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Journal of Kashgar Teachers College

Two Pictures and the Constant Nature of Hamilton's Operator in them

Cao Li-ping

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

This artcle recommends the trait of Schrodinger picture and Heisenberg picture and the relation of the state vector and the operator between two pictures.And it proves Hamilton' s operator keeping constant between Schrodinger picture and Heisenberg picture by two ways so as to show the particularity of Hamilton' s operator.

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Cambridge University Press eBooks•Oct 31, 2014•0 citations
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 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.
It is well known that the motion of a dynamical system can be pictured in two distinct ways, which Dirac names the Heisenberg picture and the Schrödinger picture. The equations of motion take different forms in the two pictures, but of course have identical physical consequences, since the motion of the system does not depend in any way on which picture we choose. It should therefore be possible to express the equations in an invariant form (independent, that is, of the picture used). It will be shown in this note that not only can this be done (equation (3)), but it can be done in such a way that it is not even necessary to introduce a picture at all (equation (3′)).
Some years have passed since the appearance of Dirac's striking interpretation of the Heisenberg-Schroedinger parallelism in the equations of motion, and yet it appears that there still remain obscure points which have led to misunderstanding. Although the Schroedinger method suffices to deal with many practical problems, the flexibility of the Dirac-Jordan representation theory, aside from its intrinsic unity and beauty, recommends it to general theoretical investigation. The present paper considers the invariant form of the equations of motion in relation to the Schroedinger-Heisenberg pictures of quantum mechanics insofar as the latter refer to the "pictured" intrinsic motion of states (vectors) and observables (Hermitian operators). The discussion is divided roughly into 5-sections: Operator and Vector Transforms, Contact Transformations, Heisenberg-Schroedinger Operators, Equations of Motion and Invariant Properties, and the Density Operator and Operator Spur. Where possible, quasi-geometrical diagrams are given to illustrate the relation and transition between the several "pictures of the motion."
This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol. The following steps are the local and global conservation laws associated with the Schrödinger equation, probabilistic interpretation of the wave function and probability distribution, the spreading of wave packets and transformation properties of wave functions. In the last part, the Heisenberg picture is outlined at an introductory level. From classical to wave mechanics In the previous chapter we have seen how to go from wave equations on space–time to Hamilton equations on phase space by means of an appropriate approximation requirement on wave solutions. In more formal language, we could say we have considered a procedure to associate a Hamiltonian ordinary differential equation on phase space starting with a partial differential equation on configuration space. This association produces a correspondence between differential operators and polynomials on phase space, which relies on the notion of symbol (see below). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is physically non-trivial because it amounts to building a wave equation from the equation on rays. From the formal point of view it appears to contain some ambiguities. Such a transition is always possible if polynomials in p have constant coefficients.