Hilbert space quantum theory

These results were obtained by using the time-dependent quantum mechanical evolution of a state vector. We have generalized these to non-equilibrium situations [16] with the given initial state in a thermodynamic equilibrium state. This theory employs the density matrix which obeys the von Neumann equation. To incorporate the thermodynamic initial condition along with the von Neumann equation, it is advantageous to go to Liouville (L) space instead of the Hilbert (H) space in which DFT is formulated. This L-space quantum theory was developed by Umezawa over the last 25 years. We have adopted this theory to set up a new action principle which leads to the von Neumann equation. Appropriate variants of the theorems above are deduced in this framework. 

Object.—Quantum statistics was discussed briefly in Chapter 12 of The Mathematics of Physics and Chemistry, and as far as elementary treatments of quantum statistics are concerned,1 that introductory discussion remains adequate. In recent years, however, a spectacular development of quantum field theory has presented us with new mathematical tools of great power, applicable at once to the problems of quantum statistics. This chapter is devoted to an exposition of the mathematical formalism of quantum field theory as it has been adapted to the discussion of quantum statistics. The entire structure is based on the concepts of Hilbert space, and we shall devote a considerable fraction of the chapter to these concepts. 

The quantum states of Schrodinger s theory constitute an example of Hilbert space, and their scalar product has a direct physical meaning. Any such example of Hilbert space, where we can actually evaluate the scalar products numerically, is called a representation of Hilbert space. We shall continue discussing the properties of abstract Hilbert space, so that all our conclusions will apply to any and every representation. 

We must next consider more precisely the connection between the description of bodily identical states by the two observers (the requirements of Postulate 1). Quite in general, in fact, a physical theory, and quantum electrodynamics in particular, is fully defined only if the connection between the description of bodily identical states by (equivalent) observers is known for every state of the system and for every pair of observers. Since the observers are equivalent every state which can be described by 0 can also be described by O. Given a bodily state of the same system, observer 0 will ascribe to it a state vector Y0> in his Hilbert space and observer O will attribute to it a state vector T0.) in his Hilbert space. The above formulation of invariance means that there exists a one-to-one correspondence between the vectors Y0> and Y0.) used by observers 0 and O to describe bodily the same state.3 This correspondence guarantees that the two Hilbert spaces are in fact isomorphic. It is, therefore, possible for the two observers to agree to describe states of the system by vectors in the same Hilbert space. A similar statement can be made for the observables there exists a one-to-one correspondence between the operators Q0 and Q0>, which observers 0 and O attribute to observables. The consistency of the theory (Postulate 2) demands, however, that the two observers make the same prediction as the outcome of the same experiment performed on bodily the same system. This requires the relation... 

Rate of change of observables, 477 Ray in Hilbert space, 427 Rayleigh quotient, 69 Reduction from functional to algebraic form, 97 Regula fold method, 80 Reifien, B., 212 Relative motion of particles, 4 Relative velocity coordinate system and gas coordinate system, 10 Relativistic invariance of quantum electrodynamics, 669 Relativistic particle relation between energy and momentum, 496 Relativistic quantum mechanics, 484 Relaxation interval, 385 method of, 62 oscillations, 383 asymptotic theory, 388 discontinuous theory, 385 Reliability, 284... 

Potential fluid dynamics, molecular systems, modulus-phase formalism, quantum mechanics and, 265—266 Pragmatic models, Renner-Teller effect, triatomic molecules, 618-621 Probability densities, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 705-711 Projection operators, geometric phase theory, eigenvector evolution, 16-17 Projective Hilbert space, Berry s phase, 209-210... 

Concepts in quantum field approach have been usually implemented as a matter of fundamental ingredients a quantum formalism is strongly founded on the basis of algebraic representation (vector space) theory. This suggests that a T / 0 field theory needs a real-time operator structure. Such a theory was presented by Takahashi and Umezawa 30 years ago and they labelled it Thermofield Dynamics (TFD) (Y. Takahashi et.al., 1975). As a consequence of the real-time requirement, a doubling is defined in the original Hilbert space of the system, such that the temperature is introduced by a Bogoliubov transformation. 

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

In the case of quantum field theory the section determines the Hilbert space of states under a certain gauge. This choice of gauge then determines the unitary representation of the Hilbert space. We may then replace the section with the fermion field /, which acts on the Fock space of states. It is then apparent that a gauge transformation A t > A t + 84 is associated with a unitary transform of the fermion field v / > v / I 8 /. The unitary transformation of the fermion... 

E. Eliav, A. Borschevsky, K.R. Shamasundar, S. Pal, U. Kaldor, Intermediate Hamiltonian Hilbert space coupled cluster method Theory and pilot application, Int. J. Quantum Chem. 109 (13) (2009) 2909. 

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... 

As an extension of Noether s theorem to quantum mechanics, the hypervirial theorem [101] derives conservation laws from invariant transformations of the theory. Consider a unitary transformation of the Schrodinger equation, U(H — F)T = U(H — = 0, and assume the variational Hilbert space closed under a... 

To stay in Hilbert space would imply the measuring/measured quantum systems to remain in an entangled state unknown to people at the Fence. But all possible changes are there anyway. To disclose them, energy must be exchanged, and consequently, entropy must vary. One is coming close to thermodynamics as soon as the description of phenomena forces to take the systems away Hilbert space arena. In other words, Hilbert space alone is not adequate to handle this type of physics because it is an abstract formalism only. This implies that actual emergence of a particular outcome cannot be accounted for by a quantum theory the space-time occurrence of one click is not predictable by the theory. 

The merger is between Special Theory of Relativity and abstract quantum states through rigged Hilbert spaces the inertial frame is used to set up an abstract configuration space in laboratory space. At the abstract level, both formalisms are required. 

It is hardly necessary to emphasize that the present review is a fairly simple exercise in linear algebra and is intended to familiarized theoretical physicists and chemists working on the quantum theory of matter with the fundamental properties of the unbounded similarity transformations as applied to N-electron systems. Special attention has been given to the change of the spectra and how it is related to the domain of the transformation applied and to the fact that the eigenfunctions may be transformed not only within the L2 Hilbert space, but also out of and into this space (see Fig. 1). 

The symmetry of a physical quantum system imposes distinctive regular structures in its associated Hilbert space. These distinctive patterns are determined purely by the group theory of the symmetry and are independent of other details of the system. The wave function ip(x) of the electron in a hydrogen atom centred at the origin, may be considered again as an illustrative example. 

Theoretical chemistry has two problems that remain unsolved in terms of fundamental quantum theory the physics of chemical interaction and the theoretical basis of molecular structure. The two problems are related but commonly approached from different points of view. The molecular-structure problem has been analyzed particularly well and eloquent arguments have been advanced to show that the classical idea of molecular shape cannot be accommodated within the Hilbert-space formulation of quantum theory [161, 2, 162, 163]. As a corollary it follows that the idea of a chemical bond, with its intimate link to molecular structure, is likewise unidentified within the quantum context [164]. In essence, the problem concerns the classical features of a rigid three-dimensional arrangement of atomic nuclei in a molecule. There is no obvious way to reconcile such a classical shape with the probability densities expected to emerge from the solution of a molecular Hamiltonian problem. The complete molecular eigenstate is spherically symmetrical [165] and resists reduction to lower symmetry, even in the presence of a radiation field. 

---