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Journal of the Korean Physical Society

Physics analysis with Leibniz’s differential operators dn

U-Rae Kim, Sungwoong Cho, Jungil Lee

Published Aug 29, 2023• 0 citations
Publisher

Abstract

Abstract We introduce a systematic approach to represent Leibniz’s nth-order differential operator $$d^n$$ d n as the ratio of an infinite product of infinitesimal difference operators to an infinitesimal parameter. Because every difference operator can be expressed as a difference of two shift operators that translate the argument of a function by finite amounts, Leibniz’s differential operator $$d^n$$ d n is eventually expressed as the infinite product of infinitesimal binomial operators consisting of the shift operators. We apply this strategy to demonstrate the derivation of the translation or time-evolution operators in quantum mechanics. This fills the logical gap in most textbooks on quantum mechanics that usually omit explicit derivations. Our approach could be employed in general physics or classical mechanics classes with which one can solve the equation of motion without prior knowledge of differential equations.

DOI
10.1007/s40042-023-00894-8

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Abstract We introduce a systematic approach to represent Leibniz’s nth-order differential operator dnd^ndn d n as the ratio of an infinite product of infinitesimal difference operators to an infinitesimal parameter. Because every difference operator can be expressed as a difference of two shift operators that translate the argument of a function by finite amounts, Leibniz’s differential operator dnd^ndn d n is eventually expressed as the infinite product of infinitesimal binomial operators consisting of the shift operators. We apply this strategy to demonstrate the derivation of the translation or time-evolution operators in quantum mechanics. This fills the logical gap in most textbooks on quantum mechanics that usually omit explicit derivations. Our approach could be employed in general physics or classical mechanics classes with which one can solve the equation of motion without prior knowledge of differential equations.
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The fundamental role of an operator is to effect a transformation. Differential operators and matrices are two important examples of how operators can be represented in a concrete way that makes quantitative calculations possible on systems that they represent. Both types of operator play important roles in quantum mechanics and the people shall meet them again. In a sense an operator interrogates a system represented by its eigenfunction, which results in a definite answer given by the eigenvalue. Operators with real eigenvalues also play a key role in quantum theory. Operators with real eigenvalues are conventionally represented by what are termed Hermitian or self-adjoint operators. Non-commutation has important consequences when it comes to information about two different parameters associated with a system. The connection between differentiation of a product, the Leibniz rule, and the commutator bracket, is indicative of the underlying algebraic structure of differential calculus.
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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.
This chapter aims to develop an appropriate notation and to facilitate the subsequent discussion of the role played by the Lie algebras. It is unlikely that one manages to express a sufficiently realistic model only in terms of the operators that span a finite-dimensional Lie algebra, but it may be possible to simulate most of the relevant features of many physical phenomena by means of such simple models. The chapter discusses well-known approximate methods for the treatment of the time-evolution equation. If the Hamiltonian operator belongs to a finite-dimensional Lie algebra, then the differential equation for the time-evolution operator is easier to solve, and even the calculation of transition probabilities and expectation values of observables may be straightforward. The most relevant feature of this application of the Lie algebraic method is the occurrence of a matrix equation of finite dimension in contrast with the expansion in a basis set of state vectors, which typically leads to matrices of infinite dimension.
In this chapter, we finally introduce time evolution. The development is carried out for finite, enumerable systems (i.e., systems whose states live in a finite dimensional, inner product space) which, of course, allows a straight forward extension to the continuum and infinite dimensional state spaces. The necessary mathematical background of unitary operators, and their properties relevant to quantum mechanics are first introduced. In particular, the fact that preservation of norm under the action of an operator implies unitarity, and that every unitary operator can be expressed as the exponential of i timesa Hermitian operator are introduced. Time evolution is then introduced in its integral form through the time evolution operator, and the Hamiltonian operator is introduced in this context. This approach, however, necessitates the restriction to conservative systems. Next, the differential form of the law of time evolution, i.e., Schrodinger equation, is derived from the time evolution operator, and this is postulated to be the general law of time evolution. We then demonstrate how, for conservative systems, the solution of the Schrodinger equation essentially reduces to solving the eigenvalue problem of the Hamiltonian. Constants of motion and Heisenberg picture have been relegated to the problem section.