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Operator ordering ambiguity
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The quantum meaning of the equivalent classical expressions pq and qp
is different due to the canonical commutation relation
xp-px=i hbar. (CCR)
For more complex classical functions, one has a much nore serious
operator ordering ambiguity, which is an obstacle for the naive
quantization of classical theories.
This ambiguity is explained by remembering that classical mechanics is
the limit hbar to zero of quantum mechanics. Thus every operator
expression G(x,p,hbar) becomes classically G(x,p,0), i.e., one loses
one degree of freedom. That one can determine the quantum theory from
the classical limit is therefor as impossible as reconstructing a
function of a single varible from its value at t=0.
On the other hand, different quantum mechanical orderings of the same
classical expression differ only by a term O(hbar) that can be found
by performing the reordering with the help of the CCR. Thus the
difference disappears in the classical limit.