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CRC Press eBooks

Time evolution of quantum systems

Dževad Belkić

Published Jan 15, 2020• 0 citations
Publisher

Abstract

In non-relativistic quantum mechanics, there exist several different ways of describing the time dependence of dynamics of physical systems. Of particular importance for quantum scattering theory are the Schrodinger, Heisenberg and Dirac pictures. For establishing a connection between quantum mechanical formalisms and classical mechanics, as well as for investigating relativistic solutions, the Heisenberg picture appears to be the most convenient. This picture of quantum mechanics is particularly adapted for the examination of free fields. The customary wave-mechanical version of quantum theory is obtained if we choose, e.g., in the framework of the Schrodinger picture to work with the coordinate representation. Possible results of measurements on a quantum system are called eigenvalues. To each of the latter values corresponds one or more eigenstates. An eigenstate is degenerate when more than one eigenstate correspond to a single eigenvalue.

DOI
10.1201/9780429146497-6

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In non-relativistic quantum mechanics, there exist several different ways of describing the time dependence of dynamics of physical systems. Of particular importance for quantum scattering theory are the Schrodinger, Heisenberg and Dirac pictures. For establishing a connection between quantum mechanical formalisms and classical mechanics, as well as for investigating relativistic solutions, the Heisenberg picture appears to be the most convenient. This picture of quantum mechanics is particularly adapted for the examination of free fields. The customary wave-mechanical version of quantum theory is obtained if we choose, e.g., in the framework of the Schrodinger picture to work with the coordinate representation. Possible results of measurements on a quantum system are called eigenvalues. To each of the latter values corresponds one or more eigenstates. An eigenstate is degenerate when more than one eigenstate correspond to a single eigenvalue.
The description of the time evolution of systems is technically known as the "Schrodinger picture." An important alternative description is the "Heisenberg picture," which indeed was the one Heisenberg used when he first formulated quantum mechanics. The Schrodinger and Heisenberg pictures represent two different but equivalent sets of rules. In other words, when applied to a given physical situation they lead to the same observable predictions. In some respects, the Heisenberg picture presents definite technical advantages. It is particularly convenient, for instance, in quantum field theory. The chapter discusses work consistently with the equivalent Schrodinger set of rules. For this reason, it describes briefly the Heisenberg formulation. The chapter shows that the expressions used in the Schrodinger picture can be carried over to the Heisenberg picture. It implies, however, that the kets associated with the possible values that can be found upon measurement of an observable are themselves time-dependent kets.
In classical mechanics the state of a system is defined by the numerical values of a set of dynamical variables such as the position coordinates and the velocities. In quantum mechanics the state of a system at an instant is represented by a unit vector in a Hilbert space, where a set of axes are the eigenvectors of a complete set of observables. The description of quantum dynamics where the wave function is time-dependent and the operators describing the observables are time-independent is known as the Schrodinger picture. The use of total time derivative of operator that depends on the time evolution of the wave function as well as on any intrinsic time-dependence in the operators simplifies the development of quantum mechanics and allows the development to more closely follow the corresponding development of classical mechanics. The difference between the Schrodinger and Heisenberg pictures is best understood as the difference in the basic set of observables.
This chapter demonstrates how one can find the time evolution of the operators from the knowledge of the time evolution of the state vector, and vice versa. It introduces a unitary operator and discusses the concept of a unitary transformation to show how one could transfer the time dependence from the state vectors to the operators. The possibility that either the state vectors or the operators can depend explicitly on time will lead to introduce the Schrodinger, Heisenberg, and interaction pictures. The chapter discusses the fundamental differences between these pictures. It also discusses the Ehrenfest theorem, which shows under which conditions quantum mechanics predicts the same results for measured physical quantities as classical physics. It illustrates unitary transformations of an arbitrary operator Â(t) which can depend explicitly on time. Quantum mechanics produces the same result as classical mechanics for a system in which particles are represented by well-localized wave functions.
This chapter briefly explains the two elementary points concerning quantum mechanics: the mean value rule and the Heisenberg picture. The Schrodinger and Heisenberg pictures are two different but equivalent sets of rules. In other words, when applied to a given physical situation they lead to the same observable predictions. In some respects the Heisenberg picture has definite technical advantages. It is particularly convenient, for instance, in quantum field theory. For describing pure cases the use of wave functions in place of kets is sometimes convenient. The mean value of an observable A and the probability that a measurement of A yields eigenvalue a k can be expressed by means of wave functions. The state vector of a system that does not undergo a measurement does not change with time, obviously this state vector is suddenly modified—also in the Heisenberg picture—when a measurement takes place.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
Quantum mechanics takes various forms. For atomic and nuclear physics, the Schrödinger picture is convenient. For quantum field theory and for correspondence with classical mechanics through Poisson brackets, the Heisenberg picture is useful. The interaction picture is useful in calculations. The Wigner phase-space is convenient for studying the uncertainty relations and their symmetries. The density matrix and probability distribution are necessary for extracting measurable dynamic quantities and are essential for ensemble averages. Mathematical instruments, such as the Lie algebra of the rotation group, the creation and annihilation operators of the Fock space and the Lie algebra governing Dirac’s two oscillators are shown to be derivable from Heisenberg’s commutation relations.