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Birkhäuser Basel eBooks

Unitary Evolution Groups

Published Jan 1, 2009• 0 citations
Publisher

Abstract

Unitary evolution groups are in one-to-one correspondence with self-adjoint operators. They are also responsible for the time evolution of quantum states, that is, the solutions of Schrhödinger equations. In this chapter such relations are described in detail, including standard examples of unitary evolution groups and infinitesimal generators. Different continuity assumptions on the unitary groups are discussed.

DOI
10.1007/978-3-7643-8795-2_6

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arXiv:1006.5183•Physica Scripta•Dec 7, 2011•2 citations
Unitary evolution groups are in one-to-one correspondence with self-adjoint operators. They are also responsible for the time evolution of quantum states, that is, the solutions of Schrhödinger equations. In this chapter such relations are described in detail, including standard examples of unitary evolution groups and infinitesimal generators. Different continuity assumptions on the unitary groups are discussed.
The Unitary Dynamical Group in Schrödinger's Picture Up to now we have been concerned with the description of a physical system at a fixed instant of time. The time evolution of quantum systems constitutes a many-faced, wide issue which, however, is somewhat tangential to the main aim of this volume. We shall touch upon it only briefly, restricting ourselves to the more familiar aspects of the problem. We assume, to begin with, that the physical system has no superselection rules, so that states and physical quantities correspond one-to-one to density operators and self-adjoint operators, respectively, on the Hilbert space. To describe the dynamical evolution of the system one has to specify the way the representatives of states and physical quantities evolve in, which is fixed in time, since the physical system is supposed to preserve its identity. There is an old tenet according to which there are two kinds of time evolution: the discontinuous, nondeterministic evolution undergone by the quantum system when a measurement is made on it, and the continuous, deterministic evolution caused by the interaction with external forces or with other quantum systems. The first kind of evolution will be dealt with in Chapter 8, where we shall find it less alarming than is often claimed.
The time-dependent Schrödinger equation forms the chapter's starting point. For a time-independent Hamiltonian, a quantum state may be time evolved with the standard, unitary time evolution operator. After obtaining this operator, an algorithm is provided by which a state may be time evolved, and measurement probabilities determined. A spin 1/2 example illustrates time evolution concretely. All this makes close contact with the discussion of unitary operators in Chapter 10. The time dependence of expectation values forms the next topic, followed by the concept of a constant of the motion, which is carefully contrasted with that of a stationary state. The energy-time uncertainty relations, which rest on a quite different conceptual footing than the ‘standard’ uncertainty relations, occupy the rest of the chapter. An example from spin 1/2 illustrates the energy-time uncertainty relations.
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.
Schrodinger wave equation, Hilbert space formalism, born’s rule for probability and hermitian operator are mainly the core of quantum theory and quantum description is generally based on Hilbert space of states, observables are described by self adjoint operators on the space of states, physical symmetries are explained by unitary transformations, and time evolution is a one-parameter group of unitary transformations on the Hilbert space of states. In this paper, I have discussed why a generalized philosophy is required before the development of quantum probability to fill up the gap.
In this paper we propose a systematic method to solve the inverse dynamical problem for a quantum system governed by the von Neumann equation: to find a class of Hamiltonians reproducing a prescribed time evolution of a pure or mixed state of the system. Our approach exploits the equivalence between an action of the group of evolution operators over the state space and an adjoint action of the unitary group over Hermitian matrices. The method is illustrated by two examples involving a pure and a mixed state.