Operator, in quantum mechanics

Table 2.1 Some Common Operators in Quantum Mechanics.

A maps real variables to real variables (Hermitian operators to Hermitian operators in quantum mechanics). 

A physicist would view the expression (10) as typical in quantum mechanics and as corresponding to the evolution operator. Equations (8) and (9) are, incidentally, very typical in gauge theory, such as in QCD. Thus, guided by our intuition, we can reformulate our chief problem as a quantum-mechanical one. In other words, the approaches to the l.h.s. of the non-Abelian Stokes theorem are analogous to the approaches to the evolution operator in quantum mechanics. There are the two main approaches to quantum mechanics, especially to the construction of the evolution operator opearator approach and path-integral approach. Both can be applied to the non-Abelian Stokes theorem successfully, and both provide two different formulations of the non-Abelian Stokes theorem. 

After having described the expression for the rate constant within the framework of classical mechanics, we turn now to the quantum mechanical version. We consider first the definition of a flux operator in quantum mechanics.2 To that end, the flux density operator (for a single particle of mass to) is defined by... 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

Note Although the commutator is a well defined and useful operator in quantum mechanics, it does not correspond to an observable quantity. Thus one need not be concerned about obtaining an imaginary expectation value. 

L has many of the. mathematical properties of the Hamiltonian operator in quantum mechanics. 

Operators in quantum mechanics are determined by their consequences. Operators map the wave function i into a new state vector ... 

The analogy of the time-evolution operator in quantum mechanics on the one hand, and the transfer matrix and the Markov matrix in statistical mechanics on the other, allows the two fields to share numerous techniques. Specifically, a transfer matrix G of a statistical mechanical lattice system in d dimensions often can be interpreted as the evolution operator in discrete, imaginary time t of a quantum mechanical analog in d — 1 dimensions. That is, G exp(—tJf), where is the Hamiltonian of a system in d — 1 dimensions, the quantum mechanical analog of the statistical mechanical system. From this point of view, the computation of the partition function and of the ground-state energy are essentially the same problems finding... 

The above expressions refer to the evolution operator in quantum mechanics. Similar expressions can be derived in the framework of statistical physics in order to establish a connection with the Langevin s formalism. 

In the nonrelativistic quantum mechanics to which we are confining ourselves, electron spin must be introduced as an additional hypothesis. We have learned that each physical property has its corresponding linear Hermitian operator in quantum mechanics. For such properties as orbital angular momentum, we can construct the quantum-mechanical operator from the classical expression by replacing p Py,Pz by the appropriate operators. Hie inherent spin angular momentum of a microscopic particle has no analog in classical mechanics, so we cannot use this method to construct operators for spin. For our purposes, we shall simply use symbols for the spin operators, without giving an explicit form for them. 

Operators such as this that yield back the original function times a constant are called eigenfunctions and the corresponding constants are known as the eigenvalues. You will find that most of the operators in quantum mechanics will be of this type. Substituting the second partial derivatives into Equation (3.3) and cancelling the signs yields... 

Postulate 2 For every physically observable variable in classical mechanics, there exists a corresponding linear, Hermitian operator in quantum mechanics. Examples are shown in Table 3.2, where the symbol indicates a quantum mechanical operator and h = h/2n. A Hermitian operator is one which satisfies Equation (3.28). 

A complete description of the dynamics of any molecular system is contained in the Hamiltonian H, which is the energy operator in quantum mechanics or the energy function in classical mechanics. In general, the Hamiltonian is a function of the electronic and nuclear degrees of freedom, as is the description of the system dynamics. This complex problem simplifies through the adoption of the Born-Oppenheimer approximation, which is the assumption that nuclear and electronic motion are independent due to their substantially different time scales and masses. This assumption allows one to first solve for the dynamics of the electrons and then obtain the forces experienced by the nuclei as determined by this fixed-electron configuration. 

Although aromaticity concept has been talked about by all chemists for almost two centuries there exists no unambiguous quantitative definition of aromaticity mainly because it is not an experimentally measurable quantity. It is also not possible to theoretically calculate it as the expectation value of a linear hermitian operator in quantum mechanics, mainly due to the same reason stated above. 

Postulate II To every observable laboratory measurement in classical mechanics there corresponds an operator in quantum mechanics. 

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