This definition of the canonical momentum ensures that one of the Euler—Lagrange equations has the form. Canonical quantization is applied, by definition, on canonical coordinates. Indeed, counterexamples exist satisfying the canonical commutation relations but not the Weyl relations. An analogous relation holds for the spin operators. In general, position and momentum are vectors of operators and their commutation relation between different components of position and momentum can be expressed as. A Modern Approach to Quantum Mechanics. These technical issues are the reason that the Stone—von Neumann theorem is formulated in terms of the Weyl relations. One can easily reformulate the Weyl relations in terms of the representations of the Heisenberg group. Although the quantity p kin is the "physical momentum", in that it is the quantity to be identified with momentum in laboratory experiments, it does not satisfy the canonical commutation relations; only the canonical momentum does that.
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