Obtaining hydrogen energy eigenstate wave functions using the Runge-Lenz vector

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 〉 with other degenerate states ∣ n , l ± 1 , m ′ 〉 . 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.


Runge-Lenz Vector and the Hydrogen energy spectrum
In the early days of the development of Quantum Mechanics, Pauli made used of the Runge-Lenz vector and successfully quantized the Hydrogen atom system in the Matrix Mechanics framework [1]. The energy spectrum obtained is identical with the one obtained from the Schrödinger equation [2]. His method is very interesting and ingenious and the consistency contributed greatly to the the development of Quantum Mechanics. Some early development along this line can be found in [3], while the recent development is well summarised in [4].
Since the Runge-Lenz vector commutes with the Hamiltonian, it can connect different degenerate states. Furthermore, as a vector, it is natural to use it to connect energy eigenstate n l m , , ñ | with other degenerate states n l m , 1,  ¢ñ | . It will be interesting to use it to obtain the corresponding wave functions and to show explicitly that they are identical to the results obtained from the Schödinger equation. As we shall see recursive relations of radial wave functions can be obtained. They are consistent with the results obtained by factorized the Schrödinger equation [5][6][7][8] (see also [9]). Nevertheless we believe that it is more natural to use the Runge-Lenz vector, a conserved vector of the system, to obtain the factorization results. The wave functions of the whole spectrum can be obtained easily. We will also briefly discuss the E>0 case and see that the corresponding wave functions can be verified. As in the Pauli analysis, we will see that group theory or symmetry plays a prominent role in the present analysis. Some early works somewhat related to this line of approach can be found in [10][11][12]. We will make a comparison later, as the discussion is rather technical. Nevertheless, as will be shown, it is worth noting that, in the present topics, this work goes beyond those early studies.
The lay our of this paper is as following. In the first section we briefly go through the derivations of the Hydrogen atom spectrum 1 and will concentrate on obtaining wave function via the Runge-Lenz vector in the next section, which is followed by a discussion and conclusion section. An appendix is added for the derivation of some relevant matrix elements using group theory.
Note that Jakob Hermann was the first to show that there exists a vector that is conserved for a special case of the inverse-square central force, and worked out its connection to the eccentricity of the orbital ellipse [14].
Hermann's work was generalized to its modern form by Johann Bernoulli in 1710 [15]. The Runge-Lenz vector satisfies the following relations: It should be noted that the Runge-Lenz vector is a conserved operator. The above relations will be useful in obtaining the Hydrogen atom energy spectrum [1].

Hydrogen atom energy spectrum
The eigenvalue equation of the Hydrogen atom Hamiltonian is given by where we only consider the E<0 case here and α is a possible quantum number. The set of the eigenstates | }with E fixed spans the degenerate space of the energy eigenstates all having the same energy. From equation (4), we know that the Hamiltonian commutes with L  and A  . Hence, it is useful to define the following matrices: and the relations in equations (3), (5) and (6) correspond to the following relations of matrices: The above equation, equation (15), implies that L i and A′ j are the generators of the O(4) group [1]. The quantization of the Hydrogen atom system can be achieved by using group theory [1]. A representation of O(4) can be expressed as a direct product of two SO(3) representations as following 2 . Defining two new sets of operators B i We can now return to the usual Dirac notation. The corresponding eigenvalue equations are . Equation (20) implies the following relation: . The energy eigenvalue E=E n can now be obtained as with n defined as n≡2b+1=1, 2, 3, K. It will be useful to define the (reduced) Bohr radius, a e c Z 4  is the fine structure constant. The energy spectrum obtained by Pauli [1] is consistent with the one obtained from the Schrödinger equation [2] and the consistency contributed significantly to the development of Quantum Mechanics in the early days.
2. Hydrogen atom energy eigenstate 2.1. Connecting degenerate states using the Runge-Lenz vector 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 , , ñ | with other degenerate states, n l m , 1,  ¢ñ | and n l m , , ¢ñ | 3 . As shown in the previous 2 It is probable that a modern reader is more familiar with the case of the SO(3,1) Lorentz group, which can be analyzed using a similar manipulation, see, for example, [16]. 3 The latter states ( n l m , , ¢ñ | ) are, however, prohibited by parity. As we shall see the corresponding matrix elements are vanishing.  ,  1  2  ,  ,  ,  1  2  ,  ,  ,   ,  1  2  ,  ,  1  4  ,  1  2 , , , . In general these eigenstates do not have specified angular momentum quantum numbers and are different from the energy eigenstates in a more familiar basis: Since the angular momentum can be obtained through the following equation, see equation (16), viewed as two independent spin operators, the n l m , , ñ | state can be constructed as in the analysis of the addition of angular momentum: From equation (16), we have A n l l n l l n l l B B n l l E , , , where we apply A − on the m=l state, which is found to be useful in obtaining the radial wave function in later discussion.
Using the familiar formula of non-vanishing matrix elements of lowering operator L − =L x −iL y suitably, we have where the derivation are shown in appendix. Note that the above results also hold for the l=0 case, where the equation implies that n l l B n l l , are vanishing as they should.
Using the above equation, the corresponding matrix elements of A − are given by Note that the matrix element for the l′=l state is vanishing as required by parity. Substitute them into equation (28), we finally obtain our master formula: A n l l Z na n l l l n l l l n l l n l l , , Note that as in the Pauli analysis, the above master formula follows from group theory or symmetry. As we shall see the above equation can provides recursive relations on degenerate states, and the relations are powerful enough to determine the wave functions.
To proceed we need to work out the left-hand-side of the above equation. Using p L L p i p 2  ´ + ´ =  , it is convenient to express the±components of the Runge-Lenz vector as where we have defined r ± ≡x±iy. Consequently, the operator (A − ) op in equation (34)

It can be further expressed as
where we have made use of equation ( It is useful to note that the spherical harmonics Y l, l has the following properties 4 : These are the recursive relations of the radial wave function R nl (r) and they are important results of this section. The above relations are consistent to the results found in [5][6][7] (see also [9]) using the factorization method. Nevertheless we believe that according to the properties of the Runge-Lenz vector, which is a conserved vector, the above derivation is the most natural way to obtain them. As we shall see shortly they are powerful enough to determine the radial wave functions. 4 The first relation follows from equation (39) , Using r n l m R r Y , , , we clearly see that the it is equivalent to equation (48).

2.2.3.
Obtaining other R r nl ( ) Once R n,n−1 (r) is known, other R n, l (r) can be obtained readily by applying the second relation of the recursive relations given in equation (47): In principle, the procedure can be carried out to obtain all R nl (r).
It will be useful to show explicitly some of the radial wave functions obtained: and compare them to those obtained by solving the Schrödinger equation directly, see for example, [18]. Indeed, it is clear that the wave functions obtained in the two approaches are consistent 5 . It is interesting that even the phase conventions match.

The radial wave function R nl satisfies the radial Schrödinger equation
Applying the recursive relations, equation (47), on R n,l /r l can bring it to R n,l±1 /r l±1 and suitably apply the relations again can bring them back to R n, l /r l . These procedures produce the following identities on R nl /r l , 6 It is clear that equation (63) is same as the above equation with E Z a n 2 n 2 2 0 2 2  m = -. 5 Note that there is a typo in the normalization factor of R 31 (r) in [18]. 6 The additional factors 1/(l+1) 2 However, now the situation is different. The above relations cannot be used to obtain energy eigenvalue as in the . Solving the Schrödinger equation now reduces to finding functions that satisfy the recursive relations in equation (70).

Discussion and conclusions
We now compare our results with those obtained in [10][11][12].
In [10], Stahlhofen and Bleuler, pointed out the relation of factor (in the factorization approach) and the Runge-Lenz vector. In their two one-page discussions on this issue [10], they defined