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The London Edinburgh and Dublin Philosophical Magazine and Journal of Science

XVII. Commutation relations in Lagrangian quantum mechanics

W. K. Burton, B. F. Touschek

Published Feb 1, 1953• 5 citations
Publisher

Abstract

Schwinger's method for deriving commutation relations is shown to fail in the case of first order equations of motion. A new method, adapted from some recent work of Peierls is discussed in connection with two examples of first order equations in point mechanics.
DOI
10.1080/14786440208520287

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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.
This work starts with classical equations of motion and sets very general quantization conditions (commutation relations). It is proved that these conditions imply that the equations of motion are equivalent to the Euler–Lagrange equations of a Lagrangian L. The result is a generalization of work by Feynman, recently reported by Dyson [Am. J. Phys. 58, 209–211 (1990)]. The Lagrangian L need not be unique. Examples are given, including classical equations that do not come from a Lagrangian and therefore cannot be quantized consistently.
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Abstract The classical and quantum theories of dynamical systems whose Lagrangians are linear in the co-ordinate derivatives are studied, with a view to clarifying a number of points in relation to Schwinger’s quantum-mechanical variational principle. In particular, linear variations of the dynamical variables are considered, and it is shown that they may be consistently included in Schwinger’s formalism. It is also found to be possible to obtain Green’s generalized commutation rules by excluding c number variations from the theory, for a suitably restricted class of Lagrangian.