Velocity of a Quantum Particle in a Classically Forbidden Region

Quantum computing researchers grappling with how particles behave in classically forbidden regions — a phenomenon central to quantum tunneling and, by extension, to quantum computing hardware like tunnel junctions and Josephson junctions — now have a clearer theoretical picture thanks to a rigorous new analysis by Christian Beck, Sheldon Goldstein, Dustin Lazarovici, Roderich Tumulka, and Nino Zanghi. The team resolves a reported experimental challenge to Bohmian mechanics, showing the challenge dissolves once two specific errors in the original analysis are corrected. The result reinforces the consistency between Bohmian mechanics and standard quantum mechanics in the regime most relevant to quantum computing tunneling devices.

What Problem Does This Paper Solve?

A 2025 experiment by Sharoglazova et al., published in Nature, appeared to show that measured particle speeds in a classically forbidden region contradicted the predictions of Bohmian mechanics. Beck and colleagues demonstrate that both speed estimates in that experiment contain errors, and that correcting those errors brings the results into exact agreement with Bohmian mechanics — eliminating the apparent conflict with standard quantum mechanics entirely.

The original experiment was elegant: it inferred the speed of quantum particles in the classically forbidden region of a potential step — a zone where classical physics forbids particle motion entirely — by studying particle populations in coupled waveguides. Because quantum mechanics allows particles to penetrate forbidden regions through tunneling, understanding how fast they travel there matters directly for quantum computing, where tunnel junctions underpin qubit designs including superconducting qubits based on Josephson junctions. When the inferred speeds disagreed sharply with Bohmian predictions, it raised the question of whether this well-established interpretational framework was experimentally falsifiable in a new way.

How Does the New Analysis Work?

Beck and colleagues perform a detailed theoretical dissection of the experimental setup, identifying two distinct errors. The first involves an unwarranted assumption about inter-waveguide tunneling time; the second involves a misapplication of the Büttiker dwell time formula. Correcting both restores full agreement between Bohmian predictions and experimental inference.

The first error concerns the assumption that the tunneling time between the two waveguides is determined solely by the transverse coupling between them, independent of any entanglement with the longitudinal degree of freedom. In the evanescent regime — the specific physical regime where the classically forbidden region exists — both standard quantum mechanics and Bohmian mechanics predict that this assumption fails. Entanglement between transverse and longitudinal motion cannot be ignored there. The analogy is instructive: assuming the coupling time is unaffected by longitudinal entanglement is like timing a swimmer's lane-to-lane transfer while ignoring that a current is simultaneously sweeping them downstream — the two motions are coupled and cannot be separated.

The second error involves the Büttiker dwell time, a formula that quantifies how long a particle spends in a defined spatial region. The authors of the original experiment applied this formula in a configuration that did not match the formula's underlying assumptions. Beck and colleagues show that a correct application yields predictions that agree exactly with Bohmian mechanics, turning an apparent falsification into a confirmation.

Why Does This Matter for Quantum Computing?

Quantum tunneling is not an abstract curiosity for quantum computing — it is load-bearing infrastructure. Superconducting qubits, the leading quantum computing hardware platform, operate through Josephson junctions where Cooper pairs tunnel through a thin insulating barrier. The dwell time and traversal speed of particles in forbidden regions directly influence junction characteristics, decoherence rates, and ultimately qubit fidelity.

The consistency of Bohmian mechanics with standard quantum mechanics in this regime matters because Bohmian mechanics provides explicit particle trajectories — a calculational and conceptual tool that researchers use to build intuition about tunneling dynamics that the standard wavefunction-only picture obscures. When Bohmian trajectories and standard quantum predictions agree, as they do here, practitioners can use whichever framework is more computationally tractable for a given problem. The explicit Bohmian trajectory calculations provided by Beck and colleagues for the two-dimensional waveguide model offer a concrete computational resource for researchers modeling tunneling in quantum computing device geometries.

What Are the Limitations and Open Questions?

The analysis by Beck and colleagues is theoretical and interpretational rather than experimental. They do not propose a new experimental protocol for directly measuring particle velocity in forbidden regions; they demonstrate that the existing protocol was misanalyzed. The fundamental question — what speed, if any, should be assigned to a quantum particle in a classically forbidden region — remains a conceptually open problem in quantum foundations, with different interpretational frameworks offering different answers.

The paper also does not address whether a correctly designed experiment could distinguish Bohmian predictions from those of other quantum interpretations in this specific regime. That remains an active area of debate. For quantum computing developers, the practical implication is that dwell time calculations for tunnel junctions need to properly account for entanglement between degrees of freedom in the evanescent regime — a subtlety that simplified models can miss.

What Comes Next?

The explicit Bohmian trajectory and dwell time calculations presented for the two-dimensional waveguide model provide a template that the quantum computing hardware community can adapt to model particle dynamics in realistic Josephson junction geometries, where tunneling physics governs device performance at the most fundamental level.

As quantum computing devices push toward lower noise and higher fidelity, the theoretical tools for understanding quantum tunneling at this level of precision will become increasingly important to hardware design.