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Equation 85 · The Entry a Relabeling Cannot Write

What does this equation mean?

∣ψ(0)⟩=∣e,g⟩|\psi(0)\rangle = |e,g\rangle

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Inputs and operations|e,grangle
Result or condition|psi(0)rangle
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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ψ\psi

Symbol psi

psi is part of the quantity the equation computes from the expression on the right.

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ee

Symbol e

e is an input to the expression that computes the quantity on the left.

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gg

Symbol g

g is an input to the expression that computes the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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What the article says around this equation

Because HSBH_{SB} conserves total excitation number, the dynamics starting from a single excitation stays confined to the two-dimensional subspace spanned by |e,g⟩\rangle (system excited, bath in its ground state) and |g,e⟩\rangle (the reverse). On resonance, both basis states carry the same bare energy, +ℏ\hbarω\omega/2 - ℏ\hbarω\omega/2 = 0 , so HfH_f restricted to this subspace is proportional to the swap operator between the two, and the Schrödinger equation solves in closed form. Starting from |ψ(0)\psi(0)⟩\rangle = |e,g⟩\rangle ,

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