Quantum mechanics loves landscapes. A particle in one dimension can sit in a single friendly well, or it can wander between two wells separated by a barrier, its fate decided by the fickle game of probabilities. In the real world, that wandering—tunneling—drives reactions, fuels coherence in qubits, and underpins the way molecules flip between states. A new study, coming out of the Universidad Autónoma Metropolitana, Unidad Iztapalapa in Mexico City, looks at a particularly clever playground for this drama: a one dimensional sextic potential whose shape you can twist with a single knob, λ. The work, led by Angelina N. Mendoza Tavera and Adrian M. Escobar Ruiz with Robin P. Sagar, uses a blend of algebraic solvability, variational methods, and information theory to explore how quantum states change when a landscape morphs from a single well into a symmetric double well. They don’t just catalog energies; they watch how information about the particle’s position and momentum reorganizes itself as tunneling turns on and off.
What makes this study stand out is not merely the math but the way it measures the system. The authors work with a quasi-exactly solvable (QES) sextic potential, a rare breed that sits between exactly solvable problems like the harmonic oscillator and fully numerical, non-solvable beasts. In this family, a tunable parameter λ reshapes the potential; for certain values the landscape has two symmetric minima, a barrier in the middle, and the possibility of a ground state that spans both wells through tunneling. Crucially, the researchers craft trial wavefunctions with exact Fourier transforms, which lets them analyze both the real-space and momentum-space information with high precision. And they tether their numerical results to known QES solutions, giving a robust map of how the system behaves as λ slides from single well to twin-well territory.
In the pages that follow, the paper becomes less about esoteric equations and more about a central idea: information, not just variance, is a sharper instrument for seeing when a quantum system changes its character. The authors verify that a universal entropic uncertainty bound—the Beckner–Bialynicki-Birula–Mycielski relation, which in one dimension says the sum of position and momentum entropies cannot drop below a fixed constant—holds across all the states and all the λ values they study. They also show that entropy-based diagnostics, including Shannon entropies in position and momentum and divergences that compare how one state’s density differs from another, illuminate tunneling, symmetry breaking, and the formation of nearly degenerate state pairs in a way standard second-moment measures do not. This is not just a pretty story; it’s a blueprint for using information theory to diagnose and engineer quantum behavior.
To credit the human side of the work, the study is conducted at the Universidad Autónoma Metropolitana, Unidad Iztapalapa, with the authors Mendoza Tavera, Escobar Ruiz, and Sagar at the helm. The central goal is to illuminate how a single tunable landscape can host a spectrum that reorganizes itself through a mix of localization, interference, and coherence, and to show how entropic tools offer a more sensitive lens than traditional variance-based metrics. In short, the work asks: what else can entropy tell us about quantum systems if we look at both where the particle is and where its momentum wants to be, as the landscape itself changes shape?
A tunable sextet stage for quantum states
The mathematical stage is a one dimensional spectral problem with a Hamiltonian H = −(1/2) d2/dx2 + VQES(x; λ), where the potential is VQES(x; λ) = 1/2 [x6 + 2×4 − 2(2λ + 1)x2]. The term in front of the x2 piece, controlled by λ, acts like a dial that reshapes the landscape from a single, confining well into a pair of symmetric wells with a barrier at the center. This is a classic setup for tunneling: as the barrier grows, energy levels begin to split into nearly degenerate pairs because a quantum particle can “borrow” energy to tunnel between the wells. The paper tracks this transition as λ sweeps from negative through zero and into the positive, with particular attention to a critical value λnc for each state n, where tunneling effects first become energetically relevant and the landscape’s double-well character becomes manifest in the spectrum. A striking feature is that the ground state begins to tunnel around λ0c ≈ 0.73295, and higher states follow with their own thresholds.
What makes this setup fertile is the hidden algebraic structure behind the QES models. For a discrete set of λ values (specifically λ = (2N+1)/2, with N a nonnegative integer), the Hamiltonian sits inside a finite-dimensional representation of a Lie algebra (sl2R), which means a portion of the spectrum and eigenfunctions can be obtained exactly. That algebraic backbone gives researchers a solid, partially analytic handle on the problem, a rare commodity in quantum mechanics where many interesting systems are beyond exact solution. The result is a system where the potential itself changes with λ, and yet a slice of the spectrum is analytically tractable, providing a precise yardstick against which to benchmark numerical methods and variational tricks.
The researchers adopt a carefully designed variational ansatz ψn(x; λ) = Qn(x; λ) e−x4/4, with Qn chosen to respect parity and asymptotic behavior. This choice achieves three things at once: it enforces the symmetry x → −x that governs the double-well problem, it captures the correct exponential falloff for large |x|, and it yields Fourier transforms that can be written analytically. That last point is gold for entropy calculations, because knowing ψ(x) and its Fourier transform Φ(p) cleanly lets you compute position and momentum entropies without numerical convolutions clouding the picture. For even n the polynomials Qn are even functions; for odd n they’re odd. The result is a family of high-accuracy, physically meaningful trial densities that can be compared to numerically exact QES results and to the Lagrange-mesh method, a state-of-the-art numerical approach used here as a benchmark.
Beyond the variational craft, the paper also emphasizes how the spectrum rearranges itself. As λ grows, energy levels begin to pair up as they dip below the central barrier and become localized in the outer wells. This leads to a recognizable pattern: adjacent states form almost-degenerate pairs (n = 0 with n = 1, n = 2 with n = 3, and so on), a hallmark of twin-well dynamics. The authors map these pairings visually in their figures and quantify them with entropic divergences, showing that the deeper the wells, the more pronounced the tunneling-induced coherence becomes. It’s a vivid reminder that in quantum systems, the geometry of the landscape and the symmetry of the states are in constant conversation.
Entropy as a flashlight on tunneling
One of the paper’s core messages is that information theory provides sharper signals than the textbook variance in many quantum scenarios. The authors bring together several entropy-based instruments: the Shannon entropies Sx and Sp for position and momentum, the Beckner–Bialynicki-Birula–Mycielski (BBM) entropic uncertainty relation, and a pair of divergences—Kullback–Leibler (KL) and Cumulative Residual Jeffreys (CRJ)—to compare how two quantum states differ in their spatial densities. The BBM relation, in particular, sets a universal bound on Sx + Sp that holds no matter what the state looks like, capturing the tradeoff between localization in space and spreading in momentum in a way the standard deviation cannot always reveal. The authors verify that this bound holds across all n and λ they study, including the intricate tunneling regime where the wavefunctions become highly structured.
As λ crosses the critical values λnc, the position-space densities reveal the signature of tunneling. Even states that start with a single, central peak gradually develop two lobes perched near the outer wells. The ground state, for instance, shifts from concentrating at the center to becoming a superposition that spans both wells. The momentum-space densities respond in kind, displaying interference patterns that reflect the coherence between the two spatially separated components. In these momentum profiles, even and odd states show characteristic fingerprints: even states tend to maintain a peak near p = 0 as long as the symmetry supports it, while odd states typically exhibit a dip at p = 0, echoing their parity. The upshot is a set of momentum-space signatures that survive even when the energy levels are nearly degenerate, a sign of the subtle coherence that tunneling creates.
But the real punch comes when the authors bring in entropy-based diagnostics. The position entropy Sx and the momentum entropy Sp respond to the landscape’s reshaping in complementary ways that the simple Δx or Δp alone can miss. Near the critical coupling, Sx often exhibits a pronounced peak, signaling maximal delocalization in real space as the state distributes itself across both wells. The momentum entropy Sp, by contrast, can dip near those transitions, indicating a tightening in momentum-space structure when the state briefly adopts a more coherent, delocalized superposition. The entropic sum St = Sx + Sp then traces the overall quantum information content, offering a holistic readout of the system’s complexity as tunneling turns on. In practice, these entropy signals provide a more nuanced map of the transition than the Heisenberg product ΔxΔp alone, which can stay stubbornly near a harmonic-oscillator-like value even as the wavefunction reorganizes itself in deeper wells.
In addition to Sx and Sp, the paper turns to KL and CRJ divergences to compare densities across states. The KL divergence captures how one density shifts relative to another near the central region, while the CRJ divergence emphasizes differences in the tails and in the classically forbidden regions where tunneling leaks probability. The authors find that CRJ is especially sensitive to the onset of pairing between states such as (n = 0, 1) and (n = 2, 3). In those pairs, the densities become nearly indistinguishable as tunneling grows, and CRJ dips toward near-zero, signaling a coherence that’s hard to spot with variance alone. KL, meanwhile, tracks the gradual convergence of the lower-energy state toward its neighbor as λ increases, reflecting the deepening symmetry between the two wells. Taken together, these measures reveal the full, subtle choreography of a quantum system undergoing a topology-driven transition.
A remarkable consistency check in the paper is the entropic BBM bound, which holds across all observed states and parameter values. The research team also demonstrates an impressive accuracy check: their variational densities differ from exact QES densities by on the order of 10−10 in cumulative residual Jeffreys divergences for certain cases, underscoring how faithful the trial functions are as density models. In other words, entropy becomes not just a diagnostic but a precise, quantitative witness to the inner life of the quantum state as the landscape is tuned.
What this means for quantum tech and future thinking
Beyond its theoretical elegance, the work offers a practical message for quantum engineers and theorists. If you care about driving and controlling tunneling in nanoscale devices, entropy provides a richer diagnostic toolkit than the usual variance-based metrics. In devices that use double-well landscapes—think quantum dots, superconducting qubits built from Josephson junctions, or engineered traps for cold atoms—the ability to diagnose when a state is delocalized across two wells, when coherence is building, or when a pair of states is effectively tunneling-degenerate can be the difference between a stable qubit and a noisy ghost. The entropic signatures described in this study could inform how these systems are tuned in real time to optimize coherence, stability, and controllability.
The authors go a step further by positioning their QES sextic model as a continuously tunable, analytically tractable reference for benchmarking quantum information tools. Because a portion of the spectrum and eigenfunctions is known exactly, this model serves as a rigorous testbed for entropy-based diagnostics, variational approaches, and numerical algorithms used in quantum information science. In other words, the work doesn’t just tell a story about a single mathematical toy; it offers a blueprint for how to test, compare, and refine the methods researchers rely on when they push quantum technologies from the chalkboard to the lab bench.
Another broader thread is the bridge this work builds between algebraic solvability and information theory. The quasi-exact solvable structure of the sextic potential—rooted in hidden Lie algebra symmetries—speaks to deep, structural aspects of quantum systems. The information-theoretic lens then translates that structure into concrete, observable fingerprints: how localized a state is in real space, how spread it is in momentum space, how strongly two states resemble or differ from one another as the landscape evolves. That synthesis, the authors suggest, could illuminate other quantum systems where topology, symmetry, and coherence intertwine, from molecular isomerization to photons in nonlinear media.
For researchers and readers curious about the frontier, this work also sketches a path forward. The authors propose extending the analysis to Wigner quasiprobability distributions, which would lay out the full phase-space picture of these states. A refined semiclassical treatment via WKB methods could yield analytical approximations to E(λ), helping build a continuous, analytic bridge across the weak- to strong-coupling regimes. And there is the tantalizing possibility of exploring complex λ domains, PT symmetry, and other algebraically rich landscapes that sit just beyond the current study’s horizon.
In the end, the paper isn’t just about a clever potential and a set of entropy calculations. It’s a reminder that the language we choose to describe quantum systems matters. Variance can tell you how wide a wavefunction is, but entropy tells you how much information it contains and how that information is reorganized as the world around it changes. When the landscape itself is a participant in the dance, the story becomes richer, more surprising, and more useful for building the quantum technologies of tomorrow.
Lead researchers and institution: The study is carried out at the Universidad Autónoma Metropolitana, Unidad Iztapalapa, Mexico City, by Angelina N. Mendoza Tavera and Adrian M. Escobar Ruiz, with Robin P. Sagar. The team’s work blends exact quasi-analytic solvability, high-precision variational methods, and information-theoretic diagnostics to chart how a tunable quantum landscape triggers tunneling, symmetry breaking, and state pairing.
Bottom line: By showing that entropy and related divergences reveal the hidden order behind quantum transitions more clearly than traditional measures, this research points toward a practical, information-centered way to understand and harness quantum coherence in tunable landscapes. It’s a window into how the information content of a quantum state reorganizes itself when the geometry of the world changes, and a reminder that the most telling stories in quantum physics often emerge from the way we measure what we know—and what we still don’t know.
