Journal of Hubei Vocational-Technical College

Conversion of Quantum-mechanical Theory between Momentum Image and Coordinate Image

Xin‐Yan Wu

Published Jan 1, 2005• 0 citations

Abstract

Representation theory is the basic content of quantum mechanics.This paper discusses the form of mechanical quantity operator and state vector of quantum mechanics under momentum image and coordinate image.At last,it gives an example to explain the superiority by using the momentum image.

Related papers

IMPROVE PERCEPTION TOWARD REPRESENTATION TRANSFORMATION IN QUANTUM MECHANICS

Qi Zhou, Lishuai GUO

In this paper, the method how to further study the representational and the representational transformation is introduced by geometry coordinates, linear algebra and physical concept. According to this representational theory which can be simply compared with coordinate theory in geometry, two simple memory methods have been provided. Firstly, the transformation formula can be written as b =S+a from representation A to representation B, in which the matrix element of the transformation matrix is <ψ|φ>, which can be regarded as the inner product between the basis vector of representation A and representation B. At the same time, this formula can be thought of as the expression of the representation B of the state vector equals the transformation matrix acting on the expression of the representation A. Secondly, the transformation formula can be written as F' =S+FS for the mechanical operator from representation A to representation B, in which the operator F and operator F' are the expression of representations A and representations B, respectively, which this formula can be thought of as the expression of the representation B of the mechanical operator equals the conjugate matrix of transformation matrix and the expression of the representation A of the mechanical operator acting on transformation matrix. Finally, an example is given to prove the superiority of representation transformation. Meanwhile, the intrinsic function under two representations is deduced and the transformation between them is realized by the Fourier transform, thus deepening the understanding of representation transformation.

Wuli yu gongcheng.Feb 1, 20250 citations

Different Coordinate Momentum Operator

Qiao Wen-hua

Correspondence of operator-ih▽ and momentum operator in the quantum mechanics is discussed,and pointed out the simple correspondence in cartesian coordinate does not apply to general curvilinear coordinate,due to Ermi nature of mechanical quantity operator,we can't give corresponding momentum operator directly through simple correspondence under general curvilinear coordinate.

Yinshan Academic JournalJan 1, 20110 citations

The physical origin of Schrödinger equation

Xiao-Bo Yan

Abstract Quantum mechanics is one of the basic theories of modern physics. Here, the famous Schrödinger equation and the differential operators representing mechanical quantities in quantum mechanics are derived, just based on the principle that the translation invariance (symmetry) of a system in Hamiltonian mechanics should be preserved in quantum mechanics. Moreover, according to the form of the differential operators, the commutation relation in quantum mechanics between the generalized coordinate and the generalized momentum can be directly obtained. We believe that the results in this paper are very useful for understanding the physical origin of quantum mechanics.

arXiv:2106.01312European Journal of PhysicsApr 9, 20212 citations

Some Elements of the Transformation Theory

Ahmad Amer

This chapter introduces the representation of quantum states and operators in an orthonormal basis associated with a linear Hermitian operator, forming the foundation of matrix mechanics. The transformation between different representations is described, highlighting how operators and state vectors change under basis transformations. The projection operator is defined, and the closure (completeness) relation is expressed in terms of projection operators on the basis states. The Dirac bra–ket notation is introduced and thoroughly explained to unify representation theory. Detailed treatments of the coordinate and momentum representations are given, including the transformation between them via Fourier analysis. Finally, explicit forms of the coordinate and momentum operators acting on wavefunctions in the coordinate space are derived. Metadata/Keywords: matrix mechanics, representation theory, Hermitian operator, projection operator, closure relation, Dirac bra–ket notation, coordinate and momentum representations

CRC Press eBooksNov 5, 20250 citations

Two Examples of the Presentational Theory of Quantum Mechanics

LI Ji-jun

This paper systematically expounds the concrete forms and relevant problems of the coordinate functor and momentum functor in the coordinates and momentum presentations by way of the δ function and Dirac notation.

Journal of Aba Teachers CollegeJan 1, 20050 citations

Operators in quantum mechanics

Victor M. Galitski, Boris Mikhailovich Karnakov, Vladimir I Kogan +1 more

Abstract Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by Hermitian linear operators that act on state vectors that belong to a Hilbert vector space. This chapter analyzes problems that deal with the basic concepts of the theory of linear operators; eigenfunctions, eigenvalues, mean values; the projection operators; quantum-mechanical representations of operators and wave-functions; and unitary operators.

Feb 28, 20132 citations