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Rational Extension of Anisotropic Harmonic Oscillator Potentials in Higher Dimensions (Original Title)

Q: Could the presence of additional bound states in the rationally extended potentials have observable consequences in physical systems?

Yes, the presence of additional bound states in rationally extended potentials could indeed lead to observable consequences in physical systems. Here's why: Modified energy spectrum: The most direct consequence is the alteration of the system's energy spectrum. The additional bound state(s) introduce new energy levels that were not present in the original potential. This could manifest in the absorption or emission spectra of atoms or molecules subjected to such potentials. Altered transition probabilities: The presence of new energy levels can affect the transition probabilities between different states. This is because the wavefunctions of the system are modified due to the rational extension, and transition probabilities depend on the overlap of these wavefunctions. Consequently, the intensity and selection rules for spectroscopic transitions could be altered. Shifted scattering properties: For systems where scattering is relevant, the additional bound states can influence the scattering phase shifts and cross-sections. This is because the bound states can resonate with the continuum states, leading to changes in the scattering behavior. Modified tunneling rates: In systems where quantum tunneling is significant, the presence of additional bound states within the potential barrier can alter the tunneling rates. This is because the bound states can act as intermediate states during the tunneling process, effectively reducing the barrier height and enhancing tunneling. Observing these effects experimentally would require careful control and preparation of the physical system. One promising avenue is with ultracold atoms in optical traps, where the trapping potential can be precisely manipulated to mimic the desired rationally extended potentials. Spectroscopic measurements or scattering experiments could then be used to probe the modified energy spectrum and transition probabilities.

Core Concepts

This paper presents a method for constructing rationally extended potentials for quantum anisotropic harmonic oscillators in higher dimensions using supersymmetric quantum mechanics, providing exact solutions in terms of exceptional orthogonal polynomials.

Abstract

Bibliographic Information:

Kumara, R., Yadav, R. K., & Khare, A. (2024). Rational Extension of Anisotropic Harmonic Oscillator Potentials in Higher Dimensions. arXiv preprint arXiv:2411.02955.

Research Objective:

This paper aims to develop a method for constructing rationally extended potentials for quantum anisotropic harmonic oscillators (QAHO) in two and higher dimensions using the principles of supersymmetric quantum mechanics (SUSY QM).

Methodology:

The authors employ the factorization method of SUSY QM, utilizing exceptional orthogonal polynomials (EOPs) to derive exact solutions for the rationally extended potentials. They begin by reviewing the rational extension of the one-dimensional harmonic oscillator on both full and half-lines. Subsequently, they extend this approach to two-dimensional QAHOs, considering various combinations of full-line and half-line oscillators along different axes. The method is then generalized to three and higher dimensions, illustrated by a detailed analysis of a 3D QAHO with two equal frequencies.

Key Findings:

The authors successfully construct rationally extended potentials for QAHOs in higher dimensions, providing explicit expressions for the potentials, eigenfunctions, and energy eigenvalues.
The eigenfunctions of the extended potentials are expressed in terms of EOPs, specifically exceptional Hermite and Laguerre polynomials.
The extended potentials are shown to be isospectral to the conventional QAHOs, implying that they share the same energy spectrum.
The degeneracy of energy levels in the extended potentials is analyzed and found to be similar to that of the corresponding conventional QAHOs.

Main Conclusions:

The paper demonstrates the effectiveness of SUSY QM in constructing rationally extended potentials for QAHOs in multiple dimensions. The use of EOPs allows for obtaining exact solutions, providing valuable insights into the behavior of these systems. The findings contribute to a deeper understanding of supersymmetric quantum mechanics and its applications in solving quantum mechanical problems.

Significance:

This research significantly contributes to the field of quantum mechanics by providing a novel method for constructing and analyzing rationally extended potentials for anisotropic harmonic oscillators in higher dimensions. The findings have potential applications in various areas of physics, including atomic, molecular, and condensed matter physics, where anisotropic harmonic oscillator models are widely used.

Limitations and Future Research:

The study primarily focuses on the construction of rationally extended potentials for unperturbed QAHOs. Future research could explore the possibility of extending the method to include perturbations and generate a broader class of extended potentials. Additionally, investigating the physical implications and applications of these extended potentials in specific physical systems could be a promising avenue for further exploration.

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Rational Extension of Anisotropic Harmonic Oscillator Potentials in Higher Dimensions

by Rajesh Kumar... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.02955.pdf

Rational Extension of Anisotropic Harmonic Oscillator Potentials in Higher Dimensions

Deeper Inquiries

How can the presented method be adapted to construct rationally extended potentials for other types of quantum potentials beyond the anisotropic harmonic oscillator?

The method presented in the paper relies heavily on the principles of Supersymmetric Quantum Mechanics (SUSY QM) and the properties of exceptional orthogonal polynomials (EOPs). To adapt this method for constructing rationally extended potentials for other quantum potentials, we need to consider the following:

Identify a solvable potential: The starting point should be a quantum potential for which the exact solutions (eigenvalues and eigenfunctions) are known. This is crucial for constructing the seed function. Examples include the Coulomb potential, the Morse potential, and the Pöschl-Teller potential.

Construct the seed function:  The seed function is generated by strategically modifying the known eigenfunctions of the starting potential. This typically involves replacing the quantum numbers with a new set of parameters (like 'm' in the paper) and potentially changing the sign of certain terms (e.g., frequencies). The goal is to create a function that is no longer normalizable but still satisfies the original Schrödinger equation with a shifted energy (factorization energy).

Determine the partner potential: Using the SUSY QM formalism, the partner potential can be derived from the seed function. This involves calculating the superpotential (W(x)) and then using it to determine the partner potential (V-(x)). The paper provides the necessary equations for this step.

Verify isospectrality and analyze properties: The resulting rationally extended potential should be isospectral to the original potential, meaning they share the same energy spectrum (except for potentially a finite number of states). This property needs to be verified. Additionally, the properties of the new potential, such as its asymptotic behavior and the presence of singularities, should be analyzed.

Generalize to higher dimensions:  The method can be extended to higher dimensions by applying the procedure separately to each dimension and then combining the results. This is demonstrated in the paper for the two-dimensional and three-dimensional anisotropic harmonic oscillator.

By following these steps, it is possible to explore the rational extension of various quantum potentials beyond the anisotropic harmonic oscillator. However, the success of this method depends on the specific potential and the ability to find suitable seed functions.

Could the presence of additional bound states in the rationally extended potentials have observable consequences in physical systems?

Yes, the presence of additional bound states in rationally extended potentials could indeed lead to observable consequences in physical systems. Here's why:

Modified energy spectrum: The most direct consequence is the alteration of the system's energy spectrum. The additional bound state(s) introduce new energy levels that were not present in the original potential. This could manifest in the absorption or emission spectra of atoms or molecules subjected to such potentials.

Altered transition probabilities: The presence of new energy levels can affect the transition probabilities between different states. This is because the wavefunctions of the system are modified due to the rational extension, and transition probabilities depend on the overlap of these wavefunctions. Consequently, the intensity and selection rules for spectroscopic transitions could be altered.

Shifted scattering properties: For systems where scattering is relevant, the additional bound states can influence the scattering phase shifts and cross-sections. This is because the bound states can resonate with the continuum states, leading to changes in the scattering behavior.

Modified tunneling rates: In systems where quantum tunneling is significant, the presence of additional bound states within the potential barrier can alter the tunneling rates. This is because the bound states can act as intermediate states during the tunneling process, effectively reducing the barrier height and enhancing tunneling.
Observing these effects experimentally would require careful control and preparation of the physical system. One promising avenue is with ultracold atoms in optical traps, where the trapping potential can be precisely manipulated to mimic the desired rationally extended potentials. Spectroscopic measurements or scattering experiments could then be used to probe the modified energy spectrum and transition probabilities.

What are the implications of the isospectrality between the extended and conventional potentials for the dynamics and properties of quantum systems?

The isospectrality between rationally extended potentials and their conventional counterparts has profound implications for the dynamics and properties of quantum systems:

Identical energy levels (mostly):  Isospectrality implies that the energy eigenvalues of the extended and conventional potentials are identical, except for a finite number of states (often just the ground state). This means that many observable quantities that depend solely on the energy spectrum, such as thermodynamic properties at equilibrium, would remain unchanged.

Different wavefunctions and spatial probabilities: Despite sharing the same energy spectrum, the extended and conventional potentials have different eigenfunctions. This means that the spatial probability distributions of the particle in each eigenstate will be different. Consequently, properties that depend on the wavefunction's shape, such as the expectation values of position or momentum, will generally differ.

Modified dynamics and time evolution: The time evolution of a quantum system is governed by its Hamiltonian, which includes the potential. Since the extended and conventional potentials are different, the time evolution of wavepackets and the dynamics of the system will generally be distinct, even if they share the same energy spectrum.

Insights into solvable potentials: The existence of isospectral partner potentials provides valuable insights into the nature of solvable potentials in quantum mechanics. It demonstrates that a single energy spectrum does not uniquely determine the potential, and there can be multiple potentials with the same (or nearly the same) energy levels but different spatial characteristics.

Potential for novel applications: The ability to engineer potentials with specific properties while maintaining isospectrality opens up possibilities for novel applications. For instance, one could design potentials with desired scattering properties or modified tunneling rates while preserving the energy spectrum of a known system.
In essence, isospectrality between rationally extended and conventional potentials highlights a fascinating aspect of quantum mechanics: while the energy spectrum provides crucial information about a system, the complete picture requires understanding the wavefunctions and their spatial distributions, which can differ significantly even for isospectral potentials.