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In the present work neural network has been developed for the prediction of Vertical Total Electron Content (VTEC) over Bhopal (23.2°N, 77.4°E & MLAT 14.2°N), an equatorial anomaly crest region. The study is based on the VTEC data recorded by a multi-frequency multi-constellation GNSS receiver installed at Bhopal during May 2016 to August 2017. Neural Network (NN) consists of artificial neurons that are used to learn data patterns adjusting weights between input, hidden and output layers by computational modelling. In order to obtain the optimal number of neurons in the hidden layer, the performance of four systems of neural network have been tested. Out of which, the fourth neural network has been used to predict VTEC with 7 input neurons in the present study. The number of hidden layer neurons is a factor that affects the performance of the trained networks. The input layer of the network includes the year, day of the year, hour of the day, geographic latitude, geographic longitude, SSN and IRI NmF2. We have compared the TEC obtained from the neural network prediction (NN TEC), GPS derived TEC (GPS TEC) and TEC from IRI-2016 model (IRI TEC) and analysis has been made. We observed that the Root Mean Square Error (RMSE) decreases with increase in the number of hidden layer neurons. The RMSE comes out to be lowest (1.96 TECU), when the number of hidden layer neurons are 41.The TEC predicted from the NN model and estimated from GPS receiver have same trend at 01:00 and 07:00 UT. The modelled NN TEC values are in good agreement with GPS TEC values for all seasons and 0.99 correlation coefficient was calculated for summer months, but IRI TEC always underestimates GPS TEC in all seasons. The NN model excellently predicted the TEC values of August 2017 by putting the credentials of August 2016 and only 0.55–32.16% relative errors were estimated between them whereas it reaches up to 93.86% with IRI TEC. Thus, the result shows that the proposed NN model can predict the diurnal and seasonal GPS TEC more accurate than IRI over anomaly crest region Bhopal. However, the inclusion of IRI-NmF2 as input layer neuron increases the network performance.

We present the probability distribution function of the nodal domain number for the right-isosceles triangle billiard. This is the first explicit result for a nonseparable plane polygonal billiard. The distribution function has a peak at $$2/\pi$$ (the same as for integrable billiards) and also at $$2\sqrt{2}/\pi$$ . The only exact results previously known in this direction correspond to integrable billiards that are separable.

Представлена функция вероятностного распределения числа областей постоянного знака для собственных функций бильярда в форме прямоугольного равнобедренного треугольника. Это первый явный результат для несепарабельного плоского многоугольного бильярда. Функция распределения имеет максимум при $2/\pi$ (как для интегрируемых бильярдов), а также при $2\sqrt{2}/\pi $. Единственные точные результаты в этом направлении, известные к настоящему времени, соответствуют интегрируемым бильярдам, которые являются сепарабельными.

Nighttime enhancement in the Vertical Total Electron Content (VTEC) of the ionosphere near equatorial ionization anomaly crest region in the Indian ionospheric sector has been studied during May 2016 to August 2017, a solar minimum period. VTEC data is obtained from a multi-frequency multi-constellation global navigation satellite system (GNSS) receiver installed at Bhopal (23.2° N, 77.4° E & MLAT 14.2° N). Only those enhancements which show VTEC enhancement ≥10% compared to the background has been considered in the study. Out of the total 105 cases of enhancement recorded during this period, 86 events occurred during pre-midnight and 19 during post-midnight hours. Enhancements in VTEC occurred in all seasons; more frequent however during summer than winter and equinox months. Some of the nighttime enhancements in VTEC resulted in the scintillations in the satellite signals as well. Peak amplitude of the enhancement maximizes in summer whereas its half amplitude duration maximizes during equinox month. The number of occurrences of nighttime enhancement in VTEC shows negative correlation with peak amplitude while they are positively correlated with half amplitude duration. We have found that the peak amplitude of nighttime enhancements in VTEC shows strong positive correlation while percentage occurrence shows strong negative correlation with solar and magnetic activities respectively. Other characteristics like half amplitude duration, time of peak and occurrence time of nighttime enhancement in VTEC show positive correlation with solar activity while they are insignificant with magnetic activity. The peak amplitude of pre-midnight enhancements in VTEC is highly correlated with the strength of equatorial electrojet strength. The results are compared with the earlier observations and possible mechanism is also discussed.

We carry out a study on various configurations admissible in a superdeformed nucleus, 152Dy. Our interest is in extracting the nature of spectral fluctuations and compare with those of random matrix theory. In particular, we study the distribution functions of ratios of the nearest-neighbor spacing of the energy levels. This does not require usage for the average level density. For several variations of shapes, the energy levels have been calculated in a self-consistent manner, and their fluctuations show that the energy levels are correlated in a manner expected for the eigenvalues of quantum systems which are classically non-integrable. Due to limited size of level sequences in each symmetry class, our results may be termed only indicative.

In this paper, we present a systematic treatment of a [Formula: see text] qubit in the presence of a time-dependent external flux. A gauge-invariant Lagrangian and the corresponding Hamiltonian are obtained. The effect of the flux noise on the qubit relaxation is obtained using the perturbation theory. Under a time-dependent drive of sinusoidal form, the survival probability, and transition probabilities have been studied for different strengths and frequencies. The driven qubit is shown to possess coherent oscillations among two distinct states for a weak to moderate strength close to resonant frequencies of the unperturbed qubit. The parameters can be chosen to prepare the system in its ground state. This feature paves the way to prolong the lifetime by combining ideas from weak measurement and quantum Zeno effect. We believe that this is an important variation of a topologically protected qubit which is tunable.

$\ensuremath{\alpha}$-Particle evaporation spectra were measured at backward angles in $^{16}\mathrm{O}+\phantom{\rule{0.16em}{0ex}}^{94,100}\mathrm{Mo}$ reactions at multiple beam energies. Inverse level-density parameter, $K$, was determined for Cd nuclei of different isospins by simulating high-energy tail of the measured $\ensuremath{\alpha}$-particle evaporation spectra with statistical model code pace2. An overall increasing behavior of the $K$ value is observed with increasing temperature in the range of 1 to 2.5 MeV. It is observed that in the temperature region below 1.8 MeV, the parameter $K$ is higher for the neutron-rich nuclei by around 1 MeV. Semiclassical calculations were performed which treat single-particle level density of neutron and proton on different footing and this can account for the isospin effects. These calculations reproduce the $K$ values as determined from the statistical model analysis of $\ensuremath{\alpha}$-particle evaporation spectra. Present results clearly demonstrate the isospin dependence of the level-density parameter as conjectured by theoretical works.

Cluster-decay half-lives ( T 1 / 2 c , th ( s ) ) of even–even nuclei are calculated employing interaction potential involving Coulomb and proximity potentials and, the obtained results are compared with the experimental ones. In order to study cluster radioactivity in even–even superheavy nuclei (SHN), we evaluate the required disintegration energies (Q c -values) utilizing periodic-orbit theory within microscopic-macroscopic formalism and hence, predict the logarithmic values of T 1 / 2 c , th 's for even–even 298−312Og and 296−310120 nuclei. Among various isotopes of daughter nuclei, products produced with minimum value of Gibbs free energy are chosen for our investigation. Also, an analysis of the branching ratios enables us to determine the dominant mode of decay among α − decay, spontaneous fission (SF) and cluster-decay in these SHN. The emissions of 90Kr from 298Og and 94Sr cluster from 296,298120 SHN are found to be the dominant mode of decay. Our results are in agreement with those obtained using UDL method except for few cases. We believe that this work can play a significant role in the experiments leading to identification of exotic nuclei.

We consider a simple harmonic oscillator with non-Hermitian term and study it classically and quantum mechanically. We conclude that this version of oscillator, which breaks parity and time reversal, displays all the features possessed by the usual harmonic oscillator. In particular, we calculate its spectrum, adiabatic invariance and Wigner functions, and show that there is a consistency between the classical and quantum descriptions.

α-decay half-lives ( T 1 / 2 α -values) of even–even and superheavy nuclei (SHN) are determined employing interaction potential involving Coulomb and proximity potentials. The obtained results are compared with the experimental ones and to assess the precision of the present model in reproducing the experimental half-lives, a comparison is drawn with the other methods as well. We require disintegration energy (Q α -values) as an input in order to predict the ( T 1 / 2 α -values) of an experimentally unknown SHN. So, utilizing periodic-orbit theory within microscopic-macroscopic formalism, we have calculated the binding energies and thereafter, Q α -values of known SHN. The results are found to be in good agreement with the experimental ones. Subsequently, we predict the logarithmic values of T 1 / 2 α ’s for even–even unknown SHN whose atomic numbers lie in 118 ⩽ Z ⩽ 126 regime and compare our results with those calculated employing other theoretical methods. Also, the study of spontaneous fission half-lives and branching ratios leads us to predict the SHN which can be identified through α-decay chains in the laboratories. We believe that this work can play a significant role in the experiments leading to the synthesis and identification of new superheavy elements.

We demonstrate that pairing phase transition (superfluid to normal) can be described quite generally in terms of the thermodynamical properties after verifying the obtained level densities with the available experimental data for [Formula: see text]- isotopes. Periodic-orbit theory conveniently connects the oscillatory part of level density to the underlying classical periodic orbits and hence, leads to the shell effects in the single-particle level density. Such methods incorporated with pairing effects can be used effectively to study the phase transitions in [Formula: see text]-isotopes. In addition to this, an interplay between pairing correlations and the shell effects has been understood by analyzing the results obtained for the critical temperatures and shell structure energies for [Formula: see text] isotopes. A relation between variation in critical temperatures caused by shell effects and the shell structure energies determined with and without pairing effects has been established. Furthermore, the systematics of the heat capacity (giving a clear signature of pairing phase transition) as function of temperature for these nuclei are investigated as well.

We study the phenomenon of neutrino spin-flavor oscillations due to solar magnetic fields. This allows us to examine how significantly the electron neutrinos produced in the solar interior undergo a resonant spin-flavor conversion. We construct analytical models for the solar magnetic field in all the three regions of the Sun. Neutrino spin-flavor oscillations in these magnetic fields are examined by studying the level crossing phenomenon and comparing the two cases of zero and non-zero vacuum mixing respectively. Results from the Borexino experiment are used to place an upper limit on the magnetic field in the solar core. Related phenomena such as effects of matter on neutrino spin transitions and differences between Dirac and Majorana transitions in the solar magnetic fields are also discussed., Comment: Published in Res.Astron.Astrophys

By proposing a model Hamiltonian in the first quantised form we investigate the chaotic dynamics of $$\alpha $$ -helical proteins by taking into account the quadrupole–quadrupole-type interaction. The dynamics is studied by deriving Hamilton’s equations of motion and by plotting the time-series evolution and phase-space trajectories. Chaotic trajectories are observed in the phase-space plots. The effect of the interaction parameters on the stability of proteins is also discussed.

Coloring the faces of 2-dimensional square lattice with $k$ distinct colors such that no two adjacent faces have the same color is considered by establishing connection between the $k$ coloring problem and a generalized vertex model. Associating the colors with $k$ distinct species of particles with infinite repulsive force between nearest neighbors of the same type and zero chemical potential $\mu$ associated with each species, the number of ways $[W(k)]^N$ for large $N$ is related to the entropy of the {\it{hard square lattice gas}} at close packing of the lattice, where $N$ is the number of lattice sites. We discuss the evaluation of $W(k)$ using transfer matrix method with non-periodic boundary conditions imposed on at least one dimension and show the characteristic Toeplitz block structure of the transfer matrix. Using this result, we present some analytical calculations for non-periodic models that remain finite in one dimension. The case $k=3$ is found to approach the exact result obtained by Lieb for the residual entropy of ice with periodic boundary conditions. Finally, we show, by explicit calculation of the contribution of subgraphs and the series expansion of $W(k)$, that the genenralized Pauling type estimate(which is based on mean field approximation) dominates at large values of $k$. We thus also provide an alternative series expansion for the chromatic polynomial of a regular square graph., Comment: 16 pages, 8 figures, Some major revisions done

The exact probability distributions of the amplitudes of eigenfunctions, $\Psi(x, y)$, of several integrable planar billiards are analytically calculated and shown to possess singularities at $\Psi = 0$; the nature of this singularity is shape-dependent. In particular, we prove that the distribution function for a rectangular quantum billiard is proportional to the complete elliptic integral, ${\bf K}(1 - \Psi ^2)$, and demonstrate its universality, modulo a weak dependence on quantum numbers. On the other hand, we study the low-lying states of nonseparable, integrable triangular billiards and find the distributions thereof to be described by the Meijer G-function or certain hypergeometric functions. Our analysis captures a marked departure from the Gaussian distributions for chaotic billiards in its survey of the fluctuations of the eigenfunctions about $\Psi = 0$.

We consider the motion of a system of free particles moving on a plane with regular hard polygonal scatterers arranged in a random manner. Calling this the Ehrenfest gas, which is known to have a zero Lyapunov exponent, we propose a finite-time exponent to characterize its dynamics. As the number of sides of the polygon goes to infinity, when polygon tends to a circle, we recover the usual Lyapunov exponent for the Lorentz gas from the exponent proposed here. To obtain this result, we generalize the reflection law of a beam of rays incident on a polygonal scatterer in a way that the formula for the circular scatterer is recovered in the limit of infinite number of vertices. Thus, chaos emerges from pseudochaos in an appropriate limit., 15 pages, 8 figures; Accepted for publication at Annals of Physics

We show that neutrino spin and spin-flavor transitions involve nonvanishing geometric phases. The geometric character of neutrino spin rotation is explored by studying the neutrino spin trajectory in the projective Hilbert space representation and its relation to the geometric phase. Analytical expressions are derived for noncyclic geometric phases. Several calculations are performed for different cases of rotating and nonrotating magnetic fields in the context of solar neutrinos and neutrinos produced inside neutron stars. Also the effects of adiabaticity, critical magnetic fields and cross boundary effects in the case of neutrinos emanating out of neutron stars are examined., Revised. PRD version

We present a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. We discuss, in particular, how a random superposition of plane waves can model chaotic eigenfunctions and highlight the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution. Various approaches to counting the nodal domains---using trace formulae, graph theory, and difference equations---are also illustrated with examples. The nodal patterns addressed pertain to waves on vibrating plates and membranes, acoustic and electromagnetic modes, wavefunctions of a "particle in a box'" as well as to percolating clusters, and domains in ferromagnets, thus underlining the diversity---and far-reaching implications---of the problem.

For planar integrable billiards, the eigenstates can be classified with respect to a quantity determined by the quantum numbers. Given the quantum numbers as $m, n$, the index which represents a class is $c = m\, \mbox{mod}\, k\,n$ for a natural number, $k$. We show here that the entire tower of states can be generated from an initially given state by application of the operators introduced here. Thus, these operators play the same role for billiards as raising and lowering operators in angular momentum algebra., Comment: 6 pages, 1 figure

Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, $\nu$, of the eigenfunctions are considered. The billiards for which the time-independent Schr\"odinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and non-separable integrable billiards, $\nu $ satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of $m\mod kn$, given a particular $k$, for a set of quantum numbers, $m, n$. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations., Comment: 13 pages, 5 figures

The idea of adiabatic invariance is presented in the context of simple classical mechanical models. The adiabatic invariant jumps across the separatrix — an attempt has been made to bring out the basic ideas underlying the Melnikov-Arnold integral. This becomes important as soon as a perturbation to a regular, stable system makes it dynamically unstable.

Semiclasically derived shell structure and pairing correlation energies have been included in the recently developed formula by Royer et al to evaluate the binding energies of several well known α−radioactive nuclei after appending the effects of deformation as well. Exploiting single-particle level density for spherically symmetric harmonic oscillator potential with spin–orbit interactions within the microscopic-macroscopic framework, Q α -values and the penetration probabilities are determined. This method has been extended to several recently detected superheavy elements, experimentally unknown ones and, also to four α−decay chains in Z = 120 isotopes. The obtained results have been compared with the available experimental and theoretical ones. A stability analysis for such nuclei has been done on the basis of the obtained half-lives (T 1/2-values) as well.

Data on nuclear level densities extracted from transmission data or gamma energy spectrum store the basic statistical information about nuclei at various temperatures. Generally this extracted data goes through model fitting using computer codes like CASCADE. However, recently established semiclassical methods involving no adjustable parameters to determine the level density parameter for magic and semi-magic nuclei give a good agreement with the experimental values. One of the popular ways to paramaterize the level density parameter which includes the shell effects and its damping was given by Ignatyuk. This damping factor is usually fitted from the experimental data on nuclear level density and it comes around 0.05 $MeV^{-1}$. In this work we calculate the Ignatyuk damping factor for various nuclei using semiclassical methods., Comment: 9 Figures, 1 Table

In this paper the classical evolution of a particle is studied which bounces back and forth in a 1D vibrating cavity such that the reflection from the wall does not change the speed of the particle. A peculiar behaviour of the particle motion can be seen where the time evolution of the motion shows superposition of linear and oscillatory behaviour. In particular, the parameter range is found in which the particle oscillates between the walls in steady state as if the wall was static and it is showed that for these parameter ranges the particle settles to this steady state for all initial conditions. It is proposed that this phenomenon can be used to bunch charged particles in short pulses where the synchronization proposed in our model should work against the space charge effect in the charged particle bunch.

We enumerate simple periodic orbits for the well-known Fermi-Pasta-Ulam (FPU) problem. Using these solutions as simple modes for the problem, we construct quantum solutions of the corresponding problem. These studies present a natural generalization of the concept of phonon in the nonlinear realm.

The ionospheric Total Electron Content (TECs), derived by dual frequency signals from the Global Positioning System (GPS) recorded near the Indian equatorial anomaly region, Bhopal (23.2°N, 77.4°E, Geomagnetic 14.2°N) were analyzed for the period of January, 2005 to February, 2008. The work deals with monthly, diurnal, solar and magnetic activity variations on night-time enhancement in TEC. From a total of 157 night-time enhancements, 75 occur during pre-midnight and 82 post-midnight hours. The occurrence of night-time enhancement in TEC is utmost during summer months, followed by equinox and winter months. The occurrence of night-time enhancement in TEC decreases with increase in solar and magnetic activities. We observed that peak size and half amplitude duration are positively correlated, while time of occurrence of night-time enhancement in TEC and time of peak enhancement are negatively correlated with solar activity. The peak size, half amplitude duration, time of peak enhancement and time of occurrence of night-time enhancement in TEC shows negative correlation with magnetic activity. The results have been compared with the earlier ones and discussed in terms of possible source mechanism responsible for the enhancement at anomaly crest region.

Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in [Ann. Phys. 351, 1-12 (2014)]. The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards., 7 pages, 2 figures

We investigate a quantum many-body system with particles moving on a circle and subject to two-body and three-body potentials. In this new class of models, that extrapolates from the celebrated Calogero-Sutherland model and a system with interactions among nearest and next-to-nearest neighbors, the interactions can be tuned as a function of range. We determine the exact ground state energy and wavefunction and obtain a part of the excitation spectrum., Comment: 8 pages, 1 figure

We argue that mass parameters appearing in the treatment of large-amplitude collective motion, be it fission or heavy-ion reactions, originate as a consequence of their relation with Lyapunov exponents coming from the classical dynamics, and, fractal dimension associated with diffusive modes coming from hydrodynamic description.

GPS satellites data obtained at Bhopal (23.16° N, 77.36° E, geomagnetic latitude 14.23° N) India were analyzed to study the TEC changes during several geomagnetic storms (−300 nT

We employ the spectral properties of cyclic random matrices to understand the approach to equilibrium of a biased random walk on a disordered lattice. The simplicity of the calculations and the flexibility with which randomness can be modelled makes it a worthwhile alternative.

A perturbative form of a circular billiard with an off-centre flux line yields a non-Hermitian Hamiltonian with real spectrum. The spacing among neighbouring levels deviates from Poisson distribution (expected for integrable systems). The eigenfunctions exhibit directional property due to broken parity and time-reversal symmetries.

We discuss the relevance of random matrix theory for pseudo-Hermitian systems, and, for Hamiltonians that break parity P and time-reversal invariance T. In an attempt to understand the random Ising model, we present the treatment of cyclic asymmetric matrices with blocks and show that the nearest-neighbour spacing distributions have the same form as obtained for the matrices with scalar entries. We also summarize the theory for random cyclic matrices with scalar entries. We have also found that for block matrices made of Hermitian and pseudo-Hermitian sub-blocks of the form appearing in Ising model depart from the known results for scalar entries. However, there is still similarity in trends even in log-log plots.

It is shown that the integral of the uncertainty of energy with respect to time is independent of the particular Hamiltonian of the quantum system for an arbitrary pseudo-unitary (and hence \( \mathcal{P}\mathcal{T} \)-) quantum evolution. The result generalizes the time-energy uncertainty principle for pseudo-unitary quantum evolutions. Further, employing random matrix theory developed for pseudo-Hermitian systems, time correlation functions are studied in the framework of linear response theory. The results given here provide a quantum brachistochrone problem where the system will evolve in a thermodynamic environment with spectral complexity that can be modelled by random matrix theory.

We construct the expression for the ground state eigenfunction of the Schrodinger equation for a particle inside an arbitrary planar triangle under the influence of a potential.

We calculate the electric dipolarizability of 7 Li nucleus within the cluster model and estimate a value of about 0.0188 fm 3 . We also discuss the possibility of observing this in the scattering of 7 Li from a 208 Pb target at energies about 30 MeV.

We present the exact solution for the Schrodinger equation for a particle inside an N-dimensional regular simplex shaped enclosure. This result extends and unifies the earlier results for equilateral triangle and K-tetrahedron billiards.

The nonlinear dynamics of intra- and inter-spine interaction models of alpha-helical proteins is investigated by proposing a Hamiltonian using the first quantized operators. Hamilton's equations of motion are derived, and the dynamics is studied by constructing the trajectories and phase space plots in both cases. The phase space plots display a chaotic behaviour in the dynamics, which opens questions about the relationship between the chaos and exciton-exciton and exciton-phonon interactions. This is verified by plotting the Lyapunov characteristic exponent curves.

A connection between random-matrix theories and exactly solvable models is discussed here. This is done in three parts: firstly, for theWigner—Dyson case; secondly, for the short-range Dyson case; and thirdly, for the pseudo-Hermitian one. The exactly solvable models are variants and extensions of Calogero—Sutherland—Moser systems.

We present a random matrix theory for systems invariant under the joint action of parity, , and time reversal, , and, more generally, for pseudo-Hermitian systems. This brings out the appearance of the metric in a systematic way so that consistency with the postulates of quantum mechanics is maintained. Here we specialize only to 2 × 2 matrices and we construct a pseudo-unitary group. With explicit examples, nearest-neighbour level-spacing distributions for various classes of ensembles are found to exhibit a degree of level repulsion different from those hitherto known. This work is not only relevant to quantum chaos, but also to two-dimensional statistical mechanics and consistent non-local relativistic theories.

We investigate if physical laws can impose limit on computational time and speed of a quantum computer built from elementary particles. We show that the product of the speed and the running time of a quantum computer is limited by the type of fundamental interactions present inside the system. This will help us to decide as to what type of interaction should be allowed in building quantum computers in achieving the desired speed., Comment: Latex file, 7 pages, no figures

We characterise the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains. The number of nodal domains has a quadratic form in terms of the quantum numbers, with a non-trivial number-theoretic factor. The patterns of the eigenfunctions follow a group-theoretic connection in a way that makes them predictable as one goes from one state to another. Extensive numerical investigations bring out the distribution functions of the mode number and signed areas. The statistics of the boundary intersections is also treated analytically. Finally, the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula. We believe that the results belong generically to non-separable systems, thus extending the previous works which are concentrated on separable and chaotic systems., 26 pages, 13 figures

We study the one dimensional β-Fermi Pasta Ulam lattice of N particles with quadratic and quartic interaction potential. Applying periodic boundary condition and considering only nearest neighbour interaction, we quantize the nonlinear normal modes and construct a wavefunction for what we believe is a primitive nonlinear analogue of a "phonon".