One-dimensional Hydrogen Atom Research Articles

The bound-state solutions of the one-dimensional pseudoharmonic oscillator

We study the bound states of a quantum mechanical system consisting of a simple harmonic oscillator with an inverse square interaction, whose interaction strength is governed by a constant $\alpha$. The singular form of this potential has doubly-degenerate bound states for $-1/4\leq\alpha<0$ and $\alpha>0$; since the potential is symmetric, these consist of even and odd-parity states. In addition we consider a regularized form of this potential with a constant cutoff near the origin. For this regularized potential, there are also even and odd-parity eigenfunctions for $\alpha\geq-1/4$. For attractive potentials within the range $-1/4\leq\alpha<0$, there is an even-parity ground state with increasingly negative energy and a probability density that approaches a Dirac delta function as the cutoff parameter becomes zero. These properties are analogous to a similar ground state present in the regularized one-dimensional hydrogen atom. We solve this problem both analytically and numerically, and show how the regularized excited states approach their unregularized counterparts.

The bound-state solutions of the one-dimensional hydrogen atom

The one-dimensional hydrogen atom is an intriguing quantum mechanics problem that exhibits several properties which have been continually debated. In particular, there has been variance as to whether or not even-parity solutions exist, and specifically whether or not the ground state is an even-parity state with infinite negative energy. We study a “regularized” version of this system, where the potential is a constant in the vicinity of the origin, and we discuss the even- and odd-parity solutions for this regularized one-dimensional hydrogen atom. We show how the even-parity states, with the exception of the ground state, converge to the same functional form and become degenerate for x &amp;gt; 0 with the odd-parity solutions as the cutoff approaches zero. This differs with conclusions derived from analysis of the singular (i.e., without regularization) one-dimensional Coulomb potential, where even-parity solutions are absent from the spectrum.

Solving the Schrodinger Equation on the Basis of Finite-Difference and Monte-Carlo Approaches

The paper presents a method of numerical solution of the Schrodinger equation, which combines the finite-difference and Monte-Carlo approaches. The resulting method was effective and economical and, to a certain extent, not improved, i.e. optimal. The method itself is formalized as an algorithm for the numerical solution of the Schrodinger equation for a molecule with an arbitrary number of quantum particles. The method is presented and simultaneously illustrated by examples of solving the one-dimensional and multidimensional Schrodinger equation in such problems: linear one-dimensional oscillator, hydrogen atom, ion and hydrogen molecule, water, benzene and metallic hydrogen.

Influence of hydrogen-like nucleus mass on electronic state

In an isolated two-body composite system, the discussion of how the change of one body affects the state of the other will help to know the relation of a single particle's mixed and pure states. Given 5 one-dimensional hydrogen-like atoms models, each Coulomb interaction potential keeps invariant, while the masses of the nuclei are different. These two-body composite systems stay in their respective entangled state, each electron stays in a mixed state. If we suppose a one-dimensional hydrogen atom model having infinite nuclear mass, the electron stays in a pure state. We select position representation. The wave function of the ground state of the atom has the form of the square root of a &lt;i&gt;δ&lt;/i&gt; function. To avoid calculation trouble of &lt;i&gt;δ&lt;/i&gt; function, we select the first excited state and the superposed state of the first and the second excited states. We compare the two pure states, the first excited state and the superposed state, with those corresponding mixed states by fidelity and &lt;i&gt;l&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt; norm coherence, and calculate the purities of the mixed states. The summations become integrations due to the position variable having a continuous eigenvalue spectrum. We investigate the changes in these quantities with the increase of the nuclear mass. The results show that the purities of the mixed states and the fidelities increase with the nuclear mass increasing. However, the coherences of the mixed states decrease with the nuclear mass increasing. This can be explained as that a mixed state with non-zero coherence may approach to a pure eigenstate, while the latter has zero coherence in the eigenspace. These mean that the greater a nuclear mass is, the closer the mixed state approaches to the corresponding pure state. Therefore, the two pure states are the approximations of the corresponding mixed states. The entangled state of the electron and the nucleus is related with the nuclear mass and the Coulomb interaction potential. So, each electron mixed state and its coherence are related with the nucleus and their Coulomb interaction potential. If the nuclear mass is great, the nucleus is approximately motionless or its state is approximately unchanged, and the Coulomb interaction potential approximates to the external Coulomb potential of the electron. The electron approximately stays in a pure state. The state and its coherence are related with the nucleus and the Coulomb interaction. From other point of view, the entangled state of the nucleus and the electron approximates to the product state of the unchanged nucleus state and the electron state. In this case, an electron mixed state approximates to its corresponding pure state, and then these states and their coherences are all related with the nucleus and the Coulomb interaction.

Classical and quantum space splitting: the one-dimensional hydrogen atom

The one-dimensional (1D) hydrogen atom with potential energy V(q) = −e2/|q|, with e the electron charge and q its position coordinate, has been a source of discussion and controversy for more than 55 years. A number of incorrect claims have been made about its spectrum; for example, that its ground state has infinite binding energy, that bound states associated with a continuum of negative energy values exist, or that anomalous non-Balmer energy levels are present in the system. Given such claims and the ongoing controversy, we have re-analysed the 1D hydrogen atom, first from a classical and then from a quantum perspective both in the coordinate and in the momentum representations. This work exhibits that certain classical properties of the system may serve to clarify the properties of the quantum problem. Using the Dirichlet boundary condition, we show that the singularity of the potential prevents any relation between the right and left sides of the origin. Hence we prove that the attractive potential V(q) acts in that case as an impenetrable barrier splitting the coordinate space into two independent regions. We show that such splitting appears both in the classical and in the quantum descriptions of the system. The analysis presented in this paper may serve as a pedagogical tool for the comparison between classical and quantum problems, as well as an illustrative example of a problem involving a singular potential that can be approached both from its position and momentum representations.

About One Method of Numerical Solution of Schrodinger’s Equation

The paper considers the method of the numerical solution of the Schrodinger equation, which can partly be attributed to the class of Monte Carlo methods. The method is presented and simultaneously illustrated by the examples of solving the one-dimensional and multidimensional Schrodinger equation in the problems of a linear one-dimensional oscillator, hydrogen atom, and benzene atom.

A geodesical approach for the harmonic oscillator

The harmonic oscillator (HO) is present in all contemporary physics, from elementary classical mechanicsto quantum field theory. It is useful in general to exemplify techniques in theoretical physics. In this work,we use a method for solving classical mechanic problems by first transforming them to a free particle formand using the new canonical coordinates to reparametrize its phase space. This technique has been used tosolve the one-dimensional hydrogen atom and also to solve for the motion of a particle in a dipolar potential.Using canonical transformations we convert the HO Hamiltonian to a free particle form which becomestrivial to solve. Our approach may be helpful to exemplify how canonical transformations may be used inmechanics. Besides, we expect it will help students to grasp what they mean when it is said that a problemhas been transformed into another completely different one. As, for example, when the Kepler problem istransformed into free (geodesic) motion on a spherical surface.

Properties of the donor impurity band in mixed valence insulators

In traditional semiconductors with large effective Bohr radius, an electron donor creates a hydrogen-like bound state just below the conduction band edge. The properties of the impurity band arising from such hydrogenic impurities have been studied extensively during the last 70 years. In this paper we consider whether a similar bound state and a similar impurity band can exist in mixed valence insulators, where the gap arises at low temperature due to strong electron-electron interactions. We find that the structure of the hybridized conduction band leads to an unusual bound state that can be described using the physics of the one-dimensional hydrogen atom. The properties of the resulting impurity band are also modified in a number of ways relative to the traditional semiconductor case; most notably, the impurity band can hold a much larger concentration without inducing an insulator-to-metal transition. We estimate the critical doping associated with this transition, and then proceed to calculate the DC and AC conductivities and the specific heat. We discuss our results in light of recent measurements on the mixed valence insulator ${\mathrm{SmB}}_{6}$ and find them to be consistent with the experiments.

Generation of carrier-envelope phase-controlled isolated attosecond pulses using chirped femtosecond excitation lasers

ABSTRACTWe present an efficient approach for producing a carrier-envelope phase controlled isolated attosecond pulse by an optimized intense driving laser pulse. High-order harmonics are produced by numerically solving the time-dependent Schrödinger equation for the one-dimensional hydrogen atom in an ultrashort laser pulse. We define an efficient cost function to optimize the laser pulse by a genetic algorithm scheme. Our approach produces single attosecond pulses with desired properties, including the carrier-envelope phase, central frequency, and duration. Also, we analyze the time–frequency profiles of the attosecond emissions to gain a deeper insight into the underlying physical mechanism.

Zeta-regularized vacuum expectation values

It has recently been shown that vacuum expectation values and Feynman path integrals can be regularized using the Fourier integral operator ζ-function, yet the physical meaning of these ζ-regularized objects was unknown. Here, we show that ζ-regularized vacuum expectations appear as continuum limits using a certain discretization scheme. Furthermore, we study the rate of convergence for the discretization scheme using the example of a one-dimensional hydrogen atom in (−π, π) which we evaluate classically, using the Rigetti quantum virtual machine and on the Rigetti 8Q quantum chip “Agave” device. We also provide the free radiation field as an example for the computation of ζ-regularized vacuum expectation values in a gauge theory.

A Distributional Approach for the One-Dimensional Hydrogen Atom

We consider the one-dimensional Hydrogen atom, with the Coulomb interaction $V(x)=\frac{\gamma}{|x|}$ ($\gamma < 0$), and use Schwartz's theory of distributions to address the non-integrable singularity at the origin. This singularity renders the interaction term $V(x)\psi(x)$ in the Schr\"odinger's equation, where $\psi (x)$ is the wave function, an ill-defined product in the ordinary sense. We replace this ill-defined product by a well defined interaction \emph{distribution}, $S[\psi, V](x)$, and by imposing that it should satisfy some fundamental mathematical and physical requirements, we show that this distribution is defined up to a 4-parameter family of contact interactions, in agreement with the method of self-adjoint extensions. By requiring that the interaction distribution be invariant under parity, we further restrict the 4-parameter family of interactions to the subfamily of all the parity invariant Coulomb interactions. Finally, we present a \emph{systematic} study of the bound states within this subfamily, addressing the frequently debated issues of the multiplicity and parity of the bound states, and the boundedness of the ground state energy.

On the symmetry of a one-dimensional hydrogen atom

We touch upon a long-standing question of the “true” one-dimensional hydrogen atom solution. From a symmetry point of view, the Kepler problem in the d ≥ 2 dimension is characterized by geometrical rotational symmetry, SO(d), as well as dynamical “accidental” SO(d + 1) symmetry. Because of the topology, these two symmetries are mutually exclusive in one dimension, regardless of the regularization employed, drawing one to a conclusion that the question of the “true” hydrogen atom in one dimension does not have an answer because a single dimension cannot support both of the symmetries of the Kepler problem. We argue our findings using a novel method to recover and classify solutions appearing in the literature according to the symmetry they respect. In particular, curious features of some of the solutions—double degeneracy and particle confinement—are directly attributed to the dynamical symmetry behind them.

Some applications of the most general form of the higher-order GUP with minimal length uncertainty and maximal momentum

In this paper, using the new type of D-dimensional nonperturbative Generalized Uncertainty Principle (GUP) which has predicted both a minimal length uncertainty and a maximal observable momentum,1 first, we obtain the maximally localized states and express their connections to [P. Pedram, Phys. Lett. B 714, 317 (2012)]. Then, in the context of our proposed GUP and using the generalized Schrödinger equation, we solve some important problems including particle in a box and one-dimensional hydrogen atom. Next, implying modified Bohr–Sommerfeld quantization, we obtain energy spectra of quantum harmonic oscillator and quantum bouncer. Finally, as an example, we investigate some statistical properties of a free particle, including partition function and internal energy, in the presence of the mentioned GUP.

Optimal control of high harmonics for the generation of double attosecond pulses by using a chirped laser pulse

An efficient scheme for the optimization of ultrashort femtosecond pulse shapes interacting with an atom to control high harmonics spectrum and double attosecond pulse generation is presented. The time-dependent Schrödinger equation of one-dimensional hydrogen atom is solved numerically to obtain electric field emission. The genetic algorithm optimization method is used to control the phase and amplitude of ultrashort excitation laser pulses to generate the desired attosecond-shaped pulses. An appropriate cost function is introduced for genetic algorithm optimization of double attosecond pulse generation. It is shown that the relative intensity of two generated pulses, their delay time and duration can be controlled in this approach. Finally, the parameters of the optimized emitted attosecond pulse are compared with those of desired pulses, and the underlying physical mechanisms are discussed in detail.

Generalized higher-order Snyder model

In this paper we consider the higher-order generalization of the Snyder algebra. We find the modified inner product for this algebra. We also discuss the one-dimensional hydrogen atom eigenvalue problem. We find the invariant phase space volume for the generalized Snyder algebra. Finally we use the density of states in three dimension to discuss the Maxwellian distribution and cosmological constant.

One-dimensional hydrogen atom

The theory of the one-dimensional (1D) hydrogen atom was initiated by a 1952 paper but, after more than 60 years, it remains a topic of debate and controversy. The aim here is a critique of the current status of the theory and its relation to relevant experiments. A 1959 solution of the Schrödinger equation by the use of a cut-off at x = a to remove the singularity at the origin in the 1/| x | form of the potential is clarified and a mistaken approximation is identified. The singular atom is not found in the real world but the theory with cut-off has been applied successfully to a range of four practical three-dimensional systems confined towards one dimension, particularly their observed large increases in ground state binding energy. The true 1D atom is in principle restored when the short distance a tends to zero but it is sometimes claimed that the solutions obtained by the limiting procedure differ from those obtained by solution of the basic Schrödinger equation without any cut-off in the potential. The treatment of the singularity by a limiting procedure for applications to practical systems is endorsed.

Schrödinger equation of a particle in a uniformly accelerated frame and the possibility of a new kind of quanta

In this paper, we have developed a formalism to obtain the Schrödinger equation for a particle in a frame undergoing a uniform acceleration in an otherwise flat Minkowski spacetime geometry. We have presented an exact solution of the equation and obtained the eigenfunctions and the corresponding eigenvalues. It has been observed that the Schrödinger equation can be reduced to a one-dimensional hydrogen atom problem. Whereas, the quantized energy levels are exactly identical with that of a one-dimensional quantum harmonic oscillator. Hence, considering transitions, we have predicted the existence of a new kind of quanta, which will either be emitted or absorbed if the particles get excited or de-excited, respectively.

Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit.

An efficient way of evolving a solution to an ordinary differential equation is presented. A finite element method is used where we expand in a convenient local basis set of functions that enforce both function and first derivative continuity across the boundaries of each element. We also implement an adaptive step-size choice for each element that is based on a Taylor series expansion. This algorithm is used to solve for the eigenpairs corresponding to the one-dimensional soft Coulomb potential, 1/sqrt[x(2)+β(2)], which becomes numerically intractable (because of extreme stiffness) as the softening parameter (β) approaches zero. We are able to maintain near machine accuracy for β as low as β = 10(-8) using 16-digit precision calculations. Our numerical results provide insight into the controversial one-dimensional hydrogen atom, which is a limiting case of the soft Coulomb problem as β → 0.

On the localization of Rydberg wave packets

The Husimi distribution function is used to study the phase-space localization of Rydberg wave packets produced in a quasi one-dimensional hydrogen atom by the impact of half-cycle pulses (HCPs). The wave packet in a single-kicked atom exhibits transient phase-space localization. A weak second time-delayed HCP is found to extend the time of localization provided it is applied when the wave packet is near the inner turning point of the classical electron trajectory and momentum-transfer vectors of the first and the second kicks have the same sign. Alternatively, application of a similar second HCP increases the atomic ionization probability if the momentum-transfer vectors of the two kicks have opposite directions.

One-dimensional hydrogen atom with minimal length uncertainty and maximal momentum

We present exact energy eigenvalues and eigenfunctions of the one-dimensional hydrogen atom in the framework of the Generalized (Gravitational) Uncertainty Principle (GUP). This form of GUP is consistent with various theories of quantum gravity such as string theory, loop quantum gravity, black-hole physics, and doubly special relativity and implies a minimal length uncertainty and a maximal momentum. We show that the quantized energy spectrum exactly agree with the semiclassical results.