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Tunnelling asymptotic limits: the smooth-barrier semiclassical limit and the exact rectangular prefactor comparison

Physical scope

Two asymptotic statements about declared one-dimensional barrier transmission data: the semiclassical limit of the smooth compactly supported barrier of the WKB regime model, in which the ratio of the declared transmission coefficient to exp(-2S/h) tends to one; and the exact thick-barrier limit of the rectangular wall model, whose profile-dependent limit prefactor is 16E(U-E)/U^2 and can differ from one, showing that the exponential law alone does not determine the prefactor and that rectangular walls are outside the smooth-barrier theorem. No measured transmission, detector, sample or apparatus is asserted.

Example

Adopt the declared one-dimensional line model The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions and the WKB regime model The declared WKB regime model: nondimensionalised line equation, turning points, phase integrals and the declared barrier transmission reading with their framework postulates (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), with units as in Dimensions, SI unit conventions and quantity types of the framework. Then:

(a) Smooth-barrier semiclassical limit. Let $V, m, ℏ, ℓ, E_{⋆}, E, W, λ, h$ be data of the WKB regime model satisfying its hypothesis (W2), with exponent $S = \int_{ξ_{a}}^{ξ_{b}} \sqrt{W - λ} d ξ$ and declared transmission coefficient $T (h)$ . At fixed dimensionless barrier data $(W, λ)$ , there are $h_{0} > 0$ , $0 \leq ε (h) \leq C_{1} h$ for $0 < h < h_{0}$ , and $0 \leq C_{0} < \infty$ , depending only on the fixed barrier data and matching sections, such that $∣ T (h) e^{2 S / h} - 1 ∣ \leq \frac{C_{0} [ε (h) + e^{- 2 S / h}]}{1 - C_{0} [ε (h) + e^{- 2 S / h}]}$ for $0 < h < h_{0}$ whenever the denominator is positive; in particular $\lim_{h ↓ 0} T (h) e^{2 S / h} = 1,$ and the physical reading of the limit is the exponential law $T \approx \exp (- \frac{2}{ℏ} \int_{x_{a}}^{x_{b}} \sqrt{2 m (V - E)} d x)$ with prefactor tending to one;

(b) Rectangular thick-barrier limit. Fix $m, ℏ, E, U$ with $U > E > 0$ and vary the wall width $a \to \infty$ (equivalently, $κ a \to \infty$ ). For the rectangular wall model excluded by the smooth theorem, take the exact scattering solution of the line model with $V (x) = U$ for $0 < x < a$ and $V = 0$ otherwise, $U > E > 0$ , $k = \sqrt{2 m E} / ℏ$ , $κ = \sqrt{2 m (U - E)} / ℏ$ and exact transmission $T_{r e c t} = [1 + \frac{U^{2} \sinh^{2} (κ a)}{4 E (U - E)}]^{- 1};$ then, in this fixed-parameter limit, $\frac{T_{r e c t}}{e^{- 2 κ a}} ⟶ \frac{16 E (U - E)}{U^{2}} = \frac{16 k^{2} κ^{2}}{(k^{2} + κ^{2})^{2}},$ so in the rectangular model the exponential law holds with the profile-dependent prefactor $16 E (U - E) / U^{2}$ ; the exponent $2 κ a$ equals the classical WKB exponent $2 B / ℏ$ of the rectangular barrier, and the limit shows that the exponential alone does not determine the prefactor;

(c) Reading of the comparison. Statement (a) is a theorem about smooth compactly supported barriers with separated simple turning points, and its prefactor-one conclusion uses the Airy matching of that class; statement (b) is an exact model computation for a piecewise constant wall, whose sharp discontinuities are precisely why it is excluded from (a). Neither statement asserts a measured transmission, a detector response, a wave packet or an apparatus; both are conditional statements about the declared one-dimensional models.

Verification

Given: the WKB regime model with hypothesis (W2) and its declared transmission coefficient; the rectangular wall data $a > 0$ , $U > E > 0$ and constants $k, κ$ ; the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited suppliers.

[L1] Smooth-barrier theorem. For every smooth compactly supported $C^{5}$ barrier with exactly two separated simple turning points, the there exist $h_{0} > 0$ and fixed constants such that the declared transmission coefficient obeys the quantitative bound displayed in part (a) for $0 < h < h_{0}$ with positive denominator, and the limit $\lim_{h ↓ 0} T (h) e^{2 S / h} = 1$ (Separated smooth-barrier tunnelling: the declared transmission coefficient is exp(-2S/h) up to explicit small-h and barrier-thickness errors).

[L2] Exact rectangular transmission. For the piecewise constant wall $V = U$ on $(0, a)$ and $V = 0$ otherwise with $U > E > 0$ , the declared scattering solution has the incident reference amplitude $A_{i n} = 1 m^{- 1 / 2}$ and the unique transmission asymptotics $ψ = A_{i n} t e^{i k x}$ for $x > a$ , where $t$ is the dimensionless amplitude ratio $t = 2 e^{- i k a} / D$ with $D = 2 \cosh (κ a) + i (κ / k - k / κ) \sinh (κ a)$ , and $T_{r e c t} = ∣ t ∣^{2} = [1 + U^{2} \sinh^{2} (κ a) / (4 E (U - E))]^{- 1}$ (Rectangular barrier: scattering solution, flux balance and transmission clauses (a), (c), The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions).

[L3] Elementary limits. With $m, ℏ, E, U$ fixed and $a \to \infty$ , $κ = \sqrt{2 m (U - E)} / ℏ$ is fixed and $κ a \to \infty$ . $\sinh (x) = (e^{x} - e^{- x}) / 2$ , so $\sinh^{2} (κ a) e^{- 2 κ a} \to 1 / 4$ ; $E = ℏ^{2} k^{2} / (2 m)$ and $U - E = ℏ^{2} κ^{2} / (2 m)$ , so $16 E (U - E) / U^{2} = 16 k^{2} κ^{2} / (k^{2} + κ^{2})^{2}$ .

Proof technique: direct; read the smooth-barrier limit from the page's theorem and compute the rectangular thick-barrier limit from the exact closed form by elementary asymptotics.

Given: the data of the Example and [L1]-[L3].

1.1L1givenalgebra

[Smooth limit] Applying [L1] at fixed $(W, λ)$ for $0 < h < h_{0}$ with positive denominator gives the quantitative bound and, because the right-hand side tends to $0$ as $h ↓ 0$ at fixed barrier data, the limit $\lim_{h ↓ 0} T (h) e^{2 S / h} = 1$ . The physical exponent identity $S / h = (1 / ℏ) \int_{x_{a}}^{x_{b}} \sqrt{2 m (V - E)} d x$ is the endpoint-extended identity of clause (c) of the WKB regime model: the scaled-line lemma applies on closed subintervals strictly between the turning points, and continuity extends the integrals to their endpoints. This is (a).

1.2L2L3algebra

[Rectangular limit] Put $A : = U^{2} / (4 E (U - E)) > 0$ and $t : = κ a$ . By [L2], $T_{r e c t} = [1 + A \sinh^{2} t]^{- 1}$ , hence $T_{r e c t} e^{2 t} = [e^{- 2 t} + A \sinh^{2} (t) e^{- 2 t}]^{- 1}$ . With $m, ℏ, E, U$ fixed and $a \to \infty$ , [L3] gives the denominator limit $A / 4 > 0$ , so $T_{r e c t} e^{2 t} \to 4 / A = 16 E (U - E) / U^{2}$ . Substituting $E = ℏ^{2} k^{2} / (2 m)$ and $U - E = ℏ^{2} κ^{2} / (2 m)$ gives $16 E (U - E) / U^{2} = 16 k^{2} κ^{2} / (k^{2} + κ^{2})^{2}$ , as asserted in (b).

2.1step 1.1step 1.2L1L2givenalgebra∎

[Exponent comparison and scope] For the rectangular wall the classical WKB exponent is $\frac{2}{ℏ} \int_{0}^{a} \sqrt{2 m (U - E)} d x = 2 κ a$ , which equals the exponent of the limit in step 1.2, so the two models share the exponential law but can differ in the prefactor; the rectangular prefactor $16 y (1 - y)$ , with $y = E / U \in (0, 1)$ , equals one exactly when $y = (2 \pm \sqrt{3}) / 4$ . Statement (a) is restricted to the smooth class by its hypotheses, and statement (b) is the exact wall computation in the fixed-parameter thick-wall limit. No measured transmission, detector, packet or apparatus entered either computation. This is (c).

Remarks

Why the prefactors can differ. For the smooth barrier the Airy matching across the two simple turning points produces the transmitted amplitude with prefactor $1 + O (ε)$ ; the wall's discontinuities replace the Airy matching by exact algebraic matching, whose limit carries the profile-dependent factor $16 E (U - E) / U^{2}$ . Writing $y = E / U \in (0, 1)$ , this factor equals one when $y = (2 \pm \sqrt{3}) / 4$ and differs from one for other admissible ratios; the example compares model limits and claims no measured value.
Regime of validity. The limit in (a) is taken in the model's semiclassical parameter $h$ at fixed barrier data; the limit in (b) is taken in the wall thickness $κ a$ at fixed $E, U$ . Both are asymptotic statements about the declared one-dimensional models.

Tunnelling asymptotic limits: the smooth-barrier semiclassical limit and the exact rectangular prefactor comparison

Example

Verification

Remarks

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