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Equation 97 · Part 2 · The Entry a Relabeling Cannot Write

Symbol t^ast

ΔEΛ(t∗)=+ℏω/2\Delta E_\Lambda(t^\ast) = +\hbar\omega/2
t∗t^\ast

What this part means

tat^ast is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Its job in the formula

tat^ast is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

The passage around this formula

At t=0 this equals -ℏ\hbarω\omega/2 , exactly the bath’s own ground-state energy, matching the uncorrelated-bath limit derived above. At the swap time t∗t^\ast = π\pi/(2g) , the state is |g,e⟩\rangle up to an overall phase — the excitation has moved entirely into the bath — and Δ\Delta EΛ(t∗)E_\Lambda(t^\ast) = +ℏ\hbarω\omega/2 , exactly the bath’s excited-state energy with ⟨\langle HSBH_{SB}⟩\rangle = 0 at that instant, matching the closed form by direct…

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An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the article section

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