarity requirements placed on a phase operator, one can refer to [133]). In 1927, Dirac proposed a quantum mechanical operator tf), defined in terms of the creation and destruction operators [134], but London [135] showed that this is not Hermitean. (A further source is [136].) Another candidate, e is not unitary. 

Traditionally, for molecular systems, one proceeds by considering the electronic Hamiltonian which consists of the quantum mechanical operators for the kinetic energy of the electrons, their mutual Coulombic repulsions, and... 

Quantum mechanics is cast in a language that is not familiar to most students of chemistry who are examining the subject for the first time. Its mathematical content and how it relates to experimental measurements both require a great deal of effort to master. With these thoughts in mind, the authors have organized this introductory section in a manner that first provides the student with a brief introduction to the two primary constructs of quantum mechanics, operators and wavefunctions that obey a Schrodinger equation, then demonstrates the application of these constructs to several chemically relevant model problems, and finally returns to examine in more detail the conceptual structure of quantum mechanics. 

The eigenfunctions of a quantum mechanical operator depend on the coordinates upon which the operator acts these functions are called wavefunetions... 

Many physical properties of a molecule can be calculated as expectation values of a corresponding quantum mechanical operator. The evaluation of other properties can be formulated in terms of the "response" (i.e., derivative) of the electronic energy with respect to the application of an external field perturbation. 

The results given above are, as stated, general. Any and all angular momenta have quantum mechanical operators that obey these equations. It is convention to designate specific kinds of angular momenta by specific letters however, it should be kept in mind... 

Hamiltonian quantum mechanical operator for energy, hard sphere assumption that atoms are like hard billiard balls, which is implemented by having an infinite potential inside the sphere radius and zero potential outside the radius Hartree atomic unit of energy... 

Relationship Between Physical Transformations and Quantum Mechanical Operators.—In order to obtain information concerning the symmetry... 

This two general Cl function expressions, along with the results obtained in the section 5.1 above, permit to compute the expected value form of any quantum mechanical operator in a most complete general way. 

The second postulate states that a physical quantity or observable is represented in quantum mechanics by a hermitian operator. To every classically defined function A(r, p) of position and momentum there corresponds a quantum-mechanical linear hermitian operator A(r, (h/i)V). Thus, to obtain the quantum-mechanical operator, the momentum p in the classical function is replaced by the operator p... 

The position, momentum, and energy are all dynamical quantities and consequently possess quantum-mechanical operators from which expectation values at any given time may be determined. Time, on the other hand, has a unique role in non-relativistic quantum theory as an independent variable dynamical quantities are functions of time. Thus, the uncertainty in time cannot be related to a range of expectation values. 

The quantum-mechanical operators for the components of the orbital angular momentum are obtained by replacing px, Py, Pz in the classical expressions (5.2) by their corresponding quantum operators. 

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... 

If we replace the z-component of the classical angular momentum in equation (6.87) by its quantum-mechanical operator, then the Hamiltonian operator Hb for the hydrogen-like atom in a magnetic field B becomes... 

As an example, consider the quantum mechanical operator for the linear momentum in one dimension,... 

Formula (58) shows that the angular momentum operator for the photon consists of two terms. The first term is identical with the usual quantum-mechanical operator L for the orbital angular momentum in the momentum... 

The quantum-mechanical state is represented in abstract Hilbert space on the basis of eigenfunctions of the position operator, by F(q, t). If the eigenvectors of an abstract quantum-mechanical operator are used as a basis, the operator itself is represented by a diagonal square matrix. In wave-mechanical formalism the position and momentum matrices reduce to multiplication by qi and (h/2ni)(d/dqi) respectively. The corresponding expectation values are... 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

For conservative systems with time-independent Hamiltonian the density operator may be defined as a function of one or more quantum-mechanical operators A, i.e. g= tp( A). This definition implies that for statistical equilibrium of an ensemble of conservative systems, the density operator depends only on constants of the motion. The most important case is g= [Pg.463]

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