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CRC Press eBooks

Alternative Formulations

Bernard d’Espagnat

Published Mar 5, 2018• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9780429501449-6

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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.
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.
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.
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.
In this chapter, the authors describe the Heisenberg picture of quantum mechanics by formally accomplishing a time-dependent transformation of state vectors and operators of the Schrodinger picture. Any new picture of quantum mechanics will be acceptable, if the Schrodinger state vectors and operators are subjected to a transformation, which does not alter: the spectrum of the eigenvalues of the operators and the scalar product of an arbitrary state vector with the eigenstate vectors. These two conditions are easily fulfilled by means of the unitary transformations. The authors show that a unitary operator represents an isomorph mapping of a certain unitary vector space onto itself. Since the physical quantities measured in the experiment are, in fact, a measure of the scalar product, as implied by the definition of the probability W through the scalar product in, it is clear that the predictions based upon the new, Heisenberg picture must be identical to those obtained in the Schrodinger picture.
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 outlines the governing equations in non-relativistic quantum physics. The Schrödinger equation (the Schrödinger picture) looms large. Its intuitive simplicity and elegance render it particularly appealing to aspiring engineers and materials scientists. The other two equivalent pictures presented are the Heisenberg picture and the interaction picture. The Heisenberg picture emphasizes time-dependent operators, and the interaction picture splits the so-called interaction Hamiltonian from the total Hamiltonian of the system considered. Much attention has been paid in this chapter to establish relationships between the pictures, as meaningful as possible at this level of treatment.