Bound States of Spherically Symmetric Potentials
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Bound States of Spherically Symmetric Potentials∗ Heat Capacity Calculations Chandan Kumar
We solve the time-independent Schr¨odinger equation for spherically symmetric potentials. First, we consider simple cases of a particle on a ring and a particle on a sphere to illustrate the degeneracy arising due to symmetry. We then consider three different spherically symmetric potentials: (i) spherical well potential, (ii) isotropic three-dimensional harmonic oscillator, and (iii) spherically confined isotropic three-dimensional harmonic oscillator. Our discussion mainly focuses on the energy levels of the bound states and the associated degeneracies. Finally, we calculate the heat capacity of endohedral fullerenes using two simple models—particle in a spherical box and confined harmonic oscillator.
Chandan Kumar is currently pursuing his PhD at IISER Mohali under the guidance of Prof. Arvind. His research interests are quantum optics, continuous variable quantum information theory, and mathematical physics.
1. Introduction One of the major applications of quantum mechanics is in studying the electronic structure of molecules. The first step involves solving the Schr¨odinger equation for the electronic molecular Hamiltonian. For many-particle systems, the Schr¨odinger equation becomes quite complicated, and in many cases, analytical solutions cannot be obtained. In such circumstances, one resorts to numerical methods to solve the Schr¨odinger equation. Alternatively, a molecular system can be studied by setting up simple quantum models that retain important features. Keywords Schrodinger ¨ equation, degener-
In recent years, researchers have shown an increased interest in quantum modelling of molecular systems. In this article, we are interested in the quantum modelling of molecular systems
∗
acy, quantum models.
Vol.25, No.11, DOI: https://doi.org/10.1007/s12045-020-1071-2
RESONANCE | November 2020
1491
GENERAL ARTICLE
Figure 1.
Relation between Cartesian coordinates P(x, y, z) and its spherical coordinates P(r, θ, φ): x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ.
z P θ
O y
ϕ
x
with spherical symmetry. For example, fullerene, a spheroidal molecule, can be modelled as a sphere for studying its properties. We solve the Schr¨odinger equation for different potentials with spherical symmetry. We obtain the energy levels of the bound states and their degeneracies, which is required to calculate absorption, electrical, and thermodynamic properties. The material in this paper is organized as follows. Section 2 treats the case of a particle on a ring and a particle on a sphere. Section 3 discusses spherical well potential, isotropic three-dimensional harmonic oscillator, and spherically confined harmonic oscillator. Section 4 presents the quantum modelling of endohedral fullerene and heat capacity calculations. Finally, Section 5 provides some concluding remarks.
2. Schr¨odinger Equation in Spherical Coordinates In three dimensions, the time-independent Schr¨odinger equation in the Cartesian coordinates is
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