Stochastic Geometric Mechanics by Sergio Albeverio Ana Bela Cruzeiro & Darryl Holm

Author:Sergio Albeverio, Ana Bela Cruzeiro & Darryl Holm
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

A Lagrangian for this particle is the following function of and :

The momentum conjugated to takes the slightly unusual form

and the Hamiltonian of our particle is the following function of and :

Applying the most elementary quantisation procedure to this Hamiltonian , we obtain the Hamiltonian operator

acting on , and we can write the Schrödinger equation

which governs the time evolution of the wave function of our particle.

1.2.3 The Gauge Indeterminacy

All this procedure is very standard, except that it involves the electromagnetic potential rather than the electromagnetic field, and really depends on the particular choice which was made, as the expression of the Hamiltonian shows.

Suppose now that we change our choice of electromagnetic potential A , replacing it by for some function . This means that we replace V by and by . Then, as one checks by a short computation, the Hamiltonian H becomes a new operator , related to H by the equation

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