Manuscript Em03-F@1 · Em03-F-v1.tex · proposition · 0d4d148b

Proposition: Calculated packet displacement

Outgoing scalar packet identification and normality · lines 863–888 · proof included or linked

Em03-F@1#prop:displacement, Em03-F@1#eq:alpha

The conditional outgoing packet amplitude is

α_{s, h} = - i s g c_{h} .

Consequently,

ω_{s} (Q_{h}) = 2 Re α_{s, h}, ω_{s} (P_{h}) = 2 Im α_{s, h} .

These are calculated from the bulk profile and scattering map.

Proof. The Weyl displacement identity gives

W (r)^{†} a (v) W (r) = a (v) + i ⟨ v, r ⟩ .

Insert $r = - s g k_{F}$ . The initial state is centered, so its conditional mean is (eq:alpha). Taking the quadrature combinations proves the result.

Displayed formulas in this passage

Em03-F@1#eq:alpha · line 866
$α_{s, h} = - i s g c_{h} .$
unlabeled · line 873
$ω_{s} (Q_{h}) = 2 Re α_{s, h}, ω_{s} (P_{h}) = 2 Im α_{s, h} .$
unlabeled · line 882
$W (r)^{†} a (v) W (r) = a (v) + i ⟨ v, r ⟩ .$

Nearby in this section

section: Outgoing scalar packet identification and normality