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Journal of Physics A Mathematical and Theoretical

Newton-equivalent Hamiltonians describing a plane rotator and their quantization

Francesco Calogero

Published May 17, 2012• 3 citations
Publisher

Abstract

In the context of classical mechanics, Newton-equivalent Hamiltonians are those that yield the same Newtonian equations of motion ('acceleration equal force'). Various such Hamiltonians are discussed, all of them characterizing a rotator in the plane. Some of them are well known, while others are perhaps new. Their quantization is also discussed. Other Hamiltonians yielding analogous motions, and related open problems, are mentioned.

DOI
10.1088/1751-8113/45/22/225203

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In the context of classical mechanics, Newton-equivalent Hamiltonians are those that yield the same Newtonian equations of motion ('acceleration equal force'). Various such Hamiltonians are discussed, all of them characterizing a rotator in the plane. Some of them are well known, while others are perhaps new. Their quantization is also discussed. Other Hamiltonians yielding analogous motions, and related open problems, are mentioned.
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.
Classical mechanics is introduced in order to lay the foundation for quantum mechanics. Newton's laws are given and used to develop various equations of motion. The modern concept of energy is introduced and used to develop the equation of motion. Generalised coordinates are detailed and used to illustrate the usefulness of the appropriate coordinate system for the problem at hand. Brief introductions to the Lagrangian and Hamiltonian alternatives to Newtonian mechanics are given, primarily to show where it is that the quantum mechanical Hamiltonian comes from. Classical angular momentum mechanics is developed and the parallels between linear and rotational quantities is drawn.
The Classical Newton-Lagrange equations of motion represent the fundamental physical law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses a description of classical dynamics and presents some irresponsible speculations about its quantization by introducing a certain canonical two-form ?. By its construction ? embodies kinetic energy and forces acting within the system (not their potential). A new type of variational principle employing differential two-form ? is introduced. Variation is performed over “umbilical surfaces“ instead of system histories. It provides correct Newton-Lagrange equations of motion. The quantization is inspired by the Feynman path integral approach. The quintessence is to rearrange it into an “umbilical world-sheet“ functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, the new approach reduces to the standard quantum mechanics. As an example, Quantum Mechanics with friction is analyzed in detail.
By working directly from Newtonian equations of motion and representing acceleration ẍ as [(x,H), H], families of Hamiltonians may be constructed which are functions of the energy and which, while describing the same classical motions, are not connected by canonical transformations. One dimensional examples are given, and the formal rules of quantization executed when H is (energy)12 to show that the quantum motion does not come out right. The reason is that the commutation rule (x,p) = iħ does not have universal standing; quantum theory in employing this rule is apparently requiring not merely that H be a generator of motion, but also that it be a very particular one, viz., the energy when the system is conservative.