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CRC Press eBooks

Equation of Motion

Ahmad Amer

Published Nov 5, 2025• 0 citations
Publisher

Abstract

This chapter explores the Schrödinger and Heisenberg pictures in quantum mechanics, emphasizing their mathematical equivalence and physical consistency. The two formulations are shown to be connected by a unitary transformation. The Heisenberg equation of motion is derived, and the time evolution of operators—particularly those not explicitly dependent on time—is expressed in terms of their initial values. It is demonstrated that scalar products, expectation values, matrix elements, and commutation relations remain invariant under the change of picture. Drawing a close analogy with classical mechanics, Dirac's insight is presented: the quantum mechanical counterpart of the classical Poisson bracket is the operator commutator divided by iℏ . The chapter concludes by deriving the Lorentz force law within the quantum framework using commutator algebra. Metadata/Keywords: Schrödinger picture, Heisenberg picture, unitary transformation, Heisenberg equation of motion, Poisson bracket, Lorentz force, time evolution, expectation values

DOI
10.1201/9781003677109-9

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This chapter explores the Schrödinger and Heisenberg pictures in quantum mechanics, emphasizing their mathematical equivalence and physical consistency. The two formulations are shown to be connected by a unitary transformation. The Heisenberg equation of motion is derived, and the time evolution of operators—particularly those not explicitly dependent on time—is expressed in terms of their initial values. It is demonstrated that scalar products, expectation values, matrix elements, and commutation relations remain invariant under the change of picture. Drawing a close analogy with classical mechanics, Dirac's insight is presented: the quantum mechanical counterpart of the classical Poisson bracket is the operator commutator divided by iℏ . The chapter concludes by deriving the Lorentz force law within the quantum framework using commutator algebra. Metadata/Keywords: Schrödinger picture, Heisenberg picture, unitary transformation, Heisenberg equation of motion, Poisson bracket, Lorentz force, time evolution, expectation values
The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
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 The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
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.
We introduce the Poisson bracket operator, which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this, we derive the quantum canonical equation, which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, this equation is applicable to ccc-number canonical variables and then coincides with the canonical equation in classical mechanics. Therefore, the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to nonstandard system where the Heisenberg equation is not defined. As an example, we consider the application to the system where ccc-number and qqq-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.