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The declared single-channel resonance: Lorentzian factor, half-height points and resonance-value cross section

Physical scope

The declared single-channel resonance model with l=0, E_r>0 and arbitrary Gamma>0: its resonance-centre cross section and positive-energy ratio, and the auxiliary full-real-axis Lorentzian half-height points and width. Each half-height point has a physical reading only if positive; no Gamma<2E_r restriction is imposed on the model. Conditional algebra only; no producing potential, decay law or measured width or count is asserted.

Example

Setup. Adopt the declared single-channel resonance model The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading with channel label $l = 0$ , mass $m > 0$ , action constant $ℏ > 0$ and parameters $E_{r} > 0$ , $Γ > 0$ ; read the phase shift and channel cross section by The resonance model's phase-shift law, unitarity and channel cross section clauses (a)-(c). Put $k_{r} : = k (E_{r}) = ℏ^{- 1} \sqrt{2 m E_{r}}$ and let $L (E)$ on $R$ denote the auxiliary Lorentzian factor of that theorem.

Verification

Given: the declared model data and the statements of The resonance model's phase-shift law, unitarity and channel cross section.

1.1givenalgebra

Since $E_{r} > 0$ , the theorem can be evaluated at the physical energy $E_{r}$ , giving $\sin^{2} δ_{0} (E_{r}) = L (E_{r}) = 1$ and $σ_{0} (E_{r}) = 4 π / k_{r}^{2} = 4 π ℏ^{2} / (2 m E_{r})$ in $m^{2}$ . The real-axis factor has $L (E_{r} \pm Γ / 2) = 1 / 2$ . The upper point is always positive; the lower point is positive exactly if $Γ < 2 E_{r}$ , equals the excluded boundary $0$ if $Γ = 2 E_{r}$ , and is negative if $Γ > 2 E_{r}$ . Only positive points have physical phase and cross-section readings.

2.1step 1.1givenalgebra

For every physical $E > 0$ , with $x : = E - E_{r}$ and $y : = Γ / 2 > 0$ , dividing by the positive resonance-centre value yields $σ_{0} (E) / σ_{0} (E_{r}) = (k_{r} / k (E))^{2} y^{2} / (x^{2} + y^{2}) = (E_{r} / E) L (E)$ . At the always-positive upper point the ratio is $E_{r} / (2 (E_{r} + y))$ ; at the lower point it is $E_{r} / (2 (E_{r} - y))$ only when $E_{r} - y > 0$ . If that lower point is zero or negative, the real-axis identity $L (E_{r} - y) = 1 / 2$ is still valid but no physical ratio is evaluated there. The factor $E_{r} / E$ is energy dependent and need not vary slowly for arbitrary $Γ$ .

3.1step 2.1algebra

Take the declared parameters $E_{r} = 4 Γ$ (both in joules). Then at $E = E_{r} + Γ / 2 = (9 / 2) Γ$ the Lorentzian factor is $1 / 2$ and the ratio of step 2.1 is $(4 / (9 / 2)) (1 / 2) = 4 / 9$ ; at $E = (7 / 2) Γ$ the factor is again $1 / 2$ and the ratio is $(4 / (7 / 2)) (1 / 2) = 4 / 7$ . The first value is strictly below $1 / 2$ and the second strictly above, so this is the explicit asymmetry produced by the $1 / E$ factor, so these factor half-height locations are not the locations where $σ_{0} (E) = σ_{0} (E_{r}) / 2$ ; no maximum of $σ_{0}$ is used in this comparison.

4.1givenstep 1.1step 2.1step 3.1algebra∎

On the full real axis, $L$ reaches $1$ at $E_{r}$ and $1 / 2$ exactly at $E_{r} \pm Γ / 2$ , so its full width at half maximum is $Γ$ . On the positive physical axis both half-height locations are present exactly when $Γ < 2 E_{r}$ ; for $Γ \geq 2 E_{r}$ the lower location is excluded. This qualifies the physical interpretation without narrowing the allowed $E_{r} > 0$ , $Γ > 0$ parameters or changing the real-axis width identity. The example of step 3.1 keeps both half-height points positive and shows that the cross-section ratios there differ from $1 / 2$ .

Remarks

No decay claim. The width parameter $Γ$ is an energy parameter of the declared factor; the pair does not identify $ℏ / Γ$ with a lifetime and does not derive an exponential decay law.
No maximum claim. Because of the factor $1 / E$ in the ratio computed in step 2.1, the pair makes no statement that $σ_{0}$ attains its maximum at $E_{r}$ ; only the auxiliary Lorentzian factor's full-real-axis half-height width is asserted, with positive-energy locations qualified above.

Depends on

Dependency tree · 2 levels

Sources

J. Binney and D. Skinner, The Physics of Quantum Mechanics, Section 12.6.1, printed page 280 (PDF page 288)