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In this page, I write exact solutions of Energy Representation of Uda Equation (1).
Any unentangled solution is functional Φ such that
where ∀n ∈ Z; cn ∈ C.
The quantum history represented by this solution is the same history as represented by wave function ψ such that
ψ(n, t) ＝ exp(-iωtn + cn) ＝ λn exp(-iωnt)
in the old grammar.
Here, λn ≡ exp(cn).
If A†A|n＞ ＝ n|n＞, Σn=0∞ ψ(n, t)|n＞ ＝ Σn=0∞ λnexp(-iωnt)|n＞, Σn=0∞ ψ(n, 0)|n＞ ＝ Σn=0∞ λn|n＞.
Of course, any superposition of unentangled solutions is also a solution.
Moreover, any entangled solution perhaps is a superposition of unentangled solutions.
It is functional Φ such that
where ∀n ∈ Z; an ∈ R and bn ∈ R and f(a0, b0, a1, b1, a2, b2, ・・・) ∈ C.
cn = an + ibn, λn = exp(an + ibn).
Normalization condition for initial state is that
To let the integral converge, b and ν is restricted.
However, I do not put such a condition because there is no ambiguity even if the integral diverges.
The content of this page was developed in Number Representation of Uda Equation @ Quantum History Theory @ Problems.
|Author Yuichi Uda, Write start at 2019/11/08/17:22JST, Last edit at 2019/11/22/18:45JST|