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Journal of University of Science and Technology Liaoning

Theorem of unitary operators of transformations relating to three pictures in quantum mechanics

Shoushan Gao

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

The concept of picture in quantum mechanics is discussed,the definition of three pictures are given,the relationship between picture and representation are analyzed.A theorem of picture relating to hamiltonian decomposition is given,finally,the unitary operators of common transformations relating to three pictures in quantum mechanics are calculated by using this theorem.

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SIAM Journal on Applied Mathematics•Sep 1, 1973•78 citations
The concept of picture in quantum mechanics is discussed,the definition of three pictures are given,the relationship between picture and representation are analyzed.A theorem of picture relating to hamiltonian decomposition is given,finally,the unitary operators of common transformations relating to three pictures in quantum mechanics are calculated by using this theorem.
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.
A number of quantum states form an orthonormal and complete basis set if they are normalized and mutually orthogonal and any quantum state can be expressed as a linear combination. In classical mechanics, any two physical variables are commutable. However, this property is no longer valid for physical operators. Commutation of operators is an important property of quantum mechanics. Although there are many representations of wave functions and observables connected by unitary transformations, it is useful to distinguish certain classes of representations, called pictures, which differ in the way the time evolution of the system is treated. The Schrodinger picture is one in which the operators are time-independent but the wave function is time dependent. The virial theorem relates the kinetic energy to its potential in an energy eigenstate.
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.
Abstract In quantum mechanics courses, the concept of a reference frame is frequently hidden. In this article, we present a basic perspective to introduce the transformations of a reference frame based on unitary transformations. These unitary transformations are inspired in canonical transformation for reference frame from the perspective of classical mechanics. The manner we deal with this change in reference frame allows us to discuss some fundamental concepts in Schrödinger and Heisenberg pictures, as well as the ideas of passive and active description in quantum mechanics. These unitary transformations are inspired in a reference-frame canonical transformation from the perspective of classical mechanics. The approach we take for this reference-frame change allows for a discussion on fundamental concepts in Schrödinger and Heisenberg pictures, as well as the ideas of passive and active description in quantum mechanics. We apply this unitary transformation in five examples.
While groups of point transformations such as rotations, translations and permutations have been extensively used in quantum mechanics, we cannot say the same of the more general groups of canonical transformations. To begin with, for the latter groups we have to determine their unitary representation in appropriate spaces. But once this is achieved canonical transformations can play as important a role in quantum mechanics as point transformations. We illustrate this in relation with the harmonic oscillator in one and many dimensions and for one and many particles. We show how radial matrix elements with respect to harmonic oscillator states can be determined group theoretically both in the Schroedinger and Heisenberg pictures. We discuss the application of canonical transformations to the three body nuclear problem. We briefly outline in the concluding section applications to Coulomb problems, many body systems, accidental degeneracy, and generalized integral transforms.