# Dynamical Symmetries of the H Atom, One of the Most Important Tools of Modern Physics: SO(4) to SO(4,2), Background, Theory, and Use in Calculating Radiative Shifts

## Abstract

**:**

## 1. Introduction

#### 1.1. Objective of This Paper

#### 1.2. Outline of This Paper

#### 1.3. Brief History of Symmetry in Quantum Mechanics and Its Role in Understanding the Schrodinger Hydrogen Atom

The study of the hydrogen atom has been at the heart of the development of modern physics...theoretical calculations reach precision up to the 12th decimal place...high resolution laser spectroscopy experiments...reach to the 15th decimal place for the 1S–2S transition...The Rydberg constant is known to six parts in ${10}^{12}$ [2,3]. Today, the precision is so great that measurement of the energy levels in the H atom has been used to determine the radius of the proton.

**A**and the conserved angular momentum vector

**L**to solve for the energy spectrum of the hydrogen atom by purely algebraic means, a beautiful result, yet he did not explicitly identify that

**L**and

**A**formed the symmetry group SO(4) corresponding to the degeneracy. At this time, the degree of degeneracy in the hydrogen energy levels was believed to be ${n}^{2}$ for a state with principal quantum number n, clearly greater than the degeneracy due to rotational symmetry which is $(2\mathit{l}+1)$. The ${n}^{2}$ degeneracy arises from the possible values of the angular momentum $l=0,1,2,\dots n-1,$ and the $2l+1$ values of the angular momentum along the azimuthal axis $m=-l,-l+1,\dots 0,1,2,l+1$. The additional degeneracy was referred to as “accidental degeneracy [15].”

**L**and the Runge–Lenz vector

**A**, obeyed the commutation rules of the SO(4) [12]. His use of commutators was so early in the field of quantum mechanics, that Bargmann explained the square bracket notation he used for a commutator in a footnote [19]. He gave differential expression for the operators, adapting the approach of Lie generators in the calculation of the commutators. He linked solutions to Schrodinger’s equation in parabolic coordinates to the existence of the conserved Runge–Lenz vector and was thereby able to establish the relationship of Fock’s results to the algebraic representation of SO(4) for bound states implied by Fock and Pauli [12]. He also pointed out that the scattering states (E > 0) could provide a representation of the group SO(3,1). In a note at the end of the paper, Bargmann, who was at the University in Zurich, thanked Pauli for pointing out the paper of Hulthen and the observation by Klein that the Lie algebra of

**L**and

**A**was the same as the infinitesimal Lorentz group, which is how he referred to a Lie algebra. Bargmann’s work was a milestone demonstrating the relationship of symmetry to conserved quantities and it clearly showed that to fully understand a physical system one needed to go beyond the usual ideas of geometrical symmetry. This work was the birth, in 1936, without much fanfare, of the idea of dynamical symmetry.

**L**and

**A**and the O(4) group they form, and the other, referred to as the “global method”, first done by Fock, converts the Schrodinger equation to an integral equation with a manifest four dimensional symmetry in momentum space. They establish the equivalence of the two approaches by appealing to the solutions of the H atom in parabolic coordinates, and demonstrate that the symmetry operators in the momentum space correspond to the symmetry operators in the configuration space. As they note, the stereographic projection depends on the energy, so the statements for a SO(4) subgroup are valid only in a subspace of constant energy. They then explore the expansion of the SO(4) group to include scale changes so the energy can be changed, transforming between states of different principal quantum number, which correspond to different subspaces of SO(4). To insure that this expansion results in a group, they include other transformations, which results in the the generators forming the conformal group O(4,1). Their mathematical analysis introducing SO(4,1) is based on the projection of a p dimensional space (4 in the case of interest) on a parabaloid in p + 1 dimensions (5 dimensions). In their derivation they treat bound states in their first paper [35] and scattering states in the second paper [36].

The construction of unitary representations of non-compact groups that have the property that the irreducible representations of their maximal subgroup appear at most with multiplicity one is of certain interest for physical applications. The method of construction used here in the Coulomb potential case can be extended to various other cases. The geometrical emphasis may help to visualize things and provide a global form of the transformations.

#### 1.4. The Dirac Hydrogen Atom

## 2. Background

#### 2.1. The Relationship between Symmetry and Conserved Quantities

#### 2.2. Non-Invariance Groups and Spectrum Generating Group

#### 2.3. Basic Idea of Eigenstates of ${\left(Z\alpha \right)}^{-1}$

#### 2.4. Degeneracy Groups for Schrodinger, Dirac and Klein-Gordon Equations

## 3. Classical Theory of the H Atom

**L**, which is perpendicular to the plane of the orbit, and the Runge–Lenz vector

**A**, which goes from the focus corresponding to the center of mass and force along the semi-major axis to the perihelion (closest point) of the elliptical orbit. The conservation of

**A**is related to the fact that non-relativistically the orbits do not precess. The Hamiltonian of our bound state classical system with an energy E < 0 is [17]

**L**is the angular momentum. From Hamilton’s equation, $H=E$, so

**A**is conserved in time:

**A**and the definition of angular momentum

**A**is orthogonal to the angular momentum vector

**A**and

**L**are conserved, we can easily obtain equations for the orbits in configuration and momentum space and the eccentricity, and other quantities, all usually derived by directly solving the equations of motion.

#### 3.1. Orbit in Configuration Space

#### 3.2. The Period

#### 3.3. Group Structure SO(4)

#### 3.4. The Classical Hydrogen Atom in Momentum Space

#### 3.5. Four-Dimensional Stereographic Projection in Momentum Space

#### 3.6. Orbit in U space

#### 3.7. Classical Time Dependence of Orbital Motion

#### Remark on Harmonic Oscillator

## 4. The Hydrogenlike Atom in Quantum Mechanics; Eigenstates of the Inverse of the Coupling Constant

#### 4.1. The Degeneracy Group SO(4)

#### 4.2. Derivation of the Energy Levels

#### 4.3. Relativistic and Semi-Relativistic Spinless Particles in the Coulomb Potential and Klein–Gordon Equation

#### 4.4. Eigenstates of the Inverse Coupling Constant ${\left(Z\alpha \right)}^{-1}$

**Because of the boundedness of K, there is no continuum portion in the eigenvalue spectrum of ${\left(Z\alpha \right)}^{-1}$, the eigenvalues are discrete**. Because K is a positive definite Hermitian operator, all eigenvalues are positive, real numbers. This feature leads to a unified treatment of all states of the hydrogenlike atom as opposed to the treatment in terms of energy eigenstates in which we must consider separately the bound states and the continuum of scattering states.

**the quantum numbers, multiplicities, and degeneracies of these states $|nlm;a)$ are precisely the same as those of the usual bound energy eigenstates**. For example, there are ${n}^{2}$ eigenstates of ${\left(Z\alpha \right)}^{-1}$ that have the principal quantum number equal to n or $\left(Z\alpha \right)$ equal to $\frac{na}{m}$.

**A single value of the RMS momentum a or the energy $E=\frac{-{a}^{2}}{2m}$ applies to all the states in our complete basis**, as opposed to the usual energy eigenstates where each nondegenerate state has a different value of a. We have made this explicit by including a in the notation for the states: $|nlm;a)$. Sometimes we will write the states as $\left|nlm\right)$, provided the value of a has been specified. This behavior in which a single value of a applies to all states will prove to be very useful. In essence, it permits us to generalize from statements applicable in a subspace of Hilbert space with energy ${E}_{n}$ or energy parameter ${a}_{n}$ to the entire Hilbert space.

**By a suitable scale change or dilation, we may give the quantity a any positive value we desire**. This is affected by the unitary operator

#### 4.5. Another Set of Eigenstates of ${\left(Z\alpha \right)}^{-1}$

#### 4.6. Transformation of $\mathit{A}$ and $\mathit{L}$ to the New Basis States

#### 4.7. The $\langle {U}^{\prime}|$ Representation

#### Action of **a** and **L** on $\langle {U}^{\prime}|$

## 5. Wave Functions for the Hydrogenlike Atom

#### 5.1. Transformation Properties of the Wave Functions under the Symmetry Operations

#### 5.2. Differential Equation for the Four Dimensional Spherical Harmonics ${Y}_{nlm}\left({U}^{\prime}\right)$

#### 5.3. Energy Eigenfunctions in Momentum Space

#### 5.4. Explicit Form for the Spherical Harmonics

#### 5.5. Wave Functions in the Classical Limit

#### 5.5.1. Rydberg Atoms

#### 5.5.2. Wave Functions in the Semi-Classical Limit

#### 5.6. Quantized Semiclassical Orbits

#### 5.7. Four-Dimensional Vector Model of the Atom

## 6. The Spectrum Generating Group SO(4,1) for the Hydrogenlike Atom

#### 6.1. Motivation for Introducing the Spectrum Generating Group Group SO(4,1)

#### 6.2. Casimir Operators

#### 6.3. Relationship of the Dynamical Group SO(4,1) to the Conformal Group in Momentum Space

## 7. The Group SO(4,2)

#### 7.1. Motivation for Introducing SO(4,2)

#### 7.2. Casimir Operators

#### 7.3. Some Group Theoretical Results

#### 7.4. Subgroups of SO(4,2)

#### 7.5. Time Dependence of SO(4,2) Generators

#### 7.6. Expressing the Schrodinger Equation in Terms of the Generators of SO(4,2)

## 8. SO(4,2) Calculation of the Radiative Shift for the Schrodinger Hydrogen Atom

#### 8.1. Generating Function for the Shifts

#### 8.2. The Shift between Degenerate Levels

## 9. Conclusions and Future Research

## Funding

## Acknowledgments

## Conflicts of Interest

## References and Notes

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