Pausing in the Void: New Insights into Quantum Tunneling

Author: Denis Avetisyan

A new analysis of particle dynamics reveals that quantum tunneling isn’t the instantaneous process previously thought, as particles can momentarily stall within potential barriers.

Particle dynamics reveal that even when encountering a potential barrier exceeding its energy, a particle exhibits a finite probability of transmission via quantum tunneling, momentarily decelerating within the barrier-a behavior akin to a boat navigating a weir below its water level-and ultimately propagating through with velocity dependent on the barrier’s width $a$, as modeled by an incident wave packet characterized by a full width at half maximum of $t_{\mathrm{FWHM}}=1.14\,\mathrm{ns}$ and a flat-top duration of $t_{\mathrm{flat}}=0.86\,\mathrm{ns}$, with the time evolution of velocity within the barrier elucidating this dependence.

This review examines the time-dependent Bohmian velocity of wave packets during quantum tunneling, challenging conventional interpretations of tunneling time and dwell time calculations.

Despite its foundational role in diverse phenomena, the dynamics of quantum tunneling remain surprisingly subtle, particularly concerning the velocity of particles within the potential barrier. In ‘Instantaneous velocity during quantum tunnelling’, we analyze the temporal evolution of tunneling particles, revealing a continuous relaxation of velocity from an initial value, potentially approaching zero within wide barriers-a finding that resolves the paradox of finite density coexisting with seemingly static particles. Our work demonstrates that defining an effective speed based on probability density, rather than current, can yield misleading results and establishes a clear dynamical picture of tunneling flow. Could a deeper understanding of this internal velocity pave the way for controlling or manipulating tunneling processes in future quantum technologies?

Unveiling Quantum Barriers: The Paradox of Particle Motion

At the quantum scale, a particle’s journey isn’t simply dictated by whether it possesses enough energy to overcome an opposing force, as classical physics would suggest. Instead, particles encounter potential barriers – regions where, according to established energy principles, they shouldn’t be able to exist. These barriers aren’t necessarily physical walls, but rather areas of repulsive force or insufficient energy. The challenge arises because quantum mechanics describes particles not as localized points, but as wave functions – probability distributions indicating the likelihood of finding a particle in a given location. This wave-like nature allows a portion of the particle’s probability amplitude to ‘leak’ through the barrier, even if its energy is less than the barrier’s height. The extent of this leakage is determined by the barrier’s width and height, fundamentally altering the particle’s behavior and defying intuitive expectations about motion and confinement. This phenomenon highlights a crucial departure from classical mechanics, where a particle lacking sufficient energy would be completely stopped by such a barrier.

The realm of quantum mechanics presents a stark departure from everyday experience, particularly concerning how particles interact with barriers. Unlike classical physics, where a particle lacking sufficient energy cannot overcome an obstacle, quantum particles exhibit wave-like behavior that allows for a phenomenon called quantum tunnelling. This isn’t a matter of simply going over or through a barrier, but rather a non-zero probability of the particle appearing on the other side, even if it doesn’t possess the classically required energy. This arises because the particle is described by a wave function, which extends into the barrier region; the square of this wave function represents the probability of finding the particle at a given location. Consequently, even though the probability is diminished within the barrier, it isn’t zero, leading to a measurable tunnelling rate. This effect isn’t merely theoretical; it’s fundamental to processes like nuclear fusion in stars, radioactive decay, and even the operation of certain semiconductor devices, demonstrating the profound implications of abandoning classical intuition when exploring the quantum world.

The peculiar behavior of quantum particles necessitates a departure from classical descriptions of motion, demanding a framework centered on the wave function. Unlike classical mechanics, which predicts a particle will always be stopped by an insurmountable barrier, quantum mechanics describes particles as existing as probability waves, extending even into classically forbidden regions. This wave function, mathematically represented by $ \Psi $, doesn’t define a particle’s definite location but rather the probability of finding it at a given point. Consequently, even when encountering a barrier, there’s a non-zero probability that the particle’s wave function will ‘leak’ through, resulting in the phenomenon of quantum tunneling. It is through the wave function, and the Schrödinger equation that governs its evolution, that physicists can accurately predict and understand the probabilistic nature of particle behavior at the quantum scale, abandoning deterministic trajectories for probability distributions.

The time evolution of the effective potential at varying positions demonstrates a dependence on the energy difference Δ, while the steady-state velocity is modulated by barrier width and spatial location, as described by a hyperbolic function.

Governing Quantum Evolution: The Time-Dependent Schrödinger Equation

The Time-Dependent Schrödinger Equation, expressed as $i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = \hat{H} \Psi(\mathbf{r}, t)$, is a central postulate of quantum mechanics governing the evolution of a particle’s quantum state. Here, $\Psi(\mathbf{r}, t)$ represents the time-dependent wave function, describing the probability amplitude of finding a particle at position $\mathbf{r}$ at time $t$. The operator $\hat{H}$ is the Hamiltonian, representing the total energy of the system. Solving this partial differential equation yields the wave function at any given time, allowing prediction of the particle’s behavior and properties. The equation fundamentally describes how the quantum state, and therefore all measurable properties of a particle, changes continuously over time, given an initial quantum state and the system’s Hamiltonian.

Direct analytical solutions to the Time-Dependent Schrödinger Equation are limited to a small number of idealized systems. For most realistic potentials, the equation’s complexity necessitates the use of numerical approximation techniques. The Split-Step Fourier Method is a commonly employed algorithm that leverages the Fourier transform to efficiently propagate the wave function in time. This method exploits the fact that the Schrödinger equation can be separated into kinetic and potential energy terms; each term is applied sequentially in the Fourier and real spaces, respectively, providing a computationally effective means of simulating quantum evolution. The accuracy of these numerical solutions is dependent on parameters like the time step and grid spacing, requiring careful consideration for reliable results.

Numerical methods for solving the Time-Dependent Schrödinger Equation, such as the Split-Step Fourier Method, exhibit increased computational efficiency when applied to one-dimensional systems. Reducing the dimensionality from three to one significantly decreases the number of variables and computational points required to discretize the wave function, $ \Psi(x,t) $, thereby lowering the computational cost. This simplification allows researchers to focus on analyzing particle dynamics – specifically, the probability density and current – without being overwhelmed by the complexities of higher-dimensional calculations, enabling detailed investigations into phenomena like tunneling, scattering, and wave packet propagation.

Analysis of probability density evolution in main and auxiliary waveguides with Δ = -0.1 meV reveals a steady-state particle velocity of 2292 km/s in the auxiliary waveguide, contrasting with the zero velocity observed during flat-top pulse interaction.

Tracing Particle Paths: Bohmian Mechanics and the Velocity Field

The Bohmian Velocity, denoted as $v_B$, represents a fundamental component of Bohmian mechanics and is calculated as the gradient of the phase of the wave function, $\Psi$, divided by the mass, $m$, of the particle: $v_B = \frac{\hbar}{m} \nabla S$, where $S$ is the phase and $\hbar$ is the reduced Planck constant. This velocity is not the conventional quantum mechanical velocity operator expectation value; instead, it is a definite velocity assigned to each particle at each instant in time. Crucially, the Bohmian Velocity is entirely determined by the wave function itself, meaning the particle’s motion is guided by the quantum potential derived from $\Psi$. Unlike the probabilistic interpretations of standard quantum mechanics, the Bohmian Velocity provides a specific, deterministic trajectory for each particle, even though the initial particle position is subject to the probabilistic distribution defined by $|\Psi|^2$.

In Bohmian mechanics, the velocity of a particle is not determined by initial conditions alone, but by a velocity field derived from the wave function, $\Psi$. This field dictates the particle’s trajectory, resolving the inherent probabilistic nature of standard quantum mechanics with a deterministic model. Given the wave function and the particle’s initial position, the velocity, $v(x,t)$, is calculated as $v(x,t) = \frac{\hbar}{m} \frac{\Im(\nabla \Psi(x,t))}{\Psi(x,t)}$, where $\hbar$ is the reduced Planck constant, $m$ is the particle’s mass, and $\Im$ denotes the imaginary part. This allows for the precise prediction of a particle’s position at any given time, effectively removing randomness from the quantum process while still reproducing all empirically verified quantum predictions.

The Continuity Equation, a fundamental principle in Bohmian mechanics, mathematically guarantees the conservation of probability density as a particle’s wave function evolves over time. Expressed as $ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0 $, where $\rho$ represents the probability density and $v$ is the particle velocity, the equation states that any change in probability density within a given volume is solely accounted for by the flow of probability through the surface of that volume. This ensures that the total probability of finding the particle somewhere in the universe remains constant, upholding a key tenet of quantum mechanics and demonstrating the internal logical consistency of the Bohmian interpretation by preventing unphysical probability loss or gain.

The Quantum Potential: Sculpting the Effective Force Landscape

The quantum potential, a cornerstone of the de Broglie-Bohm interpretation, emerges not from a conventional force field, but directly from the shape of the particle’s guiding wave function. This potential is proportional to the second derivative – the curvature – of the wave function, meaning regions where the wave bends sharply exert a stronger influence on the particle’s motion. Critically, this isn’t an additional force added to the classical potential; rather, it modifies the effective potential landscape experienced by the particle. A flat or slowly varying wave function generates a negligible quantum potential, leaving the particle behaving largely as classical physics predicts. However, a highly curved wave function creates a significant quantum potential, which can either enhance or diminish the classical potential, and even create forces acting in opposition to it. This interplay between the classical potential and the quantum potential dictates the particle’s trajectory, leading to behaviors impossible to explain through classical mechanics alone, such as non-local interactions and trajectories that aren’t solely determined by the immediate forces at a given point.

The Effective Potential represents the total potential energy experienced by a quantum particle, arising from the combination of the classical potential and the Quantum Potential. This summation isn’t merely additive; the Quantum Potential, derived from the wave function’s curvature, can significantly alter the shape of the total potential landscape. Consequently, a particle doesn’t simply respond to familiar forces, but to this combined, often non-intuitive, effective force. This leads to behaviors drastically different from classical physics, such as particles accelerating towards the regions of higher potential, or tunnelling through barriers that would be impenetrable according to classical mechanics. The Effective Potential, therefore, dictates the particle’s motion and probability distributions in ways that challenge conventional understanding, highlighting the fundamentally different nature of quantum dynamics.

The behavior of a particle encountering a potential barrier is significantly shaped by its steady-state velocity, experimentally determined to be 2292 km/s within the auxiliary waveguide. This velocity isn’t simply a measure of kinetic energy; it’s intricately linked to the effective potential – the sum of the classical potential and the quantum potential arising from the wave function’s curvature. Consequently, the particle’s ability to tunnel through the barrier-a distinctly quantum mechanical phenomenon-is directly modulated by this effective potential. A modified effective potential alters the shape and height of the barrier as ‘experienced’ by the particle, influencing the probability of finding it on the other side even when classically forbidden. This means that even with a constant applied energy, the tunneling rate can be dramatically changed by the quantum environment, highlighting the crucial role of the wave function in governing particle dynamics.

Towards Complex Systems: Simulating Reality with Coupled Waveguides

The two-waveguide system provides a remarkably effective analogue for simulating particle transport in substantially more complex and realistic scenarios. This approach simplifies the challenges associated with modeling intricate potential landscapes, allowing researchers to focus on fundamental aspects of particle dynamics without being overwhelmed by computational complexity. By carefully controlling the characteristics of the waveguides – their geometry, material properties, and the coupling between them – it becomes possible to mimic the behavior of particles traversing barriers and exploring complex potentials. This model isn’t merely a simplification; it offers a controlled environment for investigating phenomena like quantum tunneling, resonance effects, and the influence of disorder, providing insights that are directly applicable to diverse fields, including condensed matter physics, materials science, and even biological systems where particle transport plays a crucial role. The elegance of this approach lies in its ability to translate complex problems into a manageable framework, enabling a deeper understanding of particle behavior in a wide range of physical systems.

The interplay between waveguides is critically determined by their coupling strength, a parameter that dictates how readily a particle can traverse the potential barrier separating them. A stronger coupling effectively narrows the barrier, increasing the probability of tunneling – a quantum mechanical phenomenon where particles pass through classically forbidden regions. Conversely, weaker coupling presents a more substantial barrier, diminishing tunneling probabilities and forcing the particle to spend more time within the potential well. This dynamic isn’t merely a quantitative shift; it fundamentally alters the particle’s behavior, influencing its velocity, dwell time, and even the effective shape of the potential landscape it experiences. Investigations reveal that manipulating this coupling strength offers a precise method for controlling particle transport, with implications for designing nanoscale devices and understanding fundamental quantum processes like electron transfer in biological systems.

Recent computational modeling of coupled waveguides has yielded a precise determination of particle velocity during tunneling, revealing that the particle’s speed within the potential barrier can approach 0 km/s after approximately 1.71 nanoseconds. This finding directly addresses a decades-long debate concerning the duration of the tunneling process – whether it’s instantaneous or finite. Traditional interpretations struggled to define a meaningful ‘dwell time’ for a particle traversing a classically forbidden region, but the simulation data provides a quantifiable measure of velocity reduction. The observed deceleration suggests the particle isn’t simply appearing on the other side of the barrier, but rather experiences a transient period of extremely low, yet non-zero, velocity within it. This challenges conventional understandings of quantum tunneling and provides empirical support for models proposing a finite tunneling time, potentially reshaping interpretations of fundamental quantum phenomena and offering insights into the dynamics of particle transport in complex systems.

The exploration of particle behavior within potential barriers, as detailed in this work, echoes a fundamental tenet of wave mechanics. Louis de Broglie observed, “Every material particle also has an associated wave, and the wave is linked to the particle.” This resonance is evident in the study’s examination of the Bohmian velocity and the wave packet’s traversal of the barrier. The nuanced understanding of particles momentarily ceasing motion within the barrier isn’t a contradiction of de Broglie’s proposition, but rather a detailed manifestation of it. It reveals how the wave-like nature dictates even the seemingly static moments during tunneling. Good architecture is invisible until it breaks, and only then is the true cost of decisions visible.

Where Do the Shadows Fall?

The investigation of instantaneous velocity within the tunneling regime exposes a subtle, yet critical, point: the assumption of continuous, unimpeded motion is likely flawed. Particles, it seems, do not simply traverse the barrier, but experience internal dynamics – momentary hesitations – that conventional dwell-time calculations often obscure. This isn’t merely a technical refinement; it suggests a deeper structural problem with how the concept of ‘time’ is applied to quantum systems. Systems break along invisible boundaries – if one cannot account for internal states within the barrier, pain is coming in the form of paradoxical results.

Future work must address the effective potential itself. The standard approach treats it as a purely geometric construct, but this analysis implies it’s a dynamical entity, shaped by the particle’s internal velocity and, crucially, its momentary stalls. A more holistic treatment – one that incorporates the wave packet’s evolution within the barrier’s influence – is needed. The challenge isn’t simply to measure tunneling time, but to define what ‘time’ even means for a particle experiencing such non-intuitive dynamics.

Ultimately, this research pushes against the tendency to treat quantum mechanics as a collection of isolated phenomena. The behavior within the barrier isn’t separate from the initial and final states; it’s an integral part of a unified process. Anticipating weaknesses requires recognizing that elegance lies not in complexity, but in uncovering the simplest, most complete description of the whole system – a description that acknowledges the shadows, not just the light.

Original article: https://arxiv.org/pdf/2512.16385.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/