Operators and Quantum Mechanics

In the first example in Section 3.1, we fonnd that d/dz]g z) = -3z g(z) for every function g, and we concluded that [z, d/dz] = -3z. In contrast, the eigenvalue equation Af x) = kf(x) [Eq. (3.14)] does not hold for every fnnction /(x), and we cannot conclude from this equation that A = k. Thns the fact that d/dx)e = 2e does not mean that the operator d/dx equals multiplication by 2. 

Sir William Rowan Hamilton (1805-1865) devised an alternative form of Newton s equations of motion involving a function H, the Hamiltonian function for the system. For a system where the potential energy is a function of the coordinates only, the total energy remains constant with time that is, E is conserved. We shall restrict ourselves to such conservative systems. For conservative systems, the classical-mechanical Hamiltonian function turns out to be simply the total energy expressed in terms of coordinates and conjugate momenta. For Cartesian coordinates x, y, z, the conjugate 

Let us find the classical-mechanical Hamiltonian function for a particle of mass m moving in one dimension and subject to a potential energy V(x). The Hamiltonian function is equal to the energy, which is composed of kinetic and potential energies. The familiar form of the kinetic energy, mv, will not do, however, since we must express the Hamiltonian as a function of coordinates and momenta, not velocities. Since = Px/m, the form of the 

The time-independent Schrodinger equation (3.1) indicales that, corresponding to the Hamiltonian function (3.20), we have a quantum-mechanical operator 

Each Cartesian component of linear momentum p is replaced by the operator 

Consider some examples. The operator corresponding to the x coordinate is multiplication by x  

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

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. 

In the system of quantum dipoles, dipole and momentum variables have to be replaced by the quantum operators, and quantum statistical mechanics has to be applied. Now, the kinetic energy given by Eq. 9 does affect the thermal average of quantity that depends on dipole variables, due to non-commutivity of dipole and momentiun operators. According to the Pl-QMC method, a quantum system of N dipoles can be approximated by P coupled classical subsystems of N dipoles, where P is the Trotter munber and this approximation becomes exact in the limit P oo. Each quantiun dipole vector is replaced by a cychc chain of P classical dipole vectors, or beads , i.e., - fii -I-. .., iii p, = Hi,I. This classical system of N coupled chains... 

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. 

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. 

The interaction operator for quantum mechanical and classical mechanical subsystems is given by [13]... 

Since the density matrix is Hermitian, we obtain the property of polarization moments which is analogous to the classical relation (2.15) fq = (—1 ) (f-q) and tp = (—l) 3( g). The adopted normalization of the tensor operators (5.19) yields the most lucid physical meaning of quantum mechanical polarization moments fq and p% which coincides, with accuracy up to a normalizing coefficient that is equal for polarization moments of all ranks, with the physical meaning of classical polarization moments pq, as discussed in Chapter 2. For a comparison between classical and quantum mechanical polarization moments of the lower ranks see Table 5.1. 

In Eq. (3.20), (a =x, y, z) are the components of the total angular momentum operator with respect to the molecule-fixed axes. The quantity is classically/p = (bT/bojp) and quantum mechanically... 

As mentioned in the Introduction, magnetic exchange is both electrostatic and quantum mechanical in nature. It is electrostatic because the relevant energies are related to the energy costs of overlapping electron densities. It is quantum mechanical because of the fundamental requirement that the total wavefimction of two electrons must be antisymmetric to the exchange of both the spin and spatial coordinates of the two electrons. The wavefimction is separable into a product of spatial wavefimction V (ri, r2) that is a function of the positions r and ri of the two electrons, and a spin coordinate wavefimction /(ai, ai), where ai is the Pauli matrix for the spin operator Si = lioi /2. Both lr r, ri) and x (fri, cri) can be symmetric or antisymmetric individually but the fundamental... 

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