Quantum Uncertainty Principle Research Articles

New Physics – The Negation of Einstein’s Theories of Relativity

From ontological considerations alone, a new revolutionary quantum dynamical-materialist dialectical (Quantum Dialectics or QD) perspective of space-time-matter-motion achieves a new quantitative mass-energy-velocity relation for any ponderable fundamental particles. This new relation, based on the quantum uncertainty principle, gives the mass m of any free fundamental particle at its origin, in terms of the inverse cube of its velocity v. This quantitative relation is different and is a negation of the one derived from mechanical-metaphysical (Classical Mechanics or CM) approach since Isaac Newton and its relativistic post-quantum formulation of Albert Einstein. This new mass-energy-velocity relation is in conformity with known knowledge and is appropriately substantiated by the recent reports that “high-energy photons of gamma radiation from a distant galaxy arrived at Earth four minutes after lower-energy photons, although they were apparently emitted at the same time”; and another saying that “the time delays of gamma-ray photons are inversely proportional to their energy”. These findings would contradict Einstein's theory of relativity, which says that all photons (particles of light) must move at the same speed c. QD considers matter-antimatter particle pairs in their elementary virtual forms and kinetic motion as an eternal quantum dynamical process of existence and non-existence (Being-Nothing) arising from abstract space and abstract time in a contradiction to each other. The virtual particles can transition to real particles through quantum tunnelling or when equivalent energy becomes available. Any existence in this view is a contradiction; vulnerable to change, motion, evolution, development etc. The aggregation of the elementary particles and their motions mediated by chance and necessity, give rise to the phenomenology of this infinite, eternal and ever-changing universe. The CM approach on the contrary is based on epistemology and considers that matter (ordinary + light) and motion have an esoteric origin from the mystery of the single act of creation by God; of a finite universe in the finite past; where the premise of conservation laws and “ex nihilo nihil fit” prevail. In its relativistic version, this approach makes a fundamental difference between ponderable matter and light and in fact question the existence of particulate matter. Two axiomatic assumptions, one by Newton of one-sided universal gravitational attraction and the other by Einstein that the velocity of light c is an absolute and universal constant has brought in unphysical objects and phenomena into scientific discourse leading to century-long confusion and scholastic debates among the physicists; with no end in sight!

Harmonic oscillator in the context of the extended uncertainty principle

At large-scale distances where the space-time is curved due to gravity, a nonzero minimal uncertainty in the momentum, [Formula: see text], is being estimated to emerge. The presence of minimal uncertainty in momentum allows a modification to the quantum uncertainty principle, which is known as the extended uncertainty principle (EUP). In this work, we handle the harmonic oscillator problem in the EUP scenario and obtain analytical exact solutions in classical and semi-classical domains. In the classical context, we establish the equations of motion of the oscillator and show that the EUP-corrected frequency is depending on the energy and deformation parameter. In the semi-classical domain, we derive the energy eigenvalue levels and demonstrate that the energy spectrum depends on [Formula: see text], as the feature of hard confinement. Finally, we investigate the impact of the EUP on the harmonic oscillator’s thermodynamic properties by using the EUP-corrected partition functions in the classical limit in the (A)dS backgrounds.

A Hessian-Free Method to Prevent Zero-Point Energy Leakage in Classical Trajectories.

The problem associated with the zero-point energy (ZPE) leak in classical trajectory calculations is well known. Since ZPE is a manifestation of the quantum uncertainty principle, there are no restrictions on energy during the classical propagation of nuclei. This phenomenon can lead to unphysical results, such as forming products without the ZPE in the internal vibrational degrees of freedom (DOFs). The ZPE leakage also permits reactions below the quantum threshold for the reaction. We have developed a new Hessian-free method, inspired by the Lowe-Andersen thermostat model, to prevent energy dipping below a threshold in the local-pair (LP) vibrational DOFs. The idea is to pump the leaked energy to the corresponding local vibrational mode taken from the other vibrational DOFs. We have applied the new correction protocol on the ab-initio ground-state molecular dynamics simulation of the water dimer (H2O)2, which dissociates due to unphysical ZPE spilling from high-frequency OH modes. The LP-ZPE method has been able to prevent the ZPE spilling of the OH stretching modes by pumping back the leaked energy into the corresponding modes, while this energy is taken from the other modes of the dimer itself, keeping the system as a microcanonical ensemble.

Tunnelling times, Larmor clock, and the elephant in the room

A controversy surrounding the “tunnelling time problem” stems from the seeming inability of quantum mechanics to provide, in the usual way, a definition of the duration a particle is supposed to spend in a given region of space. For this reason, the problem is often approached from an “operational” angle. Typically, one tries to mimic, in a quantum case, an experiment which yields the desired result for a classical particle. One such approach is based on the use of a Larmor clock. We show that the difficulty with applying a non-perturbing Larmor clock in order to “time” a classically forbidden transition arises from the quantum Uncertainty Principle. We also demonstrate that for this reason a Larmor time (in fact, any Larmor time) cannot be interpreted as a physical time interval. We provide a theoretical description of the quantities measured by the clock.

Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable

What does it take for real-deterministic $c$-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic $c$-valued variables out of the weak values obtained via a nonperturbing weak measurement of quantum operators with a postselection over a complete set of state vectors basis, which always satisfies the Kennard-Robertson-Schr\"odinger uncertainty relation. First, we introduce an auxiliary global random variable and couple it to the imaginary part of the weak value to convert the incompatibility between the quantum operator and the basis into the fluctuation of an ``error term,'' and then superimpose it onto the real part of the weak value. We show that this class of ``$c$-valued physical quantities'' provides a real-deterministic contextual hidden-variable model for the quantum expectation value of a certain class of operators. We then show that the Schr\"odinger and the Kennard-Robertson lower bounds can be obtained separately by imposing the classical uncertainty relation to the $c$-valued physical quantities associated with a pair of Hermitian operators. Within the representation, the complementarity between two incompatible quantum observables manifests the absence of a basis wherein the error terms of the associated two $c$-valued physical quantities simultaneously vanish. Furthermore, quantum uncertainty principle is captured by a specific irreducible epistemic restriction between the two $c$-valued physical quantities, foreign in classical mechanics, constraining the allowed form of their joint distribution. We then suggest an epistemic interpretation of the two terms decomposing the $c$-valued physical quantity as the optimal estimate under the epistemic restriction and the associated estimation error, and discuss the classical limit.

A New Algorithm to Solve the Generalized Nash Equilibrium Problem

We try a new algorithm to solve the generalized Nash equilibrium problem (GNEP) in the paper. First, the GNEP is turned into the nonlinear complementarity problem by using the Karush–Kuhn–Tucker (KKT) condition. Then, the nonlinear complementarity problem is converted into the nonlinear equation problem by using the complementarity function method. For the nonlinear equation equilibrium problem, we design a coevolutionary immune quantum particle swarm optimization algorithm (CIQPSO) by involving the immune memory function and the antibody density inhibition mechanism into the quantum particle swarm optimization algorithm. Therefore, this algorithm has not only the properties of the immune particle swarm optimization algorithm, but also improves the abilities of iterative optimization and convergence speed. With the probability density selection and quantum uncertainty principle, the convergence of the CIQPSO algorithm is analyzed. Finally, some numerical experiment results indicate that the CIQPSO algorithm is superior to the immune particle swarm algorithm, the Newton method for normalized equilibrium, or the quasivariational inequalities penalty method. Furthermore, this algorithm also has faster convergence and better off-line performance.

The uncertainty property of electromagnetic field in optical thin microcavity

The uncertainty property of electromagnetic field in thin microcavity is exactly derived in this paper. The intensity of the electromagnetic field at the center of thin microcavity is indeterminate while other positions intensity are determinate. The product of the deviation of momentum and the deviation of position at the center of thin microcavity is larger than a certain constant. By introducing the energy quantization condition (E=Nħω), we verify this constant has the same physical dimension as Planck’s constantħ and the value is an integral multiple of Planck’s constant, the integer is the smallest volume element at the center of thin microcavity. Furthermore, the perimeter of the smallest volume element is an integer multiple of the wavelength of the incident electromagnetic wave, which satisfies the assumption of DE Broglie matter wave. Our results are the first to find that the uncertainty property of electromagnetic field in thin microcavity and further derived the quantum uncertainty principle through the quantization of energy.

Charge transfer in positronium–proton collisions: comparison of classical and quantum-mechanical theories

Charge transfer in positronium–proton collisions is calculated using the quantum-mechanical convergent close-coupling method and classical trajectory Monte-Carlo method. Previous calculations revealed that at low incident energy the cross section in both theories scales as , where nPs and EPs are the principal quantum number and the center-of-mass energy of the incident positronium atom, respectively. However, the quantum cross section is systematically lower than classical one in absolute magnitude. To investigate the origin of this quantum suppression effect, we compare the charge transfer probabilities as functions of the impact parameter. We show that the quantum suppression in the cross section is mainly due to the low-impact parameter behavior of the probabilities governed by the quantum uncertainty principle.

Influence of intermediated measurements on quantum statistical complexity of single driven qubit

<sec> Recently, quantum statistical complexity based quantum information theory has received much attraction. Quantum measurements can extract the information from a system and may change its state. At the same time, the method of measuring multiple quantum is an important quantum control technique in quantum information science and condensed matter physics. The main goal of this work is to investigate the influence of multiple quantum measurements on quantum statistical complexity.</sec><sec> It is a fundamental problem to understand, characterize, and measure the complexity of a system. To address the issue, a damped and linearly driven two-level system (qubit) is taken for example. The driving amplitude and dephasing intensity are considered. By using the Lindblad equation and the Born-Markov approximation, the time evolution of the system can be obtained. Under multiple intermediated measurements, the system has a complex dynamic behavior. Quantum statistical complexity <inline-formula><tex-math id="M12">\begin{document}$C$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M12.png"/></alternatives></inline-formula> at the last moment <inline-formula><tex-math id="M13">\begin{document}$\tau$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M13.png"/></alternatives></inline-formula> is studied in detail. The results show that on the whole, <inline-formula><tex-math id="M14">\begin{document}$C$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M14.png"/></alternatives></inline-formula> first increases from zero to a maximal value with <inline-formula><tex-math id="M15">\begin{document}$\tau$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M15.png"/></alternatives></inline-formula> increasing, then decreases, and finally it approaches to zero. At first, the system is in a pure state and <inline-formula><tex-math id="M16">\begin{document}$C=0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M16.png"/></alternatives></inline-formula>. Finally, the system is in a maximally mixed state due to the interaction with the environment and <inline-formula><tex-math id="M17">\begin{document}$C=0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M17.png"/></alternatives></inline-formula> again. When the number of measurements <inline-formula><tex-math id="M18">\begin{document}$N$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M18.png"/></alternatives></inline-formula> is relatively small, <inline-formula><tex-math id="M19">\begin{document}$C$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M19.png"/></alternatives></inline-formula> fluctuates with <inline-formula><tex-math id="M20">\begin{document}$\tau$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M20.png"/></alternatives></inline-formula> increasing, but when <inline-formula><tex-math id="M21">\begin{document}$N$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M21.png"/></alternatives></inline-formula> is relatively large, the fluctuations disappear. Due to the quantum Zeno effect, as <inline-formula><tex-math id="M22">\begin{document}$N$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M22.png"/></alternatives></inline-formula> is larger, the variation of <inline-formula><tex-math id="M23">\begin{document}$C$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M23.png"/></alternatives></inline-formula> with <inline-formula><tex-math id="M24">\begin{document}$\tau$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M24.png"/></alternatives></inline-formula> is similar to that for the case of no intermediated measurement. Because of the quantum superposition principle, uncertainty principle, and quantum collapse, quantum measurement can disturb the system, so quantum statistical complexity <inline-formula><tex-math id="M25">\begin{document}$C$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M25.png"/></alternatives></inline-formula> exhibits a complex behavior.</sec><sec> In the quantum realm, the complexity of a system can be transformed into a resource. The quantum state needs creating, operating, or measuring. Therefore, all our results provide a theoretical reference for the optimal controlling of quantum information process and condensed matter physics. At the same time, the number of the degrees of freedom is two for the damped and linearly driven two-level system, so this system is simple and easy to study. The complexity of such a system can be tailored by properly tuning the driving strength. Therefore, the model can be used as a typical example to study the quantum statistical complexity.</sec>

Primordial power spectra of cosmological fluctuations with generalized uncertainty principle and maximum length quantum mechanics

The existence of the cosmological particle horizon as the maximum measurable length $l_{max}$ in the universe leads to a generalization of the quantum uncertainty principle (GUP) to the form $\Delta x \Delta p \geq \frac{\hbar}{2}\frac{1}{1-\alpha\Delta x^2} $, where $\alpha\equiv l_{max}^{-2}$. The effects of this GUP on simple quantum mechanical systems has been shown recently by one of the authors\cite{Perivolaropoulos:2017rgq} to be extremely small (beyond current measurements) due to the extremely large scale of the current particle horizon. This is not the case in the Early Universe during the quantum generation of the inflationary primordial fluctuation spectrum. We estimate the effects of such GUP on the primordial fluctuation spectrum and on the corresponding spectral index. We generalize the field commutation (GFC) relation to $[\varphi(k),\pi_{\varphi}(k')]$=$i\delta(k-k')\frac{1}{1-\mu\varphi^2(k)}$, where $\mu\sim \alpha^2\equiv l_{max}^{-4}$ is a GFC parameter, $\varphi$ denotes a scalar field and $\pi_{\varphi}$ denotes its canonical conjugate momentum. We obtain the predicted primordial perturbation spectrum as $P_S(k)=P_S^{(0)}(k)\left(1+\frac{\bar{\mu}}{k}\right)$ where $\bar{\mu}\equiv\mu V_* \simeq \sqrt{\alpha}= l_{max}^{-1}$ (here $V_*\simeq l_{max}^3$ is the volume corresponding to $l_{max}$) and $P_S^{(0)}(k)$ is the standard primordial spectrum obtained in the context of the Heisenberg uncertainty principle ($\mu=0$). We show that the predicted scalar spectral index is $n_s=1-\lambda-\frac{\bar{\mu}}{k}$ where $\lambda$ is a slow-roll parameter. Using observational constraints on the scale dependence of the spectral index $n_s$ we show that the $2\sigma$ range of $\alpha$ corresponds to $l_{max}\gtrsim 10^{26} m $ which is of the same order as the current particle horizon.

Error-Disturbance Trade-off in Sequential Quantum Measurements.

We derive a state-dependent error-disturbance trade-off based on a statistical distance in the sequential measurements of a pair of noncommutative observables and experimentally verify the relation with a photonic qubit system. We anticipate that this Letter may further stimulate the study on the quantum uncertainty principle and related applications in quantum measurements.

Novel designs for digital gates based on single electron devices to overcome the traditional limitation on speed and bit error rate

Single electron devices, SEDs, are distinguished among all new nano-scale electronic devices due to their very small size, ultra-low power consumption, and consistent technology. This paper is dedicated to digital application of single electron devices. The paper introduces a method for the speed enhancement and bit error rate reduction based on a new parallelism concept of tunneling paths and reduction of tunneling wait time. Using this method, one can go beyond the theoretical limitation of these devices in any specified technology which comes from quantum uncertainty principle. Correctness of this method is shown by analytical approaches and numerical method i.e. ensemble Monte-Carlo simulation, EMC. In this way, by proposing new designs and removal of maximum speed limitation in addition to BER reduction, this paper improves the seat of SEDs among other competing nano-scale devices.

Generalized Fourier transforms based on quantum uncertainty principle

Generalized Fourier transforms based on quantum uncertainty principle

Cosmological horizons, uncertainty principle, and maximum length quantum mechanics

The cosmological particle horizon is the maximum measurable length in the Universe. The existence of such a maximum observable length scale implies a modification of the quantum uncertainty principle. Thus due to non-locality of quantum mechanics, the global properties of the Universe could produce a signature on the behaviour of local quantum systems. A Generalized Uncertainty Principle (GUP) that is consistent with the existence of such a maximum observable length scale $l_{max}$ is $\Delta x \Delta p \geq \frac{\hbar}{2}\;\frac{1}{1-\alpha \Delta x^2}$ where $\alpha = l_{max}^{-2}\simeq (H_0/c)^2$ ($H_0$ is the Hubble parameter and $c$ is the speed of light). In addition to the existence of a maximum measurable length $l_{max}=\frac{1}{\sqrt \alpha}$, this form of GUP implies also the existence of a minimum measurable momentum $p_{min}=\frac{3 \sqrt{3}}{4}\hbar \sqrt{\alpha}$. Using appropriate representation of the position and momentum quantum operators we show that the spectrum of the one dimensional harmonic oscillator becomes $\bar{\mathcal{E}}_n=2n+1+\lambda_n \bar{\alpha}$ where $\bar{\mathcal{E}}_n\equiv 2E_n/\hbar \omega$ is the dimensionless properly normalized $n^{th}$ energy level, $\bar{\alpha}$ is a dimensionless parameter with $\bar{\alpha}\equiv \alpha \hbar/m \omega$ and $\lambda_n\sim n^2$ for $n\gg 1$ (we show the full form of $\lambda_n$ in the text). For a typical vibrating diatomic molecule and $l_{max}=c/H_0$ we find $\bar{\alpha}\sim 10^{-77}$ and therefore for such a system, this effect is beyond reach of current experiments. However, this effect could be more important in the early universe and could produce signatures in the primordial perturbation spectrum induced by quantum fluctuations of the inflaton field.

Quantum preparation uncertainty and lack of information

The quantum uncertainty principle famously predicts that there exist measurements that are inherently incompatible, in the sense that their outcomes cannot be predicted simultaneously. In contrast, no such uncertainty exists in the classical domain, where all uncertainty results from ignorance about the exact state of the physical system. Here, we critically examine the concept of preparation uncertainty and ask whether similarly in the quantum regime, some of the uncertainty that we observe can actually also be understood as a lack of information (LOI), albeit a lack of quantum information. We answer this question affirmatively by showing that for the well known measurements employed in BB84 quantum key distribution (Bennett and Brassard 1984 Int. Conf. on Computer System and Signal Processing), the amount of uncertainty can indeed be related to the amount of available information about additional registers determining the choice of the measurement. We proceed to show that also for other measurements the amount of uncertainty is in part connected to a LOI. Finally, we discuss the conceptual implications of our observation to the security of cryptographic protocols that make use of BB84 states.

Statistics of library website traffic and uncertainty principle

This article considers the features in statistics of library website traffic that includes a deal of bots of about 85%. It has been assumed that the features may be interpreted in terms of quantum uncertainty principle. A conclusion has been made that application of computer technologies to provide an access to information sources and, at the same time, to log the demand for the information on the Internet has a result that dramatic distortions and possible tweaks in the figures may occur at the discretion of somebody.

Uncertainty, joint uncertainty, and the quantum uncertainty principle

Historically, the element of uncertainty in quantum mechanics has been expressed through mathematical identities called uncertainty relations, a great many of which continue to be discovered. These relations use diverse measures to quantify uncertainty (and joint uncertainty). In this paper we use operational information-theoretic principles to identify the common essence of all such measures, thereby defining measure-independent notions of uncertainty and joint uncertainty. We find that most existing entropic uncertainty relations use measures of joint uncertainty that yield themselves to a small class of operational interpretations. Our notion relaxes this restriction, revealing previously unexplored joint uncertainty measures. To illustrate the utility of our formalism, we derive an uncertainty relation based on one such new measure. We also use our formalism to gain insight into the conditions under which measure-independent uncertainty relations can be found.

The uncertainty principle enables non-classical dynamics in an interferometer.

The quantum uncertainty principle stipulates that when one observable is predictable there must be some other observables that are unpredictable. The principle is viewed as holding the key to many quantum phenomena and understanding it deeper is of great interest in the study of the foundations of quantum theory. Here we show that apart from being restrictive, the principle also plays a positive role as the enabler of non-classical dynamics in an interferometer. First we note that instantaneous action at a distance should not be possible. We show that for general probabilistic theories this heavily curtails the non-classical dynamics. We prove that there is a trade-off with the uncertainty principle that allows theories to evade this restriction. On one extreme, non-classical theories with maximal certainty have their non-classical dynamics absolutely restricted to only the identity operation. On the other extreme, quantum theory minimizes certainty in return for maximal non-classical dynamics.

Security analysis on some experimental quantum key distribution systems with imperfect optical and electrical devices

In general, quantum key distribution (QKD) has been proved unconditionally secure for perfect devices due to quantum uncertainty principle, quantum noncloning theorem and quantum nondividing principle which means that a quantum cannot be divided further. However, the practical optical and electrical devices used in the system are imperfect, which can be exploited by the eavesdropper to partially or totally spy the secret key between the legitimate parties. In this article, we first briefly review the recent work on quantum hacking on some experimental QKD systems with respect to imperfect devices carried out internationally, then we will present our recent hacking works in details, including passive faraday mirror attack, partially random phase attack, wavelength-selected photon-number-splitting attack, frequency shift attack, and single-photon-detector attack. Those quantum attack reminds people to improve the security existed in practical QKD systems due to imperfect devices by simply adding countermeasure or adopting a totally different protocol such as measurement-device independent protocol to avoid quantum hacking on the imperfection of measurement devices [Lo, et al., Phys. Rev. Lett., 2012, 108: 130503].

Schrödinger wave equation for confinement of photon in a waveguide

In the present work, electromagnetic field confinement in a subwavelength waveguide structure is obtained using concepts of quantum mechanics and uncertainty principle. Semi-macroscopic considerations of field interaction at the dielectric interfaces are used in this work. The modal field profile in the subwavelength waveguide is obtained by considering the photon as a particle in the waveguide having finite probability of tunneling. Thus, uncertainty of position is assigned to it. The momentum uncertainty is calculated from position uncertainty. Schrödinger wave equation for the photon is written by incorporating position-momentum uncertainty. The equation is solved and the field distribution in the waveguide is obtained. The field distribution and power confinement is compared with conventional waveguide theory. They were found in good agreement with each other.