Representations in Quantum Mechanics

Isomorphisms of unitary representations ought to preserve the unitary structure. When they do, they are called unitary isomorphisms of representations. 

Definition 4.12 Suppose (G, V, p and iG. W, p) are two representations of the same group. Suppose V and W are complex scalar product spaces. Suppose T V W is an homomorphism of representations. If T respects the complex scalar products, i.e.. if for all v,w V we have v, w) = Tv, where, denotes the complex scalar product on V and,  

Note that every unitary homomorphism T of representations is injective if V 7 0 e V, then 

Representations are the primary object of our mathematical analysis. In particular, the natural representation of 50(3) on introduced in this 

Representations arise naturally in quantum physics, where there is a homo- 

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Some of the articles address the difficulties students have learning particular aspects of quantum mechanics. Others describe different interpretations, formulations, and representations in quantum mechanics. Still others discuss novel applications or some of the more subtle conceptual issues in quantum mechanics. A few of the articles address the integration of workable and affordable quantum mechanics experiments into the undergraduate curriculum. 

If the Hamilton operator depends on time in a harmonic fashion, the time dependence can be eliminated by transformation into a rotating reference frame in analogy to the transformation of the Bloch equations. A representation in the rotating frame is also called interaction representation in quantum mechanics. If the time dependence is more general, the Schrodinger equation is solved for small enough time increments, during which H is approximately constant. For each of the n time increments At a solution of the form (2.2.46) applies. The complete evolution operator is the time ordered product of the incremental evolution operators. This operator is written in short hand as... 

We have deduced this formula in accordanco with the classical vector model representation. In quantum mechanics this representation is certainly still permissible, but with this difference, that the square of the magnitude of an angular momentum, with the quantum number I, is not equal to P as in Bohr s theory, Init is given by l(l + 1). This is proved in Appendix XIX (p. 302) for the orbital angular momentum ... 

As discussed subsequently, introduction of the standard p, q representation in classical mechanics, and of the Wigner-Weyl representation in quantum mechanics, defines densities p(p,q) = (p,q p) that both lie in the same Hilbert space. Thus, the essential difference between quantum and classical mechanics... 

There is one final case, which we describe very briefly here and in more detail later. The classical description can be written in a form that is quite similar to the number operator representation in quantum mechanics. An operator 0 is assigned to molecule i, which is one if / is of type a and zero otherwise. Now, however, these operators do not themselves depend on the positions and momenta they follow a dynamics that is specified by the classical Liouville operator of the system. In particular, the Liouville operator determines the conditions under which species interconversion is possible. Hence, just as in the quantum mechanical case, the problem of the specification of the precise conditions for reaction is deferred to the Liouville operator. Section VI describes how such Liouville operators can be constructed. 

The solution of time evolution problems for classical systems is facilitated by introducing a classical phase space representation that plays a role in the description of classical systems in a matmer that is formally analogous to the role played by the coordinate and momentum representations in quantum mechanics. The state vectors T ) of this representation enumerate all of the accessible phase points. The phase function /(E ) is given by / (f ) = (f I/), which can be thought to represent a component of the vector f) in the classical phase space representation. The application of the classical Liouville operator (f ) to the phase function /(f ) is defined by (f )/(f ) = (f I/), where is an abstract op-... 

Insufficient reason in risk analysis, 316 Integral curves, construction, 336 Interaction representation, 418 Interference terms in quantum mechanics, 425... 

Parameterization of the N-Atom Problem in Quantum Mechanics. II. Coupled-Angular-Momentum Spectral Representations for Four-Atom Systems. 

Since the birth of quantum theory, there has been considerable interest in the transition from quantum to classical mechanics. Because the two formulations are given in a different theoretical framework (nonlinear classical trajectories versus expectation values of linear operators), this transition is far more involved than the naive limit —> 0 suggests. By exploring the classical limit of quantum mechanics, new theoretical concepts have been developed, including path integrals [1], various phase-space representations of quantum mechanics [2], the semiclassical propagator and the trace formula [3], and the notion of quantum... 

Many-body problems in quantum mechanics are usually described by the number of particles N in the system and the probabilities of finding those particles at different locations in space. If the rank of the one-particle basis is a finite number r, an equally valid description of the system may be given by specifying the number of holes r N in the system and the probabilities of finding these holes at different locations in space. This possibility for an equivalent representation of the system by particles or holes is known as the particle-hole duality. By using the fermion anticommutation relation... 

In quantum mechanics, physically measurable quantities are represented by hermitian operators. Such operators R have matrix representations, in any basis spanning the space of functions on which the R act, that are hermitian ... 

Figures 8.6 and 8.8 are both representations that suffer from the limits of our informational technique, as they show two distinct things at the bottom, implying an initial separation between living organisms and their environment. This separation is just what the notion of co-emergence negates. The complementarity between yin and yang in classic Chinese philosophy comes to mind they also cannot be separated from each other or the complementarity between wave and particle in quantum mechanics they can be distinguished from each other only when carrying out a specihc experiment.

The notion of a group is a natural mathematical abstraction of physical symmetry. Because quantum mechanical state spaces are linear, symmetries in quantum mechanics have the additional structure of group representations. Formally, a group is a set with a binary operation that satisfies certain criteria, and a representation is a natural function from a group to a set of linear operators. 

Before showing further applications of direct-product representations to quantum mechanics, we quote without proof a theorem we will need. Let rij a and rkip be the elements of the matrices corresponding to the symmetry operation R in the two different nonequivalent irreducible representations Ta and T it can be shown that... 

D. J. Tannor To understand the role of dissipation in quantum mechanics, it is useful to consider the density operator in the Wigner phase-space representation. Energy relaxation in a harmonic oscillator looks as shown in Fig. 1, whereas phase relaxation looks as shown in Fig. 2 that is, in pure dephasing the density spreads out over the energy shell (i.e., spreads in angle) while not changing its radial distribution... 

Time evolution in quantum mechanics is described, in the Schrodinger representation, by the Schrodinger time-dependent equation... 

The plan of the paper is the following. In section 2 we introduce the elements at the basis of the Heisenberg group representation representation theory [12-14,20] that are needed to understand the alternative group-theoretical formulation of quantum mechanics. In section 3 the Heisenberg representation of quantum mechanics (with the time dependence transferred from the vectors of the Hilbert space to the operators) is used to introduce quantum observables and quantum Lie brackets within the group-theoretical formalism described in the previous section. In section 4 classical mechanics is obtained by taking the formal limit h —> 0 of quantum observables and brackets to obtain their classical counterparts. Section 5 is devoted to the derivation of... 

In the following, we indicate the time derivative of a hermitian operator B with the symbol B. In the Heisenberg representation of quantum mechanics, it obeys the Heisenberg equation of motion... 

In quantum mechanics, spin is described by an operator which acts on a spin wavefunction of the electron. In the present case this operator describes an angular momentum with two possible eigenvalues along a reference axis. The first requirement fixes commutation rules for the spin components, and the second one leads to a representation of the spin operator by 2 x 2 matrices (Pauli matrices [Pau27]). One has... 

We recall some basic results of quantum dynamics [3], First, the state of the system and the time evolution can be expressed in a generalized (Dirac) notation, which is often very convenient. The state at time t is specified by x(t)) with the representations x(-Rjf) = (R x t)) and x P,t) = (P x(t)) in coordinate and momentum space, respectively. Probability is a concept that is inherent in quantum mechanics. (R x(t)) 2 is the probability density in coordinate space, and (-P x(f) 2 is H e same quantity in momentum space. The time evolution (in the Schrodinger picture) can be expressed as... 

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. 

In doing so it proves convenient to carry out the computations in the Wigner representation. That is, as in quantum mechanics based upon the wave function, it is necessary to deal with a representation of the density operator p. The (convenient) Wigner representation pw of p is defined, for an N degree of freedom system, by... 

Our simulations are based on well-established mixed quantum-classical methods in which the electron is described by a fully quantum-statistical mechanical approach whereas the solvent degrees of freedom are treated classically. Details of the method are described elsewhere [27,28], The extent of the electron localization in different supercritical environments can be conveniently probed by analyzing the behavior of the correlation length R(fih/2) of the electron, represented as polymer of pseudoparticles in the Feynman path integral representation of quantum mechanics. Using the simulation trajectories, R is computed from the mean squared displacement along the polymer path, R2(t - t ) = ( r(f) - r(t )l2), where r(t) represents the electron position at imaginary time t and 1/(3 is Boltzmann constant times the temperature. 

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