Quantum Mechanical Perturbation Theory Research Articles

Challenges in sensemaking and reasoning in the context of degenerate perturbation theory in quantum mechanics

We discuss an investigation of student sensemaking and reasoning in the context of degenerate perturbation theory (DPT) in quantum mechanics. We find that advanced undergraduate and graduate students in quantum physics courses often struggled with expertlike sensemaking and reasoning to solve DPT problems. The sensemaking and reasoning were particularly challenging for students as they tried to integrate physical and mathematical concepts to solve DPT problems. Their sensemaking showed local coherence but lacked global consistency with different knowledge resources getting activated in different problem-solving tasks even if the same concepts were applicable. Depending upon the issues involved in the DPT problems, students were sometimes stuck in the “physics mode” or “math mode” and found it challenging to coordinate and integrate the physics and mathematics appropriately to solve quantum mechanics problems involving DPT. Their sensemaking shows the use of various reasoning primitives. It also shows that some advanced students struggled with self-monitoring and checking their answers to make sure they were consistent across different problems. Some also relied on memorized information, invoked authority, and did not make appropriate connections between their DPT problem solutions and the outcomes of experiments. Advanced students in quantum mechanics often displayed analogous patterns of challenges in sensemaking and reasoning as those that have been found in introductory physics. Student sensemaking and reasoning show that these advanced students are still developing expertise in this novel quantum physics domain as they learn to integrate physical and mathematical concepts. Published by the American Physical Society 2024

Second-order dissociation and transition of heavy quarkonia in the quark-gluon plasma

We revisit the dissociation of heavy quarkonia by thermal partons at the next-to-leading order (NLO, also known as inelastic parton scattering dissociation) in the quark-gluon plasma (QGP). Utilizing the chromoelectric dipole coupling from quantum chromodynamics (QCD) multipole expansion as an effective Hamiltonian, this has been conducted in the approach of second-order quantum mechanical perturbation theory, which allows us to systematically incorporate the bound state wave functions. Employing the quarkonium wave functions and binding energies obtained from an in-medium potential model, we then numerically evaluate the dissociation cross sections and rates for various charmonia and bottomonia, where the infrared and collinear divergences are regularized by the thermal masses of medium partons. We demonstrate that distinct from the leading order (LO, also known as gluo-dissociation) counterparts peaking at relatively low gluon energy and falling off thereafter, the NLO cross sections first grow and then nearly saturate as the incident parton energy increases, as a result of the outgoing parton carrying away the excess energy. The resulting NLO dissociation rates increase with temperature and take over from the LO counterparts toward high temperatures, similar to pertinent findings from previous studies. We also evaluate the in-medium second-order transition between different bound states, which may contribute to the total thermal decay widths of heavy quarkonia in the QGP. Published by the American Physical Society 2024

A link between static and dynamical perturbation theory

Dynamics, the physical change in time and a pillar of natural sciences, can be regarded as an emergent phenomenon when the system of interest is part of a larger, static one. This ‘relational approach to time’, in which the system’s environment provides a temporal reference, does not only provide insight into foundational issues of physics, but holds the potential for a deeper theoretical understanding as it intimately links statics and dynamics. Reinforcing the significance of this connection, we demonstrate, based on recent progress (Gemsheim and Rost 2023 Phys. Rev. Lett. 131 140202), the role of emergent time as a vital link between time-independent and time-dependent perturbation theory in quantum mechanics. We calculate first order contributions, which are often the most significant, and discuss the issue of degenerate spectra. Based on our results, we envision future applications for the calculation of dynamical phenomena based on a single pure energy eigenstate.

Research on spacecraft in orbit perception based on artificial neural networks and digital twin technology using grating arrays

In order to improve the safety of spacecraft, the research on artificial neural network and digital twin technology based on, to our best knowledge, a novel fiber Bragg grating (FBG) sensor array is proposed for intelligent sensing monitoring of spacecraft on-orbit collisions. Femtosecond FBG arrays were fabricated on the novel oxide-doped fiber by point-by-point writing technique. The femtosecond FBG is analyzed using the time-dependent perturbation theory of quantum mechanics. The FBG array can achieve high-temperature measurement of 1100 °C and large strain measurement of 15000 µε. The sensing arrays were deployed on the surface of the spacecraft. Constructed the multi-layer perceptron neural network structure and convolutional neural network structure. 1200 samples were trained. Conducted model accuracy testing. The accuracy rate is above 98%, and accuracy verification has been implemented. The digital twin model was designed based on various data such as strain and temperature of the spacecraft structure under impact monitored by FBG sensors. A precise mapping has been formed between the physical entities of spacecraft and digital twins. Empower spacecraft with functions such as self-monitoring, judgment, and response. To ensure the stable and safe operation of spacecraft.

Relativistic Perturbation Theory for Black-Hole Boson Clouds.

We develop a relativistic perturbation theory for scalar clouds around rotating black holes. We first introduce a relativistic product and corresponding orthogonality relation between modes, extending a recent result for gravitational perturbations. We then derive the analog of time-dependent perturbation theory in quantum mechanics, and apply it to calculate self-gravitational frequency shifts. This approach supersedes the nonrelativistic "gravitational atom" approximation, brings close agreement with numerical relativity, and has practical applications for gravitational-wave astronomy.

THE PROBABILISTIC MODEL OF ULTRASHORT PULSES SCATTERING ON A FREE ELECTRON

We devoted our study to the generalization of the traditional approach for the description of photon scattering on free electrons in the case of ultrashort laser pulses (USLP). In the framework of the second order of quantum mechanical perturbation theory with the use of the Klein–Nishina formula, we derived the expression for the total scattering probability during the whole time of the pulse action that is applicable in the relativistic limit. The redshift of scattered pulse spectra at the scattering angle increase in the relativistic case was studied. The trends of the total scattering probability on the USLP duration were categorized.

On the connection between perturbation theory and new semiclassical expansion in quantum mechanics

On the connection between perturbation theory and new semiclassical expansion in quantum mechanics

Perturbation theory without power series: Iterative construction of non-analytic operator spectra

It is well known that quantum-mechanical perturbation theory often gives rise to divergent series that require proper resummation. Here I discuss simple ways in which these divergences can be avoided in the first place. Using the elementary technique of relaxed fixed-point iteration, I obtain convergent expressions for various challenging ground-states wave functions, including quartic, sextic and octic anharmonic oscillators, the hydrogenic Zeeman problem, and the Herbst-Simon Hamiltonian (with finite energy but vanishing Rayleigh-Schrödinger coefficients), all at arbitarily strong coupling. These results challenge the notion that non-analytic functions of coupling constants are intrinsically “non-perturbative”. A possible application to exact diagonalization is briefly discussed.

Using Topology Concept of Mathematics and Finite Element Method of Engineering to Develop a Mathematical Model of Metabolism in medicine in order to Control Various Chronic Diseases and their Complications via Overall Health Conditions Improvement Based on GH-Method: Math-Physical Medicine (No. 453)

The author spent 11 years with more than 30,000 hours to self-study and research chronic diseases and lifestyle details in order to control his severe type 2 diabetes (T2D) and various diabetic complications. After self-studying endocrinology and food nutrition for 4 years, from 2010 to 2013, he applied the learned medicine and food knowledge combined with his background of four academic disciplines and practical experiences on mathematics, physics, engineering, and computer science (big data analytics and artificial intelligence) to develop a mathematical model of metabolism during the entire year of 2014. He continued with his development effort of several prediction tools for weight, fasting plasma glucose (FPG), postprandial plasma glucose (PPG), daily glucose, and HbA1C in 2015-2016. Starting from 2017, he started to write papers based on his findings from his research and then published them in various journals. This particular metabolism model includes 10 major categories: 4 medical conditions including weight (M1), glucose (M2), blood pressure (M3), lipids (M4); with 6 lifestyle details including exercise (M5), water intake (M6), sleep (M7), stress (M8), food and meal (M9), and life routines regularity (M10). These 10 categories contain approximately 500 input and output elements. For example, he spent 9 months on developing the stress category with 34 elements. The total number of data collected and processed during the past 10 years have already exceeded 2 million. During 2015 to 2021, for a period of 6.5 years, he applied many theories and techniques from mathematics, physics, and engineering into his medical research work. This included topology, nonlinear algebra, finite element method, linear elastic theory, wave theory, energy theory, quantum mechanics, optical physics, statistics, machine learning, big data analytics, and artificial intelligence (AI). These various theories and techniques have effectively served as his research tools in order to complete many of the medical research tasks. As a result, his collected personal data of both biomarkers and lifestyle details were utilized to illustrate the development process of building a metabolism model of medicine using topology concept of mathematics, finite element method of engineering along with other applicable theories and principles of physics (applied math) and engineering (applied physics). This paper describes the author’s selected approach of building a mathematical model of metabolism. It uses the topology concept, part of finite element engineering modeling technique, applicable theories and principles from physics and engineering, such as optical physics, energy theory, wave theory, linear elasticity, perturbation theory of quantum mechanics, and various computer science and AI tools to investigate and develop a rather complicated metabolism model in a highly quantitative manner. Optimistically and pleasantly, it has actually achieved certain high prediction accuracy in results based on a simple concept but rather general scope of “Metabolism”.

Combined Experimental and Computational Prediction of the Piezoresistivity of Alkali-Activated Inorganic Polymers

The incorporation of smart building materials into construction will improve the working life of structures and infrastructure around the globe. Unfortunately, conventional smart building materials are cost-prohibitive because of the self-sensing additives required. Alkali-activated inorganic polymers are a promising low-cost and environmentally friendly alternative that exhibit intrinsic self-sensing properties, without the need for self-sensing additives. An improved methodology has been developed to quantify the self-sensing piezoresistivity of these materials. Experimental measurements reveal a strong intrinsic piezoresistivity up to 12%. The results agree with a first-principles model of the theoretical piezoresistivity of an alkali-activated inorganic polymer from the quantum mechanical perturbation theory. This first-of-its-kind computation provides a mechanistic explanation for the origin of intrinsic piezoresistivity in inorganic polymers.

Time-Dependent Laser Cavity Perturbation Theory: Exploring Future Nano-Structured Photonic Devices in Semi-Analytic way

We present a theoretical framework, which successfully combines two different fields of photonics: i) the laser rate equations and ii) the cavity perturbation theory, focusing particularly on micro-cavity lasers with optical anisotropies. Our approach is formally analogous to quantum-mechanical time-dependent perturbation theory, in which however the gain medium and permittivity tensor distribution are perturbed instead of the Hamiltonian. Using the general vectorial Maxwell-Bloch equations as a starting point, we derive polarization-resolved coupled-mode equations, in which all relevant geometric and anisotropy-related laser parameters are imprinted in its coefficients. Closed-form coupled-mode equations offer physical insights like rate equations approaches and the precision comparable to brute-force numeric routines, thus being the time-saving alternative to finite-difference time-domain methods. The main advantage is that one calculates numerically the shapes of cold-cavity modes used to derive coupled-mode equations for one set of parameters and the broad landscape of parameters of interest is further studied in a perturbative way. This makes the method particularly interesting for semi-analytic studies of state-of-art devices such as the photonic crystal lasers, the liquid-crystal lasers or specifically spin-lasers, in which the interplay between injected spin and cavity birefrigence creates very promising platform for ultrafast data transfer technologies.

Viscoelastic and Viscoplastic Glucose Theory (VGT #22): Applying the Concepts of Viscoelasticity, Viscoplasticity and Perturbation Theories to Predict Cancer Risk Percentages using the overall Lifestyle Scores and Calculated Cancer Risk Change Rates with Corresponding Lifestyle Scores as the Respective Viscosity Factors along with Studying the Relationship between Cancer Risks and Diabetic Glucoses Based on the GH-Method: Math-Physical Medicine (No. 601)

Viscoelastic and Viscoplastic Glucose Theory (VGT #22): Applying the Concepts of Viscoelasticity, Viscoplasticity and Perturbation Theories to Predict Cancer Risk Percentages using the overall Lifestyle Scores and Calculated Cancer Risk Change Rates with Corresponding Lifestyle Scores as the Respective Viscosity Factors along with Studying the Relationship between Cancer Risks and Diabetic Glucoses Based on the GH-Method: Math-Physical Medicine (No. 601)

Viscoelastic and Viscoplastic Glucose Theory (VGT #17): Selecting 12 Cases and Applying Theories of Viscoelasticity and Viscoplasticity from Engineering and Perturbation Theory from Quantum Mechanics to Predict the Behavior of an Output Biomarker as the Strain (ε or Behavior, Symptom) by Using the Strain Rate (dε/dt) Multiplying with a Correlated Input Biomarker of Stress Value (σ or Stressor, Cause) as

The author has collected ~3 million data regarding his health condition and lifestyle details over the past 12 years. He spent the entire year of 2014 to develop a metabolism index (MI) model using topology concept, nonlinear algebra, algebraic geometry, and finite element method. This MI model contains various measured biomarkers and recorded lifestyle details along with their induced new biomedical variables for an additional ~1.5 million data. Body weight, glucose, blood pressure, heart rate, lipids, body temperature, and blood oxygen level, along with important lifestyle details, including diet, exercise, sleep, stress, water intake, and daily life routines are included in his MI database. His developed MI model has a total of 10 categories covering approximately 500 detailed elements that constitute his defined “metabolism model” which are the building blocks or root causes for diabetes and other chronic disease complications, including but not limited to cardiovascular disease (CVD), chronic heart disease (CHD), chronic kidney disease (CKD), retinopathy, neuropathy, foot ulcer, and hypothyroidism. The end result of the MI development work is a combined MI value within any selected time period with 73.5% as its dividing line between a healthy and unhealthy state. The MI serves as the foundation to many of his follow-up medical research work. During the period from 2015 to 2017, he focused his research on type 2 diabetes (T2D), especially glucoses, including fasting plasma glucose (FPG), postprandial plasma glucose (PPG), estimated average glucose (eAG), and hemoglobin A1C (HbA1C). During the following period from 2018 to 2022, he concentrated on researching medical complications resulting from diabetes, chronic diseases, and metabolic disorders which include heart problems, stroke, kidney problems, retinopathy, neuropathy, foot ulcer, diabetic skin fungal infection, hypothyroidism, and diabetic constipation, cancer, and dementia. He also developed a few mathematical risk models to calculate the probability percentages of developing various diabetic complications. Recently, he has applied theories of elasticity, plasticity, viscoelasticity, and viscoplasticity from engineering along with the theories of wave and energy from physics to conduct his research on output biomarkers (symptoms or behaviors) resulting from a suspected or identified input biomarkers (causes or stressors). As a result, he has written 16 different articles in this research area. In this article, he utilizes the perturbation theory of quantum mechanics in modern physics to develop a set of predicted output biomarkers using the identified input biomarkers from his previous research work. Next, he compares his predicted biomarkers against his measured biomarkers using their calculated correlation coefficients and prediction accuracies.

Qubit decoherence and symmetry restoration through real-time instantons

A parametrically driven quantum oscillator, stabilized by a nonlinear dissipation, exhibits a spontaneous breaking of the parity symmetry. It results in the quantum bi-stability, corresponding to a Bloch sphere of dark states. This makes such a driven-dissipative system an attractive candidate for a qubit. The parity symmetry breaking is exact both on the classical level and within the quantum mechanical perturbation theory. Here we show that non-perturbative quantum effects lead to the symmetry restoration and result in exponentially small but finite qubit decoherence rate. Technically the symmetry restoration is due to real time instanton trajectories of the Keldysh path integral, which represents the Lindbladian evolution of the driven-dissipative oscillator.

Effect of ion irradiation on thermal conductivity of phosphorene and underlying mechanism

Defects produced by ion irradiation can effectively modulate many physical properties of phosphorene. In this paper, the molecular dynamics method is used to simulate the ion irradiation process of phosphorene. The relations between the formation probability of defects and the energy of incident ions, ion species and incident angle of ions are revealed. The non-equilibrium molecular dynamics simulation is used to calculate the thermal conductivity of irradiated phosphorene. The effects of the energy of ions, the irradiation dose, the type of ions and the incident angle of ions on the thermal conductivity of phosphorene are systematically investigated. The influence of the vacancies on the phonon participation rate of phosphorene is studied by lattice dynamics method, and the spatial distribution of localized modes is demonstrated. According to the quantum-mechanical perturbation theory and bond relaxation theory, we point out that the dominant physical mechanism of vacancy defects which significantly reduce the thermal conductivity of phosphorene is the strong scattering of phonons by the low-coordinated atoms near the vacancies. This study provides a theoretical basis for tuning the heat transport properties of phosphorene by defect engineering.

Photocurrent in plasmonic nanofibers

We developed a theory of photoresponse and photocurrent in photonic nanofibers. Photonic nanofiber is a compound system doped with an ensemble of quantum dots and metallic nanoparticles, where they interact with each other via the dipole–dipole interaction. The bound states of the confined probe photons in the nanofiber hybrid are calculated using the transfer matrix method based on Maxwell’s equations. It is found that the density of states of photons in the nanofiber depends on the dipole–dipole interaction coupling. The nonradiative decay rate due to dipole–dipole interaction rates is calculated using the quantum mechanical perturbation theory. An analytical expression of the photoresponse coefficient and the photocurrent is calculated using the density matrix method. We predicted that the quenching in photocurrent is due to the dipole–dipole interaction. Furthermore, we have shown that the photoluminescence quenching increases as the strength of the dipole–dipole coupling increases. We also compared our theory with the experimental results of the photocurrent in a nanofiber doped with Al metallic nanoparticle nanodisks and Ge/Si quantum dots. A good agreement between theory and experiment is found. Our analytical expressions can be used by experimentalists to perform new types of experiments and for inventing new types of nanosensors and nanoswitches.

Time-independent perturbation theory with Lagrange multipliers

We derive the formulas for the energy and wavefunction of the time-independent Schrödinger equation with perturbation in a compact form. Unlike the conventional approaches based on Rayleigh–Schrödinger or Brillouin–Wigner perturbation theories, we employ a recently developed approach of matrix-valued Lagrange multipliers that regularizes an eigenproblem. The Lagrange-multiplier regularization makes the characteristic matrix for an eigenproblem invertible. After applying the constraint equation to recover the original equation, we find the solutions of the energy and wavefunction consistent with the conventional approaches. This formalism does not rely on an iterative way and the order-by-order corrections are easily obtained by taking the Taylor expansion. The Lagrange-multiplier regularization formalism for perturbation theory presented in this paper is completely new and can be extended to the degenerate perturbation theory in a straightforward manner. We expect that this new formalism is also pedagogically useful to give insights on the perturbation theory in quantum mechanics.

A pendulum with a broken cylindrical symmetry: normal modes and beats

A pendulum, which has the cylindrical symmetry around its suspension wire, has two degenerate normal modes. We break this symmetry by putting a small magnet in the pendulum body and by running the pendulum in the presence of an inhomogeneous magnetic field. The force due to the magnetic field gradient modifies the equilibrium position and the spatial derivatives of this force define the normal modes of the pendulum with two different oscillation frequencies. We describe this system theoretically and we test these results by a series of experiments. This system provides a classical analogue of the degenerate perturbation theory in quantum mechanics.

Scattering of ultrashort laser pulses on plasmons in a Maxwellian plasma

On the basis of equations obtained in the framework of second-order quantum-mechanical perturbation theory, the standard approach to the calculation of scattering radiation probability is extended to the case of ultrashort laser pulses. We investigate the mechanism of the appearance of plasmon peaks in the spectrum of the plasma form factor for different parameters of the problem. For the case in which scattering on plasmons dominates over scattering on electron density fluctuations caused by chaotic thermal motion, we derive analytical expressions describing the scattering probability of ultrashort laser pulses on plasmons. Together with this, we obtain a simple expression connecting the frequency of scattered radiation and the energy transmitted from the incident pulse to plasmon, and vice versa. In considering the scattering probability, our emphasis is on the dependence on the pulse duration. We assess in detail the trends of this dependence for various relations between pulse carrier frequency and plasmon energy.

Hamiltonian truncation in Anti-de Sitter spacetime

Quantum Field Theories (QFTs) in Anti-de Sitter (AdS) spacetime are often strongly coupled when the radius of AdS is large, and few methods are available to study them. In this work, we develop a Hamiltonian truncation method to compute the energy spectrum of QFTs in two-dimensional AdS. The infinite volume of constant timeslices of AdS leads to divergences in the energy levels. We propose a simple prescription to obtain finite physical energies and test it with numerical diagonalization in several models: the free massive scalar field, ϕ4 theory, Lee-Yang and Ising field theory. Along the way, we discuss spontaneous symmetry breaking in AdS and derive a compact formula for perturbation theory in quantum mechanics at arbitrary order. Our results suggest that all conformal boundary conditions for a given theory are connected via bulk renormalization group flows in AdS.