We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product.
We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product.
A non-Hermitian operator H defined in a Hilbert space with inner product ⟨ · | · ⟩ may serve as the Hamiltonian for a unitary quantum system if it is η -pseudo-Hermitian for a metric operator (positive-definite automorphism) η . The latter defines the inner product ⟨ · | η · ⟩ of the physical Hilbert space H η of the system. For situations where some of the eigenstates of H depend on time, η becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.
A non-Hermitian operator may serve as the Hamiltonian for a unitary quantum system, if we can modify the Hilbert space of state vectors of the system so that it turns into a Hermitian operator. If this operator is time-dependent, the modified Hilbert space is generally time-dependent. This in turn leads to a generic conflict between the condition that the Hamiltonian is an observable of the system and that it generates a unitary time-evolution via the standard Schr\"odinger equation. We propose a geometric framework for addressing this problem. In particular we show that the Hamiltonian operator consists of a geometric part, which is determined by a metric-compatible connection on an underlying Hermitian vector bundle, and a nongeometric part which we identify with the energy observable. The same quantum system can be locally described using a time-dependent Hamiltonian that acts in a time-independent state space and is the sum of a geometric part and the energy operator. The full global description of the system is achieved within the framework of a moderate geometric extension of quantum mechanics where the role of the Hilbert space of state vectors is played by a Hermitian vector bundle E endowed with a metric compatible connection, and observables are given by global sections of a real vector bundle that is determined by E. We examine the utility of our proposal to describe a class of two-level systems where E is a Hermitian vector bundle over a two-dimensional sphere.
ABSTRACT: The main aim of this article is to present the time-evolution as the unitary and self-adjoint operators on Hilbert spaces and to describe its application in the development of quantum mechanics. The initial value problems associated with the quantum mechanical Schrodinger equation, ( ) = ( ) in the Hilbert space is solved by the use of timeevolution. The importance of time-evolution is also seen as the operation of turning machine in quantum mechanics, which is regarded as the time-evolution of the machine control state. Time-evolution of a quantum mechanical system is observed to be unitary and self-adjoint operators of Hilbert space, since it corresponds to an observable, specifically energy position and momentum. It was observed that time-evolution of a quantum mechanical system is generated by a self-adjoint operator, called Hamiltonian, V K H ˆ ˆ ˆ , expressed by the Schrodinger equation, above.
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.
Although• quantum mechanics has achieved great success in the explanation of physical phenomena, the mathematics used in quantum mechanics is not rigorous.Unrigorous treatments occur in the case of continuous spectrum.In a Hilbert • space, let A be a self-adjoint operator.When we define the eigenvalue and the eigenelement by the ' ' relation Af=J.f, (1) ' ' h d -. -f(q)=).f(q)2m dq has solutions 2,ri f(q)=ce"Aq for any ;.., But these are not quadratically integrable.E. Hellinger and H. Weyi< 0 have already investigated the problem of continuous spectrum and introduced the " eigendifferential."But they are concerned with a special problem, i. e. bounded quadratic forms and solutions of differential equation.In respect of generality, they do not satisfy.< 2 > E. Fues< 3 > and some other writers<" have used this conception " eigendifferential " in quantum mechanics.But the treatments are either complicated or not rigorous.J. v. Neumann< 5 > has proved that in Hilbert space, for any selfadjoint operator A, there exists a resolution of identity E(;..) such that Af= J:and he has applied this theory to quantum mechanics.Although his theory gave a rigorous mathematical foundation to quantum mechanics, it did not give any interpretation to the representation theory of Dirac's form.Recently I investigated the "theory of vector-valued set functions."This theory removes the difficulties in the treatement of continuous spectrum, and gives a rigorous interpretation to the representation theory.< 5 > The vector-valued set function q(U) is a function of set U, and its functional value is an element (vector) in the Hilbert ,space, (1) E. Hellinger [l], 231-271.H. Weyl [1], 238-251.(2) T. Carlemann used this conception "eigendifferential" in the treatise of singular integral equations.Cf.T. Carlemann [1].(3) E. Fues [1], 295-303.(4) For example, L. de Broglie [2], 140-159.(5) J. v. Neumann [1], 53-121.(6) F. Maeda [2], [5], [6].A summary of the theory of vector-valued set functions is given in the first part Qf the present paper.(1) G. Temple has formulated a principle of quantum theory in which projective operators are given priority over self-adjoint operators.(C(.G. Temple [1)).But. the .whole discussion differs from that of my present paper.