A mathematical formalism of quantum mechanics says that a pure state of a quantum system corresponds to a vector of norm 1 and an observable is a self-adjoint operator on the space of states. It is of interest to describe all linear or nonlinear operators which preserve the pure states of the system. In the linear case, it is nothing more than isometries of Hilbert spaces. In the nonlinear case, this problem was open. In this paper, in the small dimensional spaces, we shall describe all quadratic plus linear operators which preserve pure states of the quantum system.
A mathematical formalism of quantum mechanics says that a pure state of a quantum system corresponds to a vector of norm 1 and an observable is a self-adjoint operator on the space of states. It is of interest to describe all linear or nonlinear operators which preserve the pure states of the system. In the linear case, it is nothing more than isometries of Hilbert spaces. In the nonlinear case, this problem was open. In this paper, in the small dimensional spaces, we shall describe all quadratic plus linear operators which preserve pure states of the quantum system.
As we knew, a mathematical formalism of a quantum mechanics says that any quantum system is
identified with some finite- or infinite-dimensional Hilbert space; pure states correspond to vectors
of norm 1; observables are self-adjoint operators on the space of states. Thus the set of all pure
states corresponds to the unit sphere in the Hilbert space [1-2]. It is of interest to describe all linear
or nonlinear operators which preserve the pure states of the system. In the linear case, it is nothing
more than isometries of Hilbert spaces [1]. In the nonlinear case, this problem was open. In this
paper we shall describe all quadratic plus linear operators which preserve pure states of the
quantum system.!
The Hilbert-Space Description By the static description of a physical system we mean the rules that assign specified mathematical objects to the states and to the physical quantities of the system, and the prescriptions for calculating the probability distribution of the possible values of every physical quantity when the state of the system is given. In the usual Hilbert-space formulation of quantum mechanics, to each physical system is attached a separable Hilbert space (generally infinite-dimensional) over the complex field. To every physical quantity is associated a linear, self-adjoint, not necessarily bounded operator on. If one deals with a strictly quantum system, then the converse is also generally assumed: every self-adjoint operator on represents some physical quan- tity. The restriction “strictly quantum” is necessary: if the system retains some nonquantum (i.e. classical) feature, so that it requires the algorithm of so-called “superselection rules” (see Chapter 5), then there are self-adjoint operators on that do not represent physical quantities. It should also be stressed that even in the strictly quantum case, most of the self-adjoint operators actually do not represent “interesting” physical quantities: only a few of them represent physical quantities that are useful and meaningful for the description of the physical system (e.g., energy, momentum, position, angular momentum). Therefore, by asserting that every self-adjoint operator on corresponds to some physical quantity we mean there is no a priori impossibility of devising such a correspondence, but we do not claim that this correspondence is present in the real work of experimental physics.
This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.
Abstract In this chapter we present the mathematical framework of describing wave functions as vectors in a Hilbert space and observables as Hermitian operators on such a vector space. We describe the mathematical formulation of these ideas for finite and infinite-dimensional cases via the Dirac notation, and discuss the example of the Hilbert space of photon polarization. Important classes of linear operators on Hilbert spaces are discussed, including projectors, Hermitian and self-adjoint operators, as well as unitary operators. As an important centre piece for the mathematical language of quantum mechanics we present the spectral theorem. We discuss the subtleties of this framework in infinite dimensions, giving a brief overview of the distributional aspects of quantum mechanics and the conceptually important problem of finding self-adjoint extensions via the Weyl-von Neumann theorem
Abstract The object of this book is to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous [A]presentation of the mathematical tools necessary for the purposes of this theory will be given, i.e., a theory of Hilbert spaces and the so-called Hermitian [or self-adjoint] operators. To this end, an accurate introduction to unbounded operators is also necessary; that is, an extension of the theory beyond its classical limits (developed by Hilbert and E. Hellinger, F. Riesz, E. Schmidt, 0. Toeplitz). The following may be said regarding the method employed in this type of treatment: as a rule, calculations should be performed with the operators themselves (which represent physical quantities) and not with the matrices, which, after the introduction of a (special and arbitrary) coordinate system in Hilbert space, result from them. This “coordinate free” (i.e., invariant) method, with its strongly geometric language, possesses noticeable formal advantages.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.