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Physical Review

Do the Equations of Motion Determine the Quantum Mechanical Commutation Relations?

Eugene Paul Wigner

Published Mar 1, 1950• 472 citations
Publisher

Abstract

The commutator of the Hamiltonian with the operator corresponding to any physical quantity gives the operator which corresponds to the time derivative of that quantity. One can ask, hence, whether the postulate, that the quantum mechanical operators obey the classical equations of motion, uniquely determines the commutation relations. The answer is found to depend on the form of the Hamiltonian and is in the negative for a free particle and for the harmonic oscillator.

DOI
10.1103/physrev.77.711

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The commutator of the Hamiltonian with the operator corresponding to any physical quantity gives the operator which corresponds to the time derivative of that quantity. One can ask, hence, whether the postulate, that the quantum mechanical operators obey the classical equations of motion, uniquely determines the commutation relations. The answer is found to depend on the form of the Hamiltonian and is in the negative for a free particle and for the harmonic oscillator.
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.
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.
It has been shown by Wigner that the equations of motion of a quantum-mechanical system with a hamiltonian H=12p2+V(q)H=\frac{1}{2}{p}^{2}+V(q)H=21​p2+V(q) need not imply the commutation relation i[p,q]=1i[p,q]=1i[p,q]=1. Particular examples are furnished by the free particle and the harmonic oscillator. On the other hand, Wigner points out that the above commutation rule does follow from the equations of motion if V(q)V(q)V(q) is of the form V=aq+bV=aq+bV=aq+b or V=aq3V=a{q}^{3}V=aq3. In the present note, this latter result is generalized to the case of an arbitrary potential V=aqn+bV=a{q}^{n}+bV=aqn+b, where nnn is an odd integer; the last-named potentials are, in fact, particular cases of those satisfying a general criterion contained in the article.
By using the formulation of quantum mechanics with classical pictures, the motion of a quantum is analysed for linear harmonic oscillator. On this occasion it is postulated that the quantum mechanical momentum of the quantum is approximately equal to the corresponding classical mechanical momentum in the region of validity of the coordinates. According to this analysis it is shown that the quantum mechanical motion well corresponds to the classical motion, and that the wave equation determines the associated wave along the whole orbit of the quantum in the configuration space and the uniqueness of the wave throughout the orbit plays an important role in quantum mechanics. In addition to the above, an attempt is made to make clear the well-known fact that classical mechanics with the quantization gives approximately the same results as quantum mechanics.
From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.
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.