Raising and lowering the parabolic eigenstates of hydrogen
C. E. Burkhardt
St. Louis Community College at Florissant Valley
3400 Pershall Road
St. Louis, MO 63135-1499
J. J. Leventhal
Department of Physics
University of Missouri - St. Louis
St. Louis, MO 63121
Abstract
Application of the angular momentum ladder operators to energy eigenfunctions for any central
potential converts the eigenfunction to one having a neighboring value of m, the quantum number
associated with the z-component of angular momentum. For the hydrogen atom it is possible to
further convert eigenfunctions that result from separation of the Schrödinger equation in
spherical coordinates to eigenfunctions having neighboring values of  the quantum number
associated with the magnitude of the total angular momentum. In this paper it is shown how to
analogously convert hydrogen atom eigenfunctions that result from separation of the Schrödinger
equation in parabolic coordinates to eigenfunctions having neighboring values of the parabolic
quantum numbers. The derivation is performed using formalism no more sophisticated than that
used to derive the properties of the ordinary angular momentum ladder operators in
undergraduate quantum mechanics courses.
PACS numbers: 03.65.Ca ; 03.65.Fd ; 03.65.Ge
1
I. Background
The action of the angular momentum ladder operators on energy eigenfunctions for central
potentials is well-known and a subject covered in most introductory quantum mechanics courses.
As a result of the spherical symmetry of the potential, the energy eigenvalues are independent of
m, the eigenvalue of the z-component of the angular momentum L̂ z . Application of the angular
momentum ladder operators, Lˆ   Lˆ x  iLˆ y , reflect this degeneracy in m by changing it while
leaving the principal quantum number n and the total angular momentum quantum number 
unaltered. In general, the energy eigenvalues depend upon both n and  so that application of L̂
does not shift either of these quantum numbers. For the hydrogen atom, however, there exist
additional, but not often studied, ladder operators that shift the values of  (and, in some cases, m
as well), but leave n intact[1]. This reflects the super symmetry of the Coulomb potential that is
responsible for the celebrated accidental degeneracy of the hydrogen atom in which the energy is
independent of  as well as m. By combining the action of both sets of ladder operators it is
possible to generate the entire set of eigenfunctions for a given n starting with the  = 0
eigenfunction.
The extra symmetry of the Coulomb potential makes it possible to separate the Schrödinger
equation parabolic as well as spherical coordinates. There are thus two sets of hydrogen atom
eigenfunctions, the common spherical eigenfunctions which we designate by the ket nm , and
the parabolic eigenfunctions, nn1n2 m . The parabolic quantum numbers are related by
n  n1  n2  m  1
1
which reduces the number of independent quantum numbers to three as for spherical coordinates.
We remark that, traditionally, the absolute value of m occurs in Equation (1)[2]. In the work
presented here, however, inclusion of the absolute value of m in Equation (1) precludes proper
2
negative values of m.
Indeed, without the absolute value m takes on its proper values
 n  1  m  n  1.
The quantum number m is the same in both coordinate systems inasmuch as L̂ z is chosen as one
of the mutually commuting operators used to effect the separation in both systems. The operator
L̂2 , which is one of the mutually commuting operators in spherical coordinates, however, is not
used in the parabolic coordinate treatment so these eigenstates are not states of definite angular
momentum. They are, however, states having definite z-component of the angular momentum.
Since  is not a "good quantum number" in parabolic coordinates it is clear that the usual angular
momentum ladder operators cannot be employed to transform one eigenfunction into another.
Nonetheless, it is possible to use generalized angular momentum algebra for this purpose as will
be demonstrated in this paper. Moreover, the methods used are no more sophisticated than those
employed in introductory quantum mechanics courses when introducing the common angular
momentum ladder operators.
II. Formalism
We begin by noting that the super symmetry of the Coulomb (or Kepler) potential manifests itself
classically as an additional constant of the motion, that is, in addition to energy and angular
momentum. This constant is known by several names, but we adopt the appellation "Lenz
vector" for simplicity. Of course, this constant of the motion must correspond to a quantum
mechanical operator that commutes with the Hamiltonian Ĥ . This operator was first introduced
by Pauli[3]. In atomic units for which the mass of the electron, the electronic charge and  are all
set to unity, the Lenz vector operator is


1
Aˆ    pˆ  Lˆ  Lˆ  pˆ  rˆ
 2
2
As defined above, Â does indeed commute with Ĥ . Moreover, it is Hermetian, as it must be
3
since it represents an observable quantity. A number of relations between  and the other


relevant operators are listed in Table I. Notice that Aˆ i , Aˆ j  iAˆ k as it would if  were truly an
"angular momentum". Pauli showed in his brilliant 1926 paper that by scaling  and forming
linear combinations with L̂ an angular momentum could be constructed. Moreover, these
operators provide a means of obtaining the energy eigenvalues,  1 / 2n 2 in atomic units. We
perform this derivation[4] because some of the results are crucial to the development of the
ladder operators that are the main focus of this paper.
We begin by defining an operator  ' as
3
 2 Hˆ Aˆ '  Aˆ
Since this operator is to operate only on eigenfunctions of the hydrogen atom Â' may be written
1 ˆ
A
2E
Aˆ ' 
4
where E is the (unknown) energy eigenvalue. The commutators involving  ' are
Lˆ , Aˆ '   iAˆ '
Aˆ ' , Aˆ '   iLˆ
i
j
5
k
i
j
k
Now define two new operators Î and K̂ as




1
Iˆ    Lˆ  Aˆ '
 2
1
Kˆ    Lˆ  Aˆ '
 2
6
Using the commutation relations above we find that
Iˆ, Kˆ   0
Iˆ , Iˆ   iIˆ et cycle
Kˆ , Kˆ   iKˆ et cycle
Iˆ, Hˆ   0  Kˆ , Hˆ 
i
j
i
k
j
4
k
7 
It is seen that Î and K̂ each obey the commutation rule that is the very definition of an "angular
momentum". Therefore, the eigenvalues of Î 2 and K̂ 2 are (in atomic units)
Iˆ 2  ii  1
and
Kˆ 2  k k  1
8
where i and k are quantum numbers that can take on only the values
9
1 3
5 7
i, k  0, ,1, , 2, , 3, , ...
2 2
2 2
We choose Iˆz and K̂ z as the components that commute with Î 2 and K̂ 2 and designate their
quantum numbers as mi and m k . The eigenfunctions of Iˆ 2 , Kˆ 2 , Iˆz
and K̂ z may thus be
designated i, mi ; k , mk . Since they consist of two independent angular momenta it is clear that
these kets represent uncoupled angular momenta. Although the quantum numbers appear to be
different, the i, mi ; k , mk
must be the parabolic eigenfunctions because they are not states of
definite total angular momentum as are the spherical eigenfunctions. Moreover, the mutually
commuting operators that spawned them contain the Lenz vector operator, an operator that was
not included (or needed) in the spherical coordinate treatment. These uncoupled eigenfunctions
are, of course, also eigenfunctions of the Hamiltonian so the quantum number n is implicit in
them. This relationship will now be derived.
Squaring Î and K̂ gives


10


11
1
Iˆ 2    Lˆ 2  Lˆ  Aˆ ' Aˆ ' Lˆ  Aˆ ' 2
 4
and
1
Kˆ 2    Lˆ 2  Lˆ  Aˆ ' Aˆ ' Lˆ  Aˆ ' 2
 4
ˆ ' Lˆ  0  Lˆ  A
ˆ ' so that
But, A
5

1
Iˆ 2    Lˆ 2  Aˆ ' 2
 4
 Kˆ 2

12
Therefore, i = k and
Iˆ 2  Jˆ 2  ii  1  k k  1
13
1 3
 2ii  1 ; i  0, ,1, , 2, ...
2 2
In terms of the modified Lenz vector  ' the relationship between  2 , L̂ and Ĥ is (see Table I)
1  2Hˆ Aˆ '
2


 Lˆ 2  2 Hˆ  0
14
Since, however
Aˆ '
2

 Lˆ 2  4 Kˆ 2
 4 Iˆ 2
15
we have
1  2Hˆ 4 Iˆ   2Hˆ  0
2
16
Applying this operator to one of the i, mi ; k , mk which we condense to i, mi , mk we have
1  2E4ii  1  2E  0
17
because i, mi , mk is an eigenfunction of all operators in Equation (16). Solving Equation (17)
for E we have
1 1
E   
2
 2  2i  1
18
Now, (2i + 1) must be an integer because i can take on all positive half-integral and integral
values. Therefore, 2i 1  n , where n = 1, 2, 3, ... Of course, n is the principal quantum number
and
En  
1
2n 2
which is the correct eigenvalue in atomic units.
6
19
To derive the forms and actions of the ladder operators we use two of the results of the derivation
of the energy, i = k and i  n  1 / 2 . Because Î and K̂ are angular momenta we know the
forms and actions of their ladder operators. Since they are identical we work only with the
operator Î . We have
20
Iˆ  Iˆx  iIˆy
Application of these operators produces the following action
Iˆ i, mi , mk   ii  1  mi mi  1 i, mi  1, mk
21
Clearly then, to use this formalism to convert the nn1n2 m into another parabolic eigenfunction
it is necessary to find the relationship between the apparently different sets of quantum numbers.
First, we note that because Lˆ z  Iˆz  Kˆ z
22
m  mi  mk
Noting the symmetry between n1 and n2 and between mi and mk we may write, using Equation (1)
23
n  2n1  2mi  1
Of course, we could have used any combination of n1, n2 and mi and mk. Equation (23) leads to
the desired relationships
n1 
n  1  m
24
i
2
n  1  m
n2 
k
2
where the n2 equation follows by symmetry. We know that  i  mi  i and, from the derivation
of the energy eigenvalues we know that 2i 1  n . Therefore,

n  1   n  1  n   n  1
2


1
2


2
25
from which we find that
0  n1  n  1
7
26
and, of course,
0  n2  n  1
27
Now, to investigate the action of the ladder operators we choose to examine application of Iˆ to
i, mi , mk to obtain
Iˆ i, mi , mk  ii  1  mi mi  1 i, mi  1, mk
28
Noting from Equation (24) that lowering mi while keeping n constant in fact raises n1, we may
cast Equation (28) in terms of n, n1 and n2.
Iˆ nn1 n2 m 

n  1 n  1   n  1  n   n  3  n  n, n
2
2

2
1
 
2
1

1
 1, n2 , m  1
29
n1  1n  n1  1 n, n1  1, n2 , m  1
For clarity we have continued to include the magnetic quantum number m in the eigenkets
although, by Equation (1), it is redundant.
By symmetry we write
Kˆ  nn1n2 m 
n2  1n  n2  1 n, n1 , n2  1, m  1
30
Note that if n1 or n2 has its maximum value, n-1, then application of Iˆ or K̂  annihilates this
ket. To lower n1 or n2 it is necessary to raise mi or mk. We obtain for Iˆ
Iˆ nn1n2 m  n1 n  n1  n, n1  1, n2 , m  1
31
with analogous equation for the application of K̂  .
III. Generation of a manifold of parabolic states
Generation of a set of parabolic states from one of the set is merely a matter of judiciously
applying the raising and lowering operators Iˆ and K̂  . For example, suppose we know the
parabolic eigenstate n n1 n2 m  n11n  3 . If we successively apply Iˆ and K̂  we obtain
8
Iˆ n11n  3  2n  2 n21n  4
32
Kˆ  n21n  4  2n  2 n22 n  5
33
Applying Iˆ to the original n11n  3 we have
Iˆ n11n  3  n  1 n01n  2
34
Clearly successive applications of these operators can generate the entire manifold of parabolic
states for a given n.
IV. The action of Â , L̂ and Âz on parabolic eigenfunctions.
By noting that
Lˆ   Iˆ  Kˆ 
35
Aˆ '  Iˆ  Kˆ 
36
and
the actions of these operators on a parabolic eigenstate may be obtained. For example, applying
Â to an arbitrary parabolic ket we have
Aˆ ' i; mi , mk  Aˆ ' nn1 n2 m
37 
 Iˆ i; mi , mk  Kˆ  i; mi , mk

i  mi i  mi  1 i; mi  1, mk 
i  mk i  mk  1 i, mi , mk  1
1
   n  1  2mi n  1  2mi  i; mi  1, mk 
 2
1
  n  1  2mk n  1  2mk  i, mi , mk  1
 2
It is common in the parabolic coordinate description[5] to combine quantum numbers and define
q  n2  n1  .
This is referred to as the electric quantum number because the parabolic
eigenfunctions are also eigenfunctions when the hydrogen atom is subjected to a constant electric
field. Together with Equation (22) then the action of Â on a parabolic eigenket is
9
1
Aˆ ' nn1 n2 m 
2
n  1  m  q n  1  m  q  n n1  1 n2 m  1
1
2

38
n  1  m  q n  1  m  q  n n1 n2  1m  1
Similarly, for the other three operators we obtain
1
Aˆ ' nn1 n2 m 
2
n  1  m  q n  1  m  q  n n1  1 n2 m  1
1
2

39
n  1  m  q n  1  m  q  n n1 n2  1m  1
and
1
Lˆ  nn1 n2 m 
2
n  1  m  q n  1  m  q  n n1  1 n2 m  1
1
2

40
n  1  m  q n  1  m  q  n n1 n2  1m  1
In addition to the above results we may also find the action of Aˆ z' , and thus Âz , on parabolic
eigenfuntions. We have
Aˆ z' nn1n 2  Iˆz i; mi ,m k  Kˆ z i; mi ,m k
41
 mi i; mi ,m k  mk i; mi ,m k
 mi  mk  i; mi ,m k
 q nn1n 2
 n2  n1  nn1n 2
Thus, parabolic eigenfunctions are eigenfunctions of Aˆ z' with eigenvalues q. The observable Âz
thus has possible values q / n . Clearly then, the commuting operators employed when solving
the hydrogen atom problem in parabolic coordinates are Ĥ , L̂ z and Âz .
V. Generation of Clebsch-Gordan coefficients.
Since the spherical eigenfunctions and the parabolic eigenfunctions each constitute a complete
set it is possible to expand an eigenfunction of one set as a linear combination of the other. As
10
noted previously, the parabolic states, in effect, contain two uncoupled angular momenta, Iˆ 2 and
K̂ 2 while the spherical states are states of coupled angular momenta. Thus, the expansion of an
eigenfunction of one set in terms of eigenfunctions of the other set requires the Clebsch-Gordan
coefficients[6]. Using the formalism developed here it is possible, in some cases, to obtain these
coefficients. Of course, the tables of Clebsch-Gordan coefficients are more useful for this
purpose, but performing the expansion using the formalism developed here can provide useful
insight.
As an example we begin with one of the states that are the same in either coordinate system.
These are
n n  1n  1
sph
 n00 n  1
42
par
where subscripts have been added to the kets for clarity. Equation (42) holds m is a good
quantum number in both coordinate systems so for the kets to be the same they must have the
same value of m. For the case in which n1  0  n2 we have only m  n  1. But for, m  n  1,
its maximum value, , must have its maximum value, that is,   n  1 . Thus, there is only one
state in each basis set that has a maximum (or minimum) value of m.
We apply L̂ to the spherical eigenfunction and L̂ in the form Lˆ   Iˆ  Kˆ  to the parabolic
eigenfunction and obtain
Lˆ n n  1n  1
 2n  1 n n  1n  2
sph
sph
43
and
Lˆ  n 0 0 n  1
par


 Iˆ  Kˆ  n 0 0 n  1

par
n  1 n 1 0 n  2 par  n  1 n 0 1 n  2 par
so that
11
44
n n  1n  2
sph

1
n 1 0 n  2
2
par

1
n 0 1 n  2
2
par
45
Successive application of Lˆ   Iˆ  Kˆ  will generate the entire set of spherical eigenfunctions
for   n  1 . Other Clebsch-Gordan coefficients can be obtained, but with considerably more
labor and are thus not within the scope of the work presented here.
12
REFERENCES
1. C. E. Burkhardt and J. J. Leventhal, "A complete set of ladder operators for the hydrogen
atom," Am. J. Phys. (submitted).
2. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms
(Springer-Verlag; Berlin; 1957). pp 228-232.
3. W. Pauli, "On the Hydrogen Spectrum from the Standpoint of the New Quantum Mechanics"
The English translation is in: Sources of Quantum Mechanics edited by B. L.d Van Der Waerden
(Dover; New York; 1967) pp 387 - 415.
4. L. I. Schiff, Quantum Mechanics (McGraw-Hill; New York; third edition; 1968). P 234.
5. L. C. Biedenharn, J. D. Louck and P. A. Carruthers, Angular Momentum in Quantum
Mechanics: Theory and Application (Addison-Wesley; Reading, MA; 1981). Pp 335 – 344.
6. D Park, "Relation between the parabolic and spherical eigenfunctions of hydrogen," Z. Phys.
159, 155 – 157 (1960).
13
Table I. List of some relations involving the quantum mechanical operators used in this work.
ˆ  Lˆ  0  Lˆ  A
ˆ
A
Lˆ , Lˆ   iLˆ
i
j
k
Lˆ , Aˆ   iAˆ
i
j
k
Aˆ , Aˆ   2iLˆ Hˆ
i
j
k


Aˆ 2  2 Lˆ2  1 Hˆ  1
14