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Exact two-level transition versus its first coefficient

Physical scope

One exactly constructed two-level finite Hamiltonian and its explicitly supplied projective measurement. The first-order squared coefficient is not called a probability, and no continuum decay or experimental data is claimed.

Example

Under the explicit pure-state preparation and unitary-evolution assumption of Exact regulated transition probabilities and controlled coefficients, take $H = C^{2}$ , $H_{0} = 0$ , $V (t) = g_{E} σ_{x}$ with $g_{E} > 0$ joules, dimensionless $λ = 1$ , initial vector $e_{1} = (1, 0)$ , and the two-outcome projective measurement $P = ∣ e_{2} ⟩ ⟨ e_{2} ∣$ , $I - P$ , where $e_{2} = (0, 1)$ . At $τ = π ℏ / (2 g_{E})$ seconds the exact model transition probability is $1$ , whereas the first-order squared transition coefficient is $π^{2} / 4 > 1$ . The latter is a coefficient-derived square, not a probability. This is an exact constructed-model calculation, not a simulated or reported experiment.

Facts & Assumptions

Given: ${A C}_{ω}$ , $g_{E} > 0$ in joules, $ℏ > 0$ in $J s$ , the two-level Hamiltonian and projective measurement in the Example, and the conditional theorem of Exact regulated transition probabilities and controlled coefficients.

[F1]

Exact regulated transition probabilities and controlled coefficients: under its stated pure-state preparation $ρ_{0} = T_{e_{1}, e_{1}}$ and model update by the constructed unitary $U (τ)$ , for $H = g_{E} σ_{x}$ and $z = g_{E} τ / ℏ$ the exact transition probability is $\sin^{2} z$ and the first-order squared coefficient is $z^{2}$ ; the coefficient-derived square need not be a probability.

[F2]

QFT example units, metric, gauge, and pairing conventions: $g_{E}$ has energy unit joule and $ℏ$ has unit $J s$ , so $g_{E} τ / ℏ$ is dimensionless when $τ$ is in seconds.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume only ${A C}_{ω}$ , as required by the conditional supplier theorem.

Verification

1.1givenF1F2A1algebra

At the selected time, $z = g_{E} τ / ℏ = π / 2$ by the definition of $τ$ [F2]. The exact transition probability from [F1] is therefore $\sin^{2} (π / 2) = 1$ , while the first-order squared coefficient is $(π / 2)^{2} = π^{2} / 4$ .

2.1step 1.1F1algebra

Since $π > 2$ , $π^{2} / 4 > 1$ . Thus this squared coefficient cannot be a probability for the specified outcome, while the exact model probability remains $1$ under the stipulated constructed model and measurement.

3.1givenF1step 1.1algebra∎

The theorem's quartic error bound specializes at this time to $∣ 1 - π^{2} / 4 ∣ \leq π^{4} / 48$ . It is a valid absolute bound, but it exceeds $1$ since $π > 3$ ; it gives no sub-unit error control at this time. No empirical uncertainty or continuum statement is inferred.

Depends on

Dependency tree · 2 levels

Sources

David Tong, Quantum Field Theory, Chapter 3 “Interacting Fields,” §3.1.1 “Dyson’s Formula”