Using quantum-classical analogies, we find that dynamical pictures of quantum mechanics have precise counterparts in classical mechanics. In particular, the Eulerian and Lagrangian descriptions of fluid dynamics in classical mechanics are the analogs of the Schroedinger and Heisenberg pictures in quantum mechanics, respectively. Similarities between classical and quantum dynamical pictures are explored within the framework of the Koopman-von Neumann formalism. These allow for a natural definition of various dynamical pictures in classical mechanics as well as the application of classical concepts to quantum dynamics. As an illustration, we use the interaction picture to find the classical evolution of an ensemble of particles of equal initial momenta and arbitrary configuration density under the action of a constant force in one dimension. As a second example, we discuss the extension of the ideas of sensitivity to initial conditions and chaos in classical mechanics to quantum mechanics.
Using quantum-classical analogies, we find that dynamical pictures of quantum mechanics have precise counterparts in classical mechanics. In particular, the Eulerian and Lagrangian descriptions of fluid dynamics in classical mechanics are the analogs of the Schroedinger and Heisenberg pictures in quantum mechanics, respectively. Similarities between classical and quantum dynamical pictures are explored within the framework of the Koopman-von Neumann formalism. These allow for a natural definition of various dynamical pictures in classical mechanics as well as the application of classical concepts to quantum dynamics. As an illustration, we use the interaction picture to find the classical evolution of an ensemble of particles of equal initial momenta and arbitrary configuration density under the action of a constant force in one dimension. As a second example, we discuss the extension of the ideas of sensitivity to initial conditions and chaos in classical mechanics to quantum mechanics.
A quantum-mechanical counterpart to the classical mechanical variation of constants method is derived, with initial values of coordinates and momenta as “constants.” Use is made of a formal operator solution for nonautonomous or autonomous systems in classical mechanics, which we published earlier, and of the correspondence between Poisson brackets and commutators. An alternative unified Lie-algebraic derivation is also given. It is shown that the Schrödinger, Heisenberg, and interaction pictures in quantum mechanics do not correspond directly to the method of classical mechanical variation of these “constants.” A fourth picture, termed “mixed interaction,” is introduced and shown to so correspond. It complements the previous three in a symmetrical manner, bearing the same relation to the Heisenberg picture that the Schrödinger picture bears to the interaction one. The group-theoretic relationship to the interaction picture is noted, as is the relation to the usual variation-of-constants method in wave mechanics. For completeness, the classical counterparts of the Heisenberg and interaction pictures are also given. The present results arose from a comparison of quantum and classical treatments of collisions.
We discuss the particle method in quantum mechanics which provides an exact scheme to calculate the time-dependent wavefunction from a single-valued continuum of trajectories where two spacetime points are linked by at most a single orbit. A natural language for the theory is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture and the particle theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle. The Euler-Lagrange equations imply a fourth-order nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates using as input the initial wavefunction. The admissible solutions are those consistent with quasi-potential flow. The effect of the superposition principle is represented via a nonclassical force on each particle. The wavefunction is computed via the standard map between the Lagrangian coordinates and the Eulerian fields, which provides the analogue in this model of Huygens principle in wave mechanics. The method is illustrated by calculating the time-dependence of a free Gaussian wavefunction. The Eulerian and Lagrangian pictures are complementary descriptions of a quantum process in that they have associated Hamiltonian formulations that are connected by a canonical transformation. The de Broglie-Bohm interpretation, which employs the same set of trajectories, should not be conflated with the Lagrangian version of the hydrodynamic interpretation. The theory implies that the mathematical results of the de Broglie-Bohm model may be regarded as statements about quantum mechanics itself rather than about its interpretation.
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.
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 similarity in form between the two real equations implied by the single-body spin-0 Schrödinger equation in the position representation (wave mechanics) and the equations of fluid mechanics with potential flow in its Eulerian formulation was first pointed out by Madelung in 1926 [1]. In this analogy, the probability density is proportional to the fluid density, and the phase of the wave function is a velocity potential. A novel feature of the quantum fluid is the appearance of quantum stresses, which are usually represented through the quantum potential. To achieve mathematical equivalence of the models, the hydrodynamic variables have to satisfy 4.1 Introduction .................................................................................................... 55 4.2 Deduction of Schrödinger’s Equation from the Lagrangian Picture Trajectory Theory ........................................................................................... 57 4.2.1 Lagrangian Picture ............................................................................. 57 4.2.2 Eulerian Picture .................................................................................. 62 4.2.3 Equivalence of First-Order and Second-Order Trajectory Laws ........64
A generalized Euler equation of fluid dynamics is derived for describing many-body states of quantum mechanics. The Eulerian Eq. can be viewed as representing the interaction of two substates, where each substate has its own velocity and pressure fields. These field quantities are given by maps of the wavefunction. For one-body systems, the Eulerian Eq. can model either a fluid or particle description of quantum states. The generalized Euler Eq. is shown to be the gradient of an equation representing the total-energy of the two substates, having two energy fields. This total-energy Eq. is a generalization of the Bernoulli Eq. of fluid dynamics. The total-energy Eq., along with a continuity-equation, is equivalent to the time-dependent Schroedinger Eq. An equation is also derived that is equivalent to the main equation of Bohmian mechanics with additional identifications: The quantum potential of Bohmian mechanics is given as a sum of a kinetic energy and pressure fields. Also, the time derivative of the wavefunction phase is replaced by an energy field. In the formalism, field quantities are identified from their placement in equations of classical mechanics and separately, by definitions that involve the wavefunction and operators of quantum mechanics. This approach yields, unintended, and unknown energy and pressure fields. These fields, however, are shown to satisfy a continuity Eq., an equation that is equivalent to the other equation of Bohmian mechanics. It is also demonstrated that energy conservation holds for both of these energy fields, if the wavefunction is a linear-combination of eigenvectors, where the eigenvectors can be nondegenerate. A detailed investigation is given on the possible behavior, or source, of an electron that has one of the velocity fields. Alternate formulae for this velocity fields are also considered.