Beyond Reality: Quantum Matter with Complex Mass

Author: Denis Avetisyan

New research explores the unusual properties of quantum systems governed by complex mass and Morse potentials, revealing unexpected states of matter.

The ground state probability density of the hydrogen molecule ($H_2H_2$) exhibits spatial confinement variations correlated with the parameter $a_{i}$, while peak probability density demonstrates a dependency on both $a_{i}$ and the complex Morse parameter, influenced by real parameters such as $V_{or} = 38266 cm^{-1}$, $a_{r} = 1.868 Å^{-1}$, and $m_{r} = 0.5039 \mu$.

This study provides an exact solution to the Schrödinger equation for a complex mass quantum system under a complex Morse potential, investigating emergent matter types and their phases.

While conventional quantum mechanics relies on Hermitian operators, exploring non-Hermitian frameworks reveals the potential for exotic quantum phenomena and emergent matter types. This work, ‘Exact Solution of Schrödinger equation for Complex Mass Quantum System under Complex Morse Potential to study emergent matter types and its phases’, presents exact solutions to the Schrödinger equation for a quantum system with complex mass under a complex Morse potential, revealing five distinct matter classifications-ranging from stable, Hermitian-like states to purely complex, non-physical regimes. The interplay between complex parameters dictates spectral characteristics and normalization conditions, elucidating a boundary between physical and non-physical solutions. Could these findings offer a theoretical framework for understanding phenomena such as dark matter, or provide insights into novel phases of quantum matter governed by complex dynamics?

Beyond Equilibrium: Exploring the Frontiers of Non-Hermitian Quantum Mechanics

The foundations of quantum mechanics rest upon the concept of the Hamiltonian, an operator representing the total energy of a system, traditionally required to be Hermitian – meaning its eigenvalues are real and correspond to measurable energy levels. This framework successfully describes systems that are closed and stable, where energy is conserved. However, many real-world systems are open, interacting with their environment and experiencing gain or loss of energy – think of lasers with stimulated emission or decaying radioactive isotopes. For these scenarios, the Hermitian constraint breaks down, leading to inaccurate predictions. The standard mathematical tools become insufficient to describe phenomena like unidirectional propagation or the amplification of waves. Consequently, a significant limitation arises in modeling these non-equilibrium systems, necessitating a broader theoretical approach that can accommodate the complexities of energy exchange and the resulting non-Hermitian behavior.

The conventional framework of quantum mechanics, reliant on Hermitian operators to describe physical observables, proves inadequate when addressing systems that aren’t isolated – those experiencing gain or loss of energy, or interacting with an external environment. To overcome these limitations, physicists are extending quantum theory into the realm of non-Hermitian mechanics. This generalization allows for complex eigenvalues – possessing both real and imaginary components – which, while initially counterintuitive, provide a powerful tool for characterizing the dynamics of open systems. The imaginary component is not merely a mathematical artifact; it directly relates to the rate of gain or loss within the system, enabling the modeling of phenomena such as lasing, decay, and the behavior of parity-time symmetric structures. Consequently, non-Hermitian quantum mechanics offers a more complete and nuanced description of reality, moving beyond the constraints of static, closed systems and providing insights into the behavior of systems constantly exchanging energy with their surroundings – a crucial step towards understanding the complex quantum world.

The transition to non-Hermitian quantum mechanics, while offering a powerful means to describe open quantum systems, presents significant interpretive hurdles. Traditionally, eigenvalues of the Hamiltonian represent measurable energy levels, and eigenstates form a complete, orthonormal basis for describing physical states. However, non-Hermitian Hamiltonians can yield complex eigenvalues – a consequence of energy gain or loss – and, crucially, may lack the property of normalizability. This means the usual probabilistic interpretation, where the square of the wavefunction’s amplitude represents probability density, breaks down; wavefunctions can ‘explode’ in certain regions of space. Physicists are actively developing new mathematical frameworks and physical principles to reconcile these non-normalizable states with observable phenomena, exploring concepts like pseudo-hermiticity and the role of exceptional points – singularities in the parameter space where eigenvalues and eigenvectors coalesce – to establish a consistent physical picture. The challenge lies in determining which aspects of the complex spectrum are physically real and how to extract meaningful, measurable predictions from a theory that deviates so fundamentally from the foundations of standard quantum mechanics.

Analysis of the hydrogen molecule's ground state real energy eigenfunction reveals permissible and non-permissible regions defined by the imaginary parameters mim and aia, demonstrating the relationship between these parameters and the molecule's energy spectrum. — Analysis of the hydrogen molecule’s ground state real energy eigenfunction reveals permissible and non-permissible regions defined by the imaginary parameters mim and aia, demonstrating the relationship between these parameters and the molecule’s energy spectrum.

Extending the Quantum Landscape: A Complex Phase Space Formalism

The extended complex phase space formalism offers a mathematical structure for analyzing the Schrödinger equation when applied to non-Hermitian Hamiltonians. Traditional quantum mechanics relies on Hermitian operators, ensuring real eigenvalues corresponding to observable energy levels; however, many physical systems exhibit non-Hermitian behavior, such as those experiencing gain and loss, or open quantum systems interacting with an environment. By allowing complex coordinates in phase space-extending beyond the usual position and momentum-the formalism enables the calculation of complex eigenvalues and corresponding eigenfunctions, even for non-Hermitian operators. This approach does not modify the underlying physical system but provides a consistent mathematical framework to describe its evolution, crucially allowing for the treatment of systems where traditional Hermitian methods fail. The complex eigenvalues represent energy levels that may exhibit decay or growth, and the complex eigenfunctions describe the corresponding states.

The extension of the complex phase space enables the determination of complex eigenvalues and corresponding eigenfunctions for systems governed by non-Hermitian Hamiltonians. Unlike Hermitian systems where eigenvalues are strictly real, non-Hermitian systems can possess complex eigenvalues of the form $E = E_R + iE_I$. The real component, $E_R$, represents the energy, while the imaginary component, $E_I$, dictates the stability of the state. A negative $E_I$ indicates a decaying state with a growth rate proportional to $|E_I|$, while a positive $E_I$ signifies an unstable, growing state. The associated eigenfunctions, though potentially non-normalizable, describe the time evolution of these unstable or decaying states, providing insights into phenomena such as resonance, decay rates, and open quantum systems.

The application of the extended complex phase space method introduces challenges in interpreting solutions due to the potential for non-normalizable wavefunctions. While standard quantum mechanics requires wavefunctions to be normalizable – meaning the integral of $|\psi(x)|^2$ over all space must be finite – complex potentials can yield solutions that do not satisfy this condition. These non-normalizable states do not represent physically realizable wavefunctions in the traditional sense; however, they are crucial for describing the dynamics of unstable or decaying systems. Their significance lies not in their probability interpretation, but in their contribution to the overall time evolution and the calculation of measurable quantities like decay rates. Specifically, the imaginary part of the complex energy eigenvalue, obtained from solving the Schrödinger equation in this extended space, directly corresponds to the decay constant $\Gamma$, governing the exponential decay of the state with time $e^{-\Gamma t}$.

The ground state energy of the hydrogen molecule, mapped across imaginary parameters, reveals permissible regions (red/gray) and non-normalizable or negative energy states (black), with energy behavior varying predictably with changes in each parameter.

Spectral Signatures: Decoding the Behavior of Quantum States

The spectral reality of a quantum system, determined by the nature of its eigenvalues, is a foundational determinant of system behavior. Eigenvalues representing the allowed energy levels of a system can be either real or complex. Real eigenvalues signify stable states with well-defined, measurable properties; these states persist over time. Conversely, complex eigenvalues indicate instability, manifesting as decay or transitions between states. The real component of a complex eigenvalue represents the energy of a quasi-stable state, while the imaginary component defines the rate of decay or the lifetime of that state. Therefore, the sign of the imaginary component is critical; a negative imaginary component corresponds to exponential growth, while a positive imaginary component indicates exponential decay, fundamentally altering the observable characteristics of the quantum state and influencing the overall stability of the system.

In quantum mechanics, the eigenvalues of a system’s Hamiltonian operator determine its possible energy levels. Real eigenvalues signify that the corresponding energy state is stable and can be directly observed as a persistent quantum state. Conversely, complex eigenvalues indicate an unstable state where the probability amplitude of the system decays over time; these states are not directly observable as persistent entities. The presence of complex eigenvalues does not preclude the existence of measurable phenomena, however, as such states can manifest as transient behavior or contribute to the properties of more stable states. The concept of quasi-stable matter arises from systems exhibiting predominantly real eigenvalues but with a finite number of complex eigenvalues, resulting in a limited lifespan before decay, yet still allowing for measurable observation within that timeframe. These quasi-stable states represent a departure from fully stable states but are not entirely ephemeral, occupying an intermediate condition between absolute stability and complete disintegration.

This research demonstrates a direct correlation between spectral reality – the condition of eigenvalues being real or complex – and the implementation of complex Morse potentials and complex mass parameters within quantum mechanical models. Specifically, analysis reveals that to maintain real spectra – corresponding to stable, observable states – the imaginary component of the mass ($mim$) must be positive. Negative or zero values for $mim$ invariably lead to complex eigenvalues, indicating unstable or decaying states. These constraints on the imaginary mass parameter are critical for defining the boundaries between stable and quasi-stable matter, and are fundamental to understanding the observed spectral properties of various quantum systems.

Ground state probability density plots for the hydrogen molecule reveal that spatial confinement and peak density are both dependent on the complex mass parameter and fixed real parameters (V₀r = 38266 cm⁻¹, aᵣ = 1.868 Å⁻¹, mᵣ = 0.5039 μ).

Navigating the Boundaries of Existence: A Quantum Landscape

The behavior of matter, as described by quantum mechanics, isn’t limited to systems governed by conventional, energy-conserving rules. Investigations utilizing non-Hermitian Hamiltonians – mathematical descriptions allowing for energy gain and loss – reveal a surprising spectrum of possibilities, extending beyond the familiar realm of stable materials. These models demonstrate that a system’s evolution isn’t necessarily bound to a determinate, physical state; instead, it can transition through regions of instability, potentially leading to divergent behavior where solutions become unphysical and the system effectively ceases to exist as defined. However, within specific parameter ranges, these same non-Hermitian frameworks can also predict remarkably stable configurations, suggesting that the boundary between existence and non-existence is far more fluid than traditionally assumed and dependent on the system’s inherent properties, offering new avenues for exploring and potentially engineering materials with tailored characteristics.

The behavior of matter at its boundaries, transitioning between stable existence and disintegration, is fundamentally governed by the interplay between the complex Morse potential and the incorporation of complex mass. This theoretical framework reveals that a system’s fate – whether it achieves a determinate, stable state, undergoes decay, or manifests entirely unphysical characteristics – is dictated by the parameters within this potential. The complex Morse potential, traditionally used to describe molecular vibrations, is modified to allow for complex-valued mass, effectively introducing gain or loss into the system. This alteration dramatically influences the wave function, allowing solutions that would be strictly forbidden in standard quantum mechanics. Consequently, the system can evolve beyond simple decay, potentially exhibiting persistent oscillatory behavior or even growth, challenging conventional notions of material stability and offering a pathway to explore exotic states of matter.

The behavior of non-Hermitian systems isn’t arbitrary; rather, it’s constrained by specific boundaries within a parameter space defined by the imaginary mass ($m_{im}$) and the attractive potential strength ($a_{ia}$). Researchers have meticulously mapped these regions, revealing that stability and the very existence of physically meaningful solutions are not guaranteed. Outside these demarcated areas, the system’s wave functions diverge, leading to unphysical predictions. Within these regions, however, the complex interplay between the imaginary mass and potential allows for stable, determinate states. This parameter space acts as a ‘rule book’ for non-Hermitian quantum mechanics, dictating which combinations of system characteristics will yield realistic, quantifiable results and which will not, offering a crucial framework for understanding matter under these unusual conditions.

The very foundation of quantum mechanics relies on the accurate description of a system’s probability density, and this becomes particularly crucial when exploring non-Hermitian systems. As parameters shift and the system navigates the boundary between stable and unstable states, the probability density-represented mathematically by the square of the wavefunction, $ |\psi|^2 $-is profoundly affected. To ensure this description remains physically meaningful, the wavefunction must adhere to the normalization condition, dictating that the integral of the probability density over all space equals one. Failure to uphold this condition results in a probability distribution that is either unphysical-representing a probability greater than unity-or entirely disconnected from a valid quantum state, rendering the entire theoretical framework untenable. This rigorous demand for normalization acts as a powerful constraint, shaping the allowed parameter space and ultimately defining the boundaries of determinacy for matter within these complex quantum landscapes.

Ground state probability density plots for the hydrogen molecule reveal that spatial confinement and peak density are modulated by the complex Morse parameter and the reduced mass.

Unlocking Quantum Potential: Analytical Tools for a New Era

Sophisticated analytical techniques are proving essential for deciphering the intricacies of non-Hermitian quantum systems. Investigations leveraging complex Morse potential analysis allow researchers to model and predict the behavior of these systems with greater accuracy, revealing how energy landscapes influence particle dynamics. Complementary to this, studies focusing on the complex mass effect – where particles exhibit an effective mass with both real and imaginary components – provide crucial insights into stability and decay rates. These tools don’t merely describe what happens within these unusual systems, but illuminate why, revealing the underlying mechanisms that govern their unique spectral properties and paving the way for targeted manipulation and potential technological applications, such as enhanced sensing or novel materials with tailored optical responses. The combined power of these analytical approaches provides a robust framework for exploring the uncharted territory of non-Hermitian physics, offering a deeper understanding of matter’s fundamental behavior.

Investigations utilizing complex Morse potential analysis and detailed examinations of the complex mass effect allow for a profound understanding of non-Hermitian quantum systems. These analytical techniques move beyond traditional quantum mechanics by characterizing the unique spectral properties – the allowed energy levels – and stability of these systems, which don’t adhere to the usual rules of symmetry. The resulting insights reveal how these systems respond to perturbations and decay over time, offering a detailed map of their dynamic behavior. By precisely determining the energies and lifetimes of quantum states, researchers can predict and control the behavior of matter under unusual conditions, potentially leading to advancements in areas like novel materials and quantum technologies. These methods uncover how non-Hermitian systems can exhibit exceptional sensitivity or robustness, offering a pathway to engineer materials with tailored properties.

The established categorization of non-Hermitian matter based on spectral properties isn’t merely a theoretical exercise; it unlocks significant potential for both fundamental discovery and technological advancement. Researchers anticipate leveraging these insights to explore previously inaccessible quantum phenomena, such as unidirectional invisibility and asymmetric tunneling, which defy conventional quantum mechanics. Furthermore, the ability to finely control and manipulate these non-Hermitian systems could pave the way for groundbreaking technologies, including highly sensitive sensors, novel lasers with enhanced directionality, and even quantum devices capable of performing computations beyond the reach of classical computers. The precise classification offered by this framework serves as a crucial roadmap for designing and implementing these future technologies, promising a new era of quantum engineering built on principles that extend beyond traditional Hermitian physics.

Recent research has established a comprehensive classification of matter, delineating five distinct classes based on the unique characteristics of their spectral properties. This categorization moves beyond traditional understandings of phases and states, instead focusing on how these materials interact with and respond to external stimuli as revealed through spectral analysis – essentially, their ‘fingerprints’ when probed with energy. Each class exhibits markedly different behaviors in terms of energy absorption and emission, influencing their stability and potential applications. This framework, built upon detailed investigations of non-Hermitian quantum systems, offers a powerful new lens for understanding material behavior and predicting novel phenomena, potentially paving the way for the design of materials with tailored properties and functionalities. The identification of these five classes represents a significant step towards a more complete and nuanced understanding of the diverse landscape of matter itself.

The ground state peak probability density and real energy eigenfunction of the hydrogen molecule are constrained by parameter values (ai, mi), with permissible ranges defined by positive probability density (blue) and normalizable energy (red) as depicted in the parameter space.

The pursuit of exact solutions, as demonstrated within this study of complex mass quantum systems, echoes a fundamental principle of elegant design: consistency is empathy. This work meticulously unravels the behavior dictated by complex Morse potentials, revealing how subtle shifts in parameters-complex mass and potential-directly influence the resulting quantum matter. Werner Heisenberg observed, “Not only does God play dice with the universe, but He throws them where we cannot see.” This resonates with the inherent complexities explored; the study navigates the often-counterintuitive realm of non-Hermitian quantum mechanics, where traditional notions of probability density and spectral reality are challenged, revealing emergent matter types and phases that defy classical expectation. The beauty lies not in simplifying these intricacies, but in achieving a deeper understanding of their interconnectedness.

Where Do We Go From Here?

The exploration of non-Hermitian quantum mechanics, as exemplified by the treatment of complex mass and potentials, has always felt less like solving equations and more like an exercise in cartography-mapping the borders of what can reasonably be considered ‘physical.’ This work, by revealing the delicate interplay between spectral reality and the parameters defining complex phase space, underscores a fundamental truth: elegance in mathematical description is not merely aesthetic preference, but a signal of deeper understanding. The emergence of exotic states, though perhaps lacking immediate empirical counterparts, compels a re-evaluation of the assumptions baked into the very definition of ‘matter.’

However, the path ahead is not without its shadows. The current formalism, while powerful, remains largely confined to simplified models. Scaling these solutions to many-body systems, or incorporating relativistic effects, presents formidable challenges. Moreover, the interpretation of probability densities in non-Hermitian contexts demands continued scrutiny; a mathematically valid solution is insufficient if its physical meaning remains elusive.

The next logical step is not simply to find more solutions, but to refine the questions. What principles govern the stability of these emergent phases? Can these complex systems be harnessed, even in principle, to perform tasks beyond the reach of their Hermitian counterparts? The pursuit of these answers, though fraught with difficulty, promises a richer, more nuanced understanding of the quantum world – and perhaps, a more profound appreciation for the limits of our own intuition.

Original article: https://arxiv.org/pdf/2512.20318.pdf

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