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arxiv: 2605.04139 · v1 · submitted 2026-05-05 · 🪐 quant-ph · hep-th

Recognition: 3 theorem links

· Lean Theorem

Tunneling from an oscillating initial state in quantum mechanics

Oliver Janssen , Matthew Kleban , Cameron Norton
Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:25 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords quantum tunnelingmetastable potential wellresonant statessemiclassical approximationprobability currenttime-dependent decay rateSchrödinger equationoscillating initial state
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The pith

A closed-form expression gives the tunneling probability current from general and oscillating initial states as a sum over resonant states computed to first subleading semiclassical order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an analytical formula for the probability current that tunnels through a barrier when an initial state decomposes into resonant states of a metastable well. All parts of the formula, including complex energies and wavefunctions, are obtained analytically to first subleading order in the semiclassical limit. For a coherently oscillating initial state the formula produces an explicit approximation to the time-dependent decay rate. Direct comparison with numerical integration of the Schrödinger equation confirms that the approximation remains accurate over the relevant times.

Core claim

We study the decay of general initial states out of a metastable potential well in quantum mechanics. We provide a closed-form expression for the probability current that tunnels through the barrier in terms of the resonant states into which the initial state can be decomposed. All ingredients in the equation are computed analytically to first subleading order in the semiclassical limit. Specializing to a coherently-oscillating initial state, we derive an approximation to the time-dependent decay rate and demonstrate its accuracy by comparing it to a numerical solution of the Schrödinger equation.

What carries the argument

Decomposition of the initial state into resonant states whose complex energies and wavefunctions are computed to first subleading semiclassical order, yielding a closed-form expression for the tunneling probability current.

If this is right

  • An explicit time-dependent decay rate follows for any coherently oscillating initial state.
  • The tunneling current is expressed solely in terms of analytically available resonant-state data.
  • The same expression applies to arbitrary initial states that admit a resonant decomposition.
  • Accuracy holds to first subleading semiclassical order without requiring full numerical time evolution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The resonant decomposition approach may extend to slowly varying time-dependent barriers outside the static metastable-well setting.
  • Preparing particular oscillating states could allow experimental tuning of effective decay rates in quantum devices.
  • Similar current expressions might appear in multi-dimensional or many-body tunneling problems once resonant modes are identified.

Load-bearing premise

The initial state admits a decomposition into resonant states whose complex energies and wavefunctions can be computed to the required semiclassical accuracy and whose higher-order corrections remain negligible for the times of interest.

What would settle it

Numerical solution of the time-dependent Schrödinger equation for a chosen oscillating initial state in a specific metastable potential, checked against the analytical current formula at times where the semiclassical expansion should be valid.

Figures

Figures reproduced from arXiv: 2605.04139 by Cameron Norton, Matthew Kleban, Oliver Janssen.

Figure 1
Figure 1. Figure 1: A generic potential V assumed in our analysis. A well (left) is shown separated from an asymptotically free region (right) by a potential barrier. The dashed line shows a representative energy E. The classical turning points for this energy are c and a on the left and right in the well, and b on the right of the barrier. The start of the true vacuum, or free region, is denoted by xT . with energy well belo… view at source ↗
Figure 2
Figure 2. Figure 2: The first 12 resonant states as found from the CAP (Complex Absorbing Potential) view at source ↗
Figure 3
Figure 3. Figure 3: The tunneling probability, 1 − P(t) (left) and probability current j(t) (right) comparing the time evolution of a coherent harmonic oscillator initial state with |α| = 1.1 via either CAP (Complex Absorbing Potential) evolution (black) or evolution with hard-wall boundary conditions (blue). For early-enough times, before the outgoing wave reaches the boundary, the plots are nearly indistinguishable. is near… view at source ↗
Figure 4
Figure 4. Figure 4: The tunneling probability 1 − P(t) (top left) and probability current j(t) (top right) as functions of time, for a coherent simple harmonic oscillator initial state in the potential well with |α| = 1.1. The black solid lines show the direct numerical result from Crank-Nicolson time evolution with the CAP (Complex Absorbing Potential) Hamiltonian, the gray dashed lines show the formula of Eq. (18) evaluated… view at source ↗
Figure 5
Figure 5. Figure 5: Numerical verification of Eqns. (33) and (42) for a coherent initial state with |α| = 1.1. Left: cumulative tunneling probability 1−P(t) for the first few oscillations obtained by integrating the numerical current (blue) and the saddle-point approximation to the current from Eq. (33) (dashed black). The horizontal red lines mark the total ∆P per oscillation cycle predicted by Eq. (42). Right: the probabili… view at source ↗
Figure 6
Figure 6. Figure 6: The cumulative tunneling probability 1 − P(t) (left) and probability current j(t) (right) for a coherent initial state with |α| = 1.1 (black) and a random-phase initial state (blue), both with the same magnitude of the expansion coefficients |cn|. The red dotted line shows the average decay e −Γ¯t with Γ =¯ P n |cn| 2Γn. 13 view at source ↗
Figure 7
Figure 7. Figure 7: The tunneling probability 1 − P(t) (top left) and probability current j(t) (top right) as functions of time, for a coherent initial state with |α| = 2. Line styles are as in view at source ↗
read the original abstract

We study the decay of general initial states out of a metastable potential well in quantum mechanics. We provide a closed-form expression for the probability current that tunnels through the barrier in terms of the resonant states into which the initial state can be decomposed. All ingredients in the equation are computed analytically to first subleading order in the semiclassical limit. Specializing to a coherently-oscillating initial state, we derive an approximation to the time-dependent decay rate and demonstrate its accuracy by comparing it to a numerical solution of the Schr\"odinger equation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript derives a closed-form expression for the probability current tunneling through a barrier from a general initial state in a metastable well, expressed via decomposition into resonant (Siegert) states with all quantities evaluated analytically to first subleading semiclassical order. Specializing to a coherently oscillating initial state, it obtains an approximation for the time-dependent decay rate and validates it by direct numerical comparison to the time-dependent Schrödinger equation.

Significance. If the central result holds, the work supplies an analytical tool for time-dependent tunneling from non-stationary states, extending semiclassical methods beyond stationary decay. Strengths include the parameter-free analytical expressions to subleading order and the concrete numerical validation for the oscillating case, which together support the utility of the resonant-state approach.

major comments (1)
  1. [Specialization to coherently-oscillating initial state] The section specializing to the coherently-oscillating initial state: the accuracy claim for the derived time-dependent decay rate rests on the resonant-state decomposition of the initial wave function being sufficiently complete and accurate inside the well. The manuscript does not report an explicit verification that the truncated sum over resonances reproduces the initial oscillating state to the same first-subleading semiclassical accuracy used for the complex energies and wave functions. Without this check, the error budget for the current expression remains uncontrolled, particularly given that resonant states are not square-integrable and completeness holds only in the rigged-Hilbert-space sense.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive overall assessment. We address the single major comment below and will revise the manuscript to incorporate the requested verification.

read point-by-point responses

  1. Referee: The section specializing to the coherently-oscillating initial state: the accuracy claim for the derived time-dependent decay rate rests on the resonant-state decomposition of the initial wave function being sufficiently complete and accurate inside the well. The manuscript does not report an explicit verification that the truncated sum over resonances reproduces the initial oscillating state to the same first-subleading semiclassical accuracy used for the complex energies and wave functions. Without this check, the error budget for the current expression remains uncontrolled, particularly given that resonant states are not square-integrable and completeness holds only in the rigged-Hilbert-space sense.

    Authors: We agree that an explicit verification of how accurately the truncated resonant-state sum reproduces the initial oscillating wave function inside the well would strengthen the error analysis. While the direct numerical comparison of the resulting tunneling current to the time-dependent Schrödinger equation already provides indirect support for the decomposition's adequacy, we will add in the revised manuscript a dedicated check: a plot and quantitative error measure (e.g., L2 difference inside the well) comparing the truncated sum (using the identical number of resonances employed for the decay-rate formula) against the exact initial state. This will be performed at the same semiclassical order used for the energies and wave functions, thereby controlling the error budget more explicitly. We note that the rigged-Hilbert-space framework is the standard setting in which the resonant-state expansion is known to be complete, and the added numerical test will demonstrate its practical convergence for the quantities of interest. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from TDSE and resonant expansion

full rationale

The paper derives the closed-form probability current directly from the time-dependent Schrödinger equation via resonant-state (Siegert) expansion, with all quantities evaluated analytically to first subleading semiclassical order. The specialization to the coherently oscillating initial state yields an approximate time-dependent decay rate that is then compared to an independent numerical solution of the Schrödinger equation. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz smuggled from prior work; the numerical benchmark supplies an external check outside the analytic expressions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on standard quantum-mechanical axioms plus the semiclassical expansion; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Any initial state in the well can be expanded in the resonant-state basis
    Invoked to write the tunneling current as a sum over resonant contributions.
  • domain assumption Semiclassical expansion to first subleading order suffices for the resonant energies and wavefunctions
    Used to obtain closed-form analytic expressions for all ingredients.

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Reference graph

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