The existence of the group of contact transformations, regarded as the so-called continuous Schrödinger group whose elements commute with Hamiltonian,is intimately related not only to the problem of degeneracy, as shown by Jauch and Hill, but also to the problem of quantization. In the present paper, it is shown that on the basis of representations of the Lie ring generated by the operators which have a certain definite correspondence with a set of independent integrals of the equations of motion in Newtonian mechanics, an energy eigenvalue problem can be constructed in terms of a complete set of the Casimir operators. It is further pointed out that this quantization procedure is, in spite of various restrictions, applicable to quantum-mechanical systems having the continuous Schrödinger group, and then the degree of degeneracy in regard to the angular momentum can be obtained simultaneously.The existence of the group of contact transformations, regarded as the so-called continuous Schrödinger group whose elements commute with Hamiltonian,is intimately related not only to the problem of degeneracy, as shown by Jauch and Hill, but also to the problem of quantization. In the present paper, it is shown that on the basis of representations of the Lie ring generated by the operators which have a certain definite correspondence with a set of independent integrals of the equations of motion in Newtonian mechanics, an energy eigenvalue problem can be constructed in terms of a complete set of the Casimir operators. It is further pointed out that this quantization procedure is, in spite of various restrictions, applicable to quantum-mechanical systems having the continuous Schrödinger group, and then the degree of degeneracy in regard to the angular momentum can be obtained simultaneously.

Journal of the Physical Society of JapanJul 1, 19562 citations

Extracting hidden symmetry from the energy spectrum

Emil A. Yuzbashyan, W. Happer, B. L. Altshuler +1 more

In this paper we revisit the problem of finding hidden symmetries in quantum mechanical systems. Our interest in this problem was renewed by nontrivial degeneracies of a simple spin Hamiltonian used to model spin relaxation in alkali-metal vapours. We consider this spin Hamiltonian in detail and use this example to outline a general approach to finding symmetries when eigenvalues and eigenstates of the Hamiltonian are known. We extract all nontrivial symmetries responsible for the degeneracy and show that the symmetry group of the Hamiltonian is SU (2). The symmetry operators have a simple meaning which becomes transparent in the limit of large spin. As an additional example we apply the method to the hydrogen atom.In this paper we revisit the problem of finding hidden symmetries in quantum mechanical systems. Our interest in this problem was renewed by nontrivial degeneracies of a simple spin Hamiltonian used to model spin relaxation in alkali-metal vapours. We consider this spin Hamiltonian in detail and use this example to outline a general approach to finding symmetries when eigenvalues and eigenstates of the Hamiltonian are known. We extract all nontrivial symmetries responsible for the degeneracy and show that the symmetry group of the Hamiltonian is SU (2). The symmetry operators have a simple meaning which becomes transparent in the limit of large spin. As an additional example we apply the method to the hydrogen atom.

arXiv:cond-mat/0210506Journal of Physics A Mathematical and GeneralFeb 26, 200311 citations

On the Problem of Degeneracy in Quantum Mechanics

J. M. Jauch, Edward L. Hill

WORLD SCIENTIFIC eBooksAug 1, 19910 citations

The Rotator

Ahmad Amer

This chapter examines the quantum mechanical rotator, a model that effectively approximates the behavior of a diatomic molecule when radial vibrations are negligible. The energy eigenvalues and the degeneracy of energy levels are derived through straightforward calculations. The s-state is presented as an example of a state with a constant probability density, independent of spatial coordinates. The p-state is also analyzed, and its eigenstates are interpreted to reveal the most probable particle trajectories. A qualitative comparison is made between these quantum predictions and the classical description of rotational motion. Metadata/Keywords: quantum rotator, diatomic molecule, energy eigenvalues, degeneracy, probability density, classical comparison, angular momentumThis chapter examines the quantum mechanical rotator, a model that effectively approximates the behavior of a diatomic molecule when radial vibrations are negligible. The energy eigenvalues and the degeneracy of energy levels are derived through straightforward calculations. The s-state is presented as an example of a state with a constant probability density, independent of spatial coordinates. The p-state is also analyzed, and its eigenstates are interpreted to reveal the most probable particle trajectories. A qualitative comparison is made between these quantum predictions and the classical description of rotational motion. Metadata/Keywords: quantum rotator, diatomic molecule, energy eigenvalues, degeneracy, probability density, classical comparison, angular momentum

CRC Press eBooksNov 5, 20250 citations

Sum of Three Squares and Particle in a Cubic Box - Degeneracies

S. Pratik Khastgir

We address the problem of determining the degeneracy of a particular energy eigenvalue corresponding to a quantum particle trapped in a cubic box. In other words, we aim to find the number of ways a specific positive integer can be written as the sum of three natural number squares. This is a classic problem in number theory. We provide a closed-form, exact solution to the problem. We do not claim to provide proofs, as we have none; instead, we build up the formulae starting from simple cases where the answer could be deduced by studying the list of degeneracies. We present ample examples that could be verified against the degeneracy calculated numerically via the brute-force method. We arrived at the solution by meticulously studying the list of degeneracies for the first five million numbers, generated by Mathematica $^{\circledR}$ .We address the problem of determining the degeneracy of a particular energy eigenvalue corresponding to a quantum particle trapped in a cubic box. In other words, we aim to find the number of ways a specific positive integer can be written as the sum of three natural number squares. This is a classic problem in number theory. We provide a closed-form, exact solution to the problem. We do not claim to provide proofs, as we have none; instead, we build up the formulae starting from simple cases where the answer could be deduced by studying the list of degeneracies. We present ample examples that could be verified against the degeneracy calculated numerically via the brute-force method. We arrived at the solution by meticulously studying the list of degeneracies for the first five million numbers, generated by Mathematica$^{\circledR}$.

math-phCross-listedarXiv:2609.23794Sep 22, 2026

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