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arXiv:2104.04577

Hydrogen Atom: Its Spectrum and Degeneracy Importance of the Laplace-Runge-Lenz Vector

Pal, Akshay, Sen, Siddhartha

Published Apr 9, 2021• 0 citations
arXivHTMLPDF

Abstract

Consider the problem: why does the bound state spectrum $E(n) < 0$, of hydrogen atom Hamiltonian $H$ have more degenerate eigenstates than those required by rotational symmetry? The answer is well known and was demonstrated by Pauli. It is due to an additional conserved vector, $\vec A$, of $H$, called the Laplace-Runge-Lenz vector, that was first discovered for planetary orbits. However, surprisingly, a direct link between degenerate eigenstates of $H$ and the physical labels that describe them is missing. To provide such a link requires, as we show, solving a subtle problem of self adjoint operators. In our discussions we address a number of conceptual historical aspects regarding hydrogen atom that also include a careful discussion of both the classical as well as the quantum vector $\vec A$.

DOI
10.48550/arxiv.2104.04577

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Consider the problem: why does the bound state spectrum E(n)<0E(n) < 0E(n)<0, of hydrogen atom Hamiltonian HHH have more degenerate eigenstates than those required by rotational symmetry? The answer is well known and was demonstrated by Pauli. It is due to an additional conserved vector, A⃗\vec AA, of HHH, called the Laplace-Runge-Lenz vector, that was first discovered for planetary orbits. However, surprisingly, a direct link between degenerate eigenstates of HHH and the physical labels that describe them is missing. To provide such a link requires, as we show, solving a subtle problem of self adjoint operators. In our discussions we address a number of conceptual historical aspects regarding hydrogen atom that also include a careful discussion of both the classical as well as the quantum vector A⃗\vec AA.
The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schrödinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate with other degenerate states . Recursive relations can be obtained and the energy eigenstate wave functions of the whole spectrum can be obtained easily. Note that the recursive relations are consistent with those used in factorizing the Schrödinger equation. Nevertheless, this analysis provides a better reasoning originated from the conserved vector, the Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a prominent role in this analysis, while the rest of the derivations are elementary.
The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schrödinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate ∣n,l,m⟩|n,l,m\rangle∣n,l,m⟩ with other degenerate states ∣n,l±1,m′⟩|n, l\pm 1,m'\rangle∣n,l±1,m′⟩. Recursive relations can be obtained and the wave functions of the whole spectrum can be obtained easily. Note that the recursive relations are consistent with those used in factorizing the Schrödinger equation. Nevertheless, the present analysis provide a better reasoning originated from the conserved vector, the Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a prominent role in the present analysis, while the rest of the derivations are mostly elementary.
Pauli first noticed the hidden SO(4) symmetry for the Hydrogen atom in the early stages of quantum mechanics [1]. Departing from that symmetry, one can recover the spectrum of a spinless hydrogen atom and the degeneracy of its states without explicitly solving Schrodinger's equation [2]. In this paper, we derive that SO(4) symmetry and spectrum using a computer algebra system (CAS). While this problem is well known [3, 4], its solution involves several steps of manipulating expressions with tensorial quantum operators, simplifying them by taking into account a combination of commutator rules and Einstein's sum rule for repeated indices. Therefore, it is an excellent model to test the current status of CAS concerning this kind of quantum-and-tensor-algebra computations. Generally speaking, when capable, CAS can significantly help with manipulations that, like non-commutative tensor calculus subject to algebra rules, are tedious, time-consuming and error-prone. The presentation also shows a pattern of computer algebra operations that can be useful for systematically tackling more complicated symbolic problems of this kind.
The main point of this paper is to examine a “hidden” dynamical symmetry connected with the conservation of Laplace-Runge-Lenz vector (LRL) in the hydrogen atom problem solved by means of non-commutative quantum mechanics (NCQM). The basic features of NCQM will be introduced to the reader, the key one being the fact that the notion of a point, or a zero distance in the considered configuration space, is abandoned and replaced with a “fuzzy” structure in such a way that the rotational invariance is preserved. The main facts about the conservation of LRL vector in both classical and quantum theory will be reviewed. Finally, we will search for an analogy in the NCQM, provide our results and their comparison with the QM predictions. The key notions we are going to deal with are non-commutative space, Coulomb-Kepler problem, and symmetry.
The hydrogen atom has historically played a foundational role in the development of quantum mechanics, where its discrete energy spectrum is conventionally derived from solutions of the Schrödinger wave equation. In this work we present an alternative formulation in which the hydrogen spectrum emerges without invoking the Schrödinger equation. We consider a semi-quantum framework in which the electron is treated as a classical particle governed by Poisson-bracket dynamics while interacting with a quantized electromagnetic field described using second quantization. The electron moves in the Coulomb potential generated by the proton and simultaneously couples to quantized electromagnetic modes through minimal coupling. The Coulomb system possesses a hidden dynamical symmetry characterized by the Runge–Lenz vector, which enlarges the rotational symmetry to an group for bound states. Within this framework, interactions between the classical particle and the quantized field induce an effective commutation structure in the particle’s phase space. Once this structure emerges, the algebra of the conserved quantities associated with the Runge–Lenz vector becomes identical to the operator algebra used in Pauli’s symmetry-based derivation of the hydrogen spectrum. Consequently, the discrete hydrogen energy levels arise naturally from the combined effects of Coulomb symmetry and particle–field interaction, offering a physically transparent interpretation of atomic quantization and suggesting that wave–particle duality may arise dynamically from interactions with quantized electromagnetic fields.
We show that operators capable of converting certain spherical hydrogen atom eigenfunctions into others by raising the angular momentum quantum number ℓ can be constructed from components of the quantum mechanical Lenz vector operator. The derivation is performed using formalism no more sophisticated than that used to derive the properties of the angular momentum ladder operators in undergraduate quantum mechanics courses. The properties of these Lenz vector operators illustrate the accidental degeneracy of the hydrogen atom because they change the quantum number ℓ without affecting n.