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# Appendix: Notational Conventions

Two conventions for this HTML version:  sq2 = sqrt(2), and we use w for omega, the frequency or energy.

Plane wave with momentum k and energy w: eikx-iw t.

 Metric d tau2 = dt2 - dx2 - dy2 - dz2 Momentum p = -i d/dx Position q = ``multiply by x'' Hamiltonian H = i d/dt Annihilator a = (q+ip)/sq2 Creator a* = (q-ip)/sq2 q = (a+a*)/sq2 p = (a-a*)/(i sq2) commutator [A,B] = AB-BA dA/dt = i[H,A] dA/dx = -i[p,A] product rule [s,AB] = [s,A]B + A[s,B]
a*a is also called the number operator, sometimes denoted N.

Coherent states, take 1:

• Coh1(c+ib) = eibq e-icp |0>
• Coh1(c+ib) = iota e-(c2+b2)/2 SUMn ((c+ib)n/ sqrt(n!)) |n>
• Coh1(c+ib) = K exp(ibx - (x-c)2)

where |iota|=1, the last equation gives Coh1 as a complex wavefunction, and K is a normalization factor.

Coherent states, final version:

• Coh(c+ib) = e-icp+ibq |0>
• Coh(c+ib) = e-(c2+b2)/2 SUMn ((c+ib)n/ sqrt(n!)) |n>
• Coh(c+ib) = K exp(ibx - (x-c)2)

If z = c+ib, then e-iHt Coh(z) = e-it/2 Coh(ze-it). Here e-it/2 represents the ``vacuum energy''.

For the full-blown version of the Baker-Campbell-Hausdorff formula, see the postscript version of these note.

Baker-Campbell-Hausdorff formula, special case: if [A,B] commutes with both A and B, then:

eA+B = eAeB e-[A,B]/2   Next:  Tensor product and direct sum Up: Schmotons Previous: John Baez: A lower

Michael Weiss

3/10/1998