Accidental Degeneracy of Hydrogen Atom Revisited

In quantum mechanics, it is generally believed that in static sense the geometric symmetry of a Hamiltonian is responsible for the degeneracy of energy levels. If this situation does not occur, we customarily claim that there is an accidental symmetry that causes the accidental degeneracy of energy levels. Sometimes, the term “hidden symmetry” is also used as the synonym of accidental symmetry in the literature. In non-relativistic quantum mechanics, the well known example of accidental degeneracy is the hydrogen atom with Coulomb 1/r potential, in which the degree of degeneracy for a bound-state of principal quantum number n is equal to n2 but not the expected 2l+1 value as anticipated from a generic central potential. Here, we ignore the spin degree of freedom.


Introduction
In quantum mechanics, it is generally believed that in static sense the geometric symmetry of a Hamiltonian is responsible for the degeneracy of energy levels. If this situation does not occur, we customarily claim that there is an accidental symmetry that causes the accidental degeneracy of energy levels. Sometimes, the term "hidden symmetry" is also used as the synonym of accidental symmetry in the literature. In non-relativistic quantum mechanics, the well known example of accidental degeneracy is the hydrogen atom with Coulomb 1/r potential, in which the degree of degeneracy for a bound-state of principal quantum number n is equal to n 2 but not the expected 2l+1 value as anticipated from a generic central potential. Here, we ignore the spin degree of freedom.
Pauli was the pioneer who introduced the Runge-Lenz operator of the hydrogen atom to explain the accidental degeneracy in the energy levels [1]. The mystery of that accidental symmetry was then disclosed by Fock, who found that all the bound-state eigenfunctions in ordinary three-dimensional space can be translated into so (4) hyperspherical harmonic functions constructed in a fictitious four-Euclidean space [2]. As a result, eigenfunctions with the same value of n, but different orbital angular momentum quantum number l, can be transformed into one another by rotations in this four-dimensional mathematical space. Later, Bargmann gave a Lie algebraic approach of the so(4) hyperspherical symmetry to demonstrate that the accidental degeneracy of the hydrogen atom is indeed due to a hidden symmetry, which is not completely realized in the original three-Euclidean space [3].
To be more specific, the Hamiltonian of hydrogen atom in ordinary three-dimensional space is of the form φ . Obviously, the potential energy and kinetic energy separately obey the so(3) rotational symmetry. Therefore, the so-called so(4) hyperspherical symmetry of the bound-state eigenfunctions is definitely not a symmetry of the potential energy, nor is it a symmetry of the kinetic energy. It is a symmetry of the entire Hamiltonian itself! In such away, rotations of so(4) hyperspherical symmetry will leave the Hamiltonian H invariant, hence the energy eigenvalues E n , too.
We remark that in reality the so(4) hyperspherical symmetry is not the maximal symmetry the Hamiltonian H can possess when it is in motion. It was discovered in the 60's that the time-dependent admits an even larger "dynamical symmetry" than the accidental symmetry realized by the corresponding time-independent one [4][5][6][7][8][9][10][11]. The reason is that in time-dependent Schrödinger equation the energy eigenvalues E n can be treated as a variable. So, there exists an additional spectrum-generating so (2,1) symmetry that can be used to vary among different energy levels [12,13]. In effect, this so(2,1) symmetry is combined with the originally so(4) hyperspherical symmetry to form an irreducible representation of an extended so (4,2) dynamical symmetry, normally known as the conformal symmetry. For complete tabulations of so (4,2) dynamical symmetry of the hydrogen atom [14,15].
In this paper, we revisit accidental symmetry of hydrogen-like atoms in the framework of time-independent Schrödinger equation. Some analytical properties associated with that symmetry are explored. Rescaled Runge-Lenz vectorial operator is introduced, to simplify the presentations. The algebra of the symmetry is briefly reviewed. Differential equations of the eigenstates, when acted on by ladder-type generators of the algebra, are inspected. Analytical properties of the ladder generators constructed in dimensionless spherical coordinates are examined, particularly for both bound states and continuous states. This results in the discovery of some unfamiliar nevertheless notable functional relations for the radial eigenfunctions of the Hamiltonian. Moreover, similar analytical properties of the radial eigenfunctions in two-dimensional hydrogen atoms are subsequently demonstrated.
The paper is organized as follows. In section 2, we briefly review the main algebraic structures of the accidental symmetry. In section 3, we analyze analytical properties of accidental symmetry and deduce several functional relations for the radial eigenfunctions. We show analytical similarity between two-dimensional and three-dimensional hydrogen atoms in section 4. Finally, section 5 contains the conclusion.

Symmetry of Coulomb Potential
In this section, we briefly review for the purpose of completeness the main algebraic structures of a charged particle in the presence of Coulomb 1/r potential. The Hamiltonian that we will consider is the hydrogen-like atom of atomic number Z, which takes the same form positive variable for the bold-plus (E>0) ase and n Z + ∈ a positive integer for the bold-minus (E<0) case. Note that, for the E>0 case, the continuous variable n is related to the wave number k of positive energy With such a choice, the eigenvalue equations for L 2 and L z are the standard ones found in quantum mechanical textbooks, while that for operator A 2 can be established from the operator identity: To further analyze algebraic properties of the symmetry, it proves convenient to introduce ladder-type operators for both L and A by i , from which we immediately arrive at the operator identities: 2 As a result, the commutation relations in eqns. (3)-(5) can be reformulated in terms of these ladder-type operators into: some other commutators not shown above are trivially zero.
The action of ladder-type operators on the designated eigenstates is again standard for L, that is, As regards to those for A, we have already in mind that it is a spin-one tensor operator. Therefore, according to general rules for addition of angular momenta, we may formally write the corresponding equations by here, the coefficients α i (l.m) and β i (l.m) (for = , ,0 i + − ) also depend on the parameter n, though it is suppressed in the expressions.
The coefficients α i (l.m) and β i (l.m) can be directly determined from the commutation relations listed in eqns. (9) and (10). For instance, applications of the first commutator of eqn. (9) give rise to the following coefficients relations Furthermore, other commutators or identities will give the explicit forms for both α 0 (l.m) and β 0 (l.m). After some manipulations, we get The relations below reflecting the symmetry of Coulomb 1/r potential can be deduced easily It is known from textbooks of quantum mechanics that the algebra of accidental symmetry of the Hamiltonian H is generated by the angular momentum operator = ( , , ) x y z L L L L and the so-called Runge-Lenz vectorial operator = ( , , ) given by [16][17][18] The commutation relations among the two sets of L and R operators become simplified, if we are tempted to work on the representation of eigenstates of the Hamiltonian. In this way, H can be directly replaced by the energy eigenvalue E of the corresponding eigenstate considered. Owing to this fact, let us define the rescaled Runge-Lenz operator that has the dimensions of  , similar to the angular momentum L does, as: In eqn. (2), because the eigenvalues E of H contains a discrete set and a continuous energy variable, we are led to distinguish these two different spectra by introducing the (±) symbol. It means that the boldfaced plus sign in the parenthesis stands for the case of continuous states (E>0). whereas the boldfaced minus sign for that of bound states (E<0). Thereinafter, we shall adopt this neat convention of the symbol (±) throughout the paper, unless it is otherwise emphasized. That is to say, if an equation appears to not have any (±) symbol, it simply means that this equation holds true for both E>0 and E<0 cases.
The convention of bold-plus for E<0 and bold-minus for E<0 in eqn. (5) is understood. Needless to say, the commutation relations in above equations describe an so(3,1) symmetry for E>0 and an so(4) symmetry for E<0. In is noted that for E<0 the so(4) algebra may be which furthermore admits another realization in terms of two sets of bosonic spin operators [19][20][21]. As a byproduct, eqn. (4) implies that the Runge-Lenz operator A transforms in the spin-one (vector) representation under three-dimensional spatial rotations.
In quantum mechanics, the symmetry of a system is encoded in the maximal set of mutually commuting operators that includes the Hamiltonian inside of it. The set is responsible for the degeneracy of energy spectrum and results in various constants of motion in dynamics. For the accidental symmetry of hydrogen atoms we considered, the commuting set can take, for example, to be

Properties of the Symmetry
The algebraic structures of Coulomb potential presented in section 2 enable us to investigate some analytical properties of the eigenstates | , > n l m in the spherical coordinates, r=(r,θ,φ). Before going into the details, let us make one more variable change to simplify our presentations. Instead of r, we define the dimensionless position vector . Since L 2 is still a good observable, it is again legal to separate ψ nlm (ρ)into the radial R nl (ρ) and angular ( , ) m l Y θ φ parts as where A ln is the normalization factor that depends on both l and n.
In the dimensionless spherical coordinates ρ=(ρ,θ,φ), the Runge-Lenz operators (A z ,A+,A-) are expressed as follows Having found eqns. (18) and (19), we now analyze on what analytical constraints that are set by algebraic eqs. (11) to (13) are imposed on the eigenfunctions ψ nlm (ρ). Let us answer this question by first taking the differential operator remember that eqn. (22) must be identical to eqn. (20). For such assertion to hold, there are in general two alternative options in the discussion of the Coulomb 1/r potential. It depends on whether E<0 or E>0, Let us discuss them separately.
1. The case of E<0. The spectrum of negative energy values is discrete and depends on the principal quantum number n. The degree of degeneracy of the n-th energy level is n 2 , for l=0 to n-1 and m=-l to l. This case corresponds to take the boldfaced minus in the symbol (±) in eqn. (22), in which the normalization factor A nl is normally chosen to satisfy the recurrence relation where A n0 is an n dependent constant, not completely fixed by the purely algebraic method. In eqn. (24), the product is replaced by unity for l=0.
Turn to eqn. (22), the recurrence relation (23) additionally determine two first-order linear differential equations that are satisfied by the radial functions R nl (ρ). They are respectively where A n0 is another n dependent constant, not completely fixed by algebraic methods. Similarly, the product have to be replaced by unity for l=0.
In the same vein, the recurrence relation (30) additionally determine two first-order linear different equations of the radial functions R nl (ρ). They are: , and ask whether we get any new properties out of the associated differential equations generated by them. To answer it, we take the differential given in eqn. (19) and act the both on the eigenfunctions ( ) nlm ψ ρ . We arrive at In two-space, analytical properties of the eigenstates | , > n m can be studied by projecting them into the plane-polar coordinates, r=(r,φ). To present the results, we likewise prefer to adopt the dimensionless position vector α where A nm is the normalization factor that depends on both n and m.
The ladder-type operators A ± in the coordinates ρ=(ρ,φ) take the forms now, acting the differential operators (44) on the eigenfunctions ψ nm (ρ) (43) and comparing the obtained result with the algebraic ones (42), we reach the following two valuable functional relations Similar to that in three-space, there are two options for the choice

Conclusion
In this paper, we study accidental symmetry of the hydrogenlike atoms, based on the time-independent Schrödinger equation. The rescaled Runge-Lenz vectorial operator is adopted to reduce cumbersome notations. The algebra described by the angular momentum L and Runge-Lenz operator A is briefly reviewed.
The representation of eigenstates for the commuting operators 2 2 { , , , } z H L L A are inspected, as well. Acted on by the ladder-type operators such as A ± and A z analytical properties of the eigenfunctions ψ nlm (ρ) constructed in the dimensionless spherical coordinates are accordingly deduced, especially for both bound states and continuous states. Some unfamiliar functional relations of the radial eigenfunctions of the Hamiltonian are established. Similar analytical properties shared by both two-dimensional and three-dimensional hydrogen atoms are illustrated and analyzed. To conclude, it seems that functional relations in both equs. (28) and (29) for the radial eigenfunctions are quite general in the theories of hydrogen-like atoms. Though it is not shown here, same conclusion will be reached for eqns. (35) and (36), too.