John Baez: Michael has  Schmotons  Michael Weiss: Now for

John Baez: My lysdexia has a variety of origins

[You, dear reader, have been shielded by your valiant editor from the sign errors and inconsistent conventions of the original newgroup thread. At least I hope I've caught them all! But just to make your editor's life difficult, John Baez here uses these very errors as a jumping off point for an informative discussion.--ed.]

My lysdexia above has a variety of origins. Indeed, this subject is littered with banana peels on which the unwary can slip, so it is probably pedagogically useful to list them.

The main reason for my slip, writing e^ipc instead of e^-ipc, was the usual symplectic switcheroo: the momentum operator p generates translations in position space, while the position operator q generates translations in momentum space. More precisely, p generates translations to the right in position space, while q generates translations to the left in momentum space. This is built into the classical Poisson brackets:

{p,q} = 1,     {q,p} = -1

and thus in the commutators of the corresponding operators:

[p,q] = -i,     [q,p] = i.

where an extra factor of -i is traditionally thrown in to further confuse the uninitiated.

Thus in the state e^{-icp + ibq} |0>, the expectation value of position is c and the expectation value of momentum is b.

Second, there is a somewhat arbitrary convention about whether we think of a point in phase space with momentum b and position c and as being the point c+ib in the complex plane, or b+ic. Time evolution for the harmonic oscillator amounts to having our complex plane rotate as time passes, and if we use c+ib the plane will rotate clockwise, while if we use b+ic the plane will rotate counterclockwise. The former means that after a time t, the point c+ib will evolve to

e^-it (c+ib),

while the latter means that after a time t, the point b+ic will evolve to

e^it (b+ic)

The latter seems nicer to me, which is another reason for my slip, but we seem to be working with the former convention.

3/10/1998