Download Lenz vector operations on spherical hydrogen atom

Lenz vector operations on spherical hydrogen atom eigenfunctions
C. E. Burkhardt
St. Louis Community College at Florissant Valley, 3400 Pershall Road, St. Louis, Missouri 63135-1499
J. J. Leventhala)
Department of Physics, University of Missouri—St. Louis, St. Louis, Missouri 63121
共Received 10 September 2003; accepted 9 April 2004兲
We show that operators capable of converting certain spherical hydrogen atom eigenfunctions into
others by raising the angular momentum quantum number ᐉ can be constructed from components of
the quantum mechanical Lenz vector operator. The derivation is performed using formalism no more
sophisticated than that used to derive the properties of the angular momentum ladder operators in
undergraduate quantum mechanics courses. The properties of these Lenz vector operators illustrate
the accidental degeneracy of the hydrogen atom because they change the quantum number ᐉ without
affecting n. © 2004 American Association of Physics Teachers.
关DOI: 10.1119/1.1758225兴
I. INTRODUCTION
It is well-known that the Kepler/Coulomb potentials endow planetary orbits and hydrogen atoms with special properties which are not present in systems subject to other central potentials. For example, a pure Keplerian orbit does not
precess. Moreover, the total energy of the system depends
only on the value of the semi-major axis and not on that of
the semi-minor axis. There exists, therefore, an infinity of
possible orbits, all having the same energy and semi-major
axis, but having different values for the semi-minor axis.
While the semi-major axis is independent of the angular momentum, the semi-minor axis is determined by the angular
momentum, so the energy is independent of the angular momentum. This classical degeneracy is the result of the same
symmetry of the 1/r potential that causes the celebrated ‘‘accidental degeneracy’’ of the hydrogen atom, i.e., the independence of the energy eigenvalues on the angular momentum
quantum number ᐉ. This symmetry is different from the spatial symmetry that exists for any central potential. It is often
referred to as a ‘‘dynamical symmetry.’’1 The spatial symmetry causes the energy to be independent of m, the quantum
number corresponding to the z-component of the angular
momentum.
Classically, the dynamical symmetry manifests itself as an
additional constant of the motion, the Lenz vector A, which
points along the major axis of the ellipse.2,3 This results in an
orbit that does not precess. Quantum mechanically, A corresponds to an additional operator  that commutes with the
Hamiltonian Ĥ as shown by Pauli in his landmark paper.4 Of
course the magnitude of the angular momentum and each of
its components also commute with Ĥ for any central potential.
Although these consequences of the dynamical symmetry
of the 1/r potential are rarely discussed at the undergraduate
level, they provide insight into a variety of classical and
quantum concepts. In this paper we will concentrate on the
quantum mechanical consequences of the classical constant
A. We will show that it is possible to derive a set of operators
consisting of components of  that transform certain spherical hydrogen eigenstates into other spherical eigenstates.
Spherical eigenstates are those that result from separation of
the Schrödinger equation in spherical coordinates and are
characterized by the quantum numbers n 共energy兲, ᐉ 共angular
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Am. J. Phys. 72 共8兲, August 2004
http://aapt.org/ajp
momentum兲, and m 共z-component of angular momentum兲.
The dynamical symmetry also makes it possible to separate
the Schrödinger equation using parabolic coordinates.5 We
also note that it is possible to take advantage of the dynamical symmetry to determine the energies of the hydrogen atom
eigenstates using operator techniques alone.1,4
II. BACKGROUND
The spherical form of any vector may be defined in terms
of the Cartesian components. For the angular momentum we
have
L̂ ⫾ ⫽L̂ x ⫾iL̂ y ,
L̂ 0 ⫽L̂ z .
共1兲
The fact that the L̂ ⫾ change only the value of m and neither ᐉ nor n when operating on a spherical eigenstate for any
central potential is a manifestation of the degeneracy associated with the spatial symmetry of a central potential. Thus,
the L̂ ⫾ are ladder operators for any central potential. The
spherical form of the Lenz vector operator may be similarly
defined as
 ⫾ ⫽ x ⫾i y
 0 ⫽ z .
共2兲
It will be shown that  ⫾ and  0 共henceforth referred to as
 z ) effect state-to-state changes in the ᐉ quantum number of
certain hydrogen atom spherical eigenfunctions, viz. states
for which 兩 m 兩 ⫽ᐉ. Because  ⫾ and  z change ᐉ without
changing the energy they reflect the accidental degeneracy of
the hydrogen atom in which the energy eigenvalues are independent of ᐉ. In this paper we will derive the properties of
 ⫾ and  z and obtain the exact expressions for the actions of
these operators on selected eigenstates without having to resort to advanced concepts.6 The derivation will be carried out
using formalism no more sophisticated than that used to derive the properties of L̂ ⫾ in most undergraduate quantum
mechanics courses. Additionally, it will be seen that, given
the ᐉ⫽0 eigenfunction for a particular n, all eigenfunctions
for that n can be obtained by judicious application of a combination of L̂ ⫾ ,  ⫾ and  z .
Using atomic units for which ប⫽1⫽e⫽m e , where e is
the electronic charge and m e the electronic mass, the Lenz
vector in classical mechanics is defined in atomic units as
© 2004 American Association of Physics Teachers
1013
Table I. List of some relations involving the quantum mechanical operators
used in this work. ⑀ i jk is the Levi–Civita symbol defined as ⫹1 for an even
permutation of the i, j, k and ⫺1 for an odd permutation; it vanishes if two
or three indices are the same.
⫺2Â z 冑ᐉ 共 ᐉ⫹1 兲 ⫺m 共 m⫹1 兲 兩 nᐉ; 共 m⫹1 兲 典 .
ᐉ 共 ᐉ⫹1 兲 ⫹2 共 ᐉ⫹1 兲 ⫽ 共 ᐉ⫹1 兲共 ᐉ⫹2 兲 .
共3兲
共4兲
共5兲
where we have inserted a semi-colon between quantum numbers in the ket on the right hand side for clarity. Equation 共5兲
is, of course, also valid for spherical hydrogen atom eigenfunctions since the Coulomb potential is a central potential.
For convenience, a number of relations between the various operators to be used in this work are compiled in Table I.
Some require lengthy algebraic manipulations, but the derivation of all is straightforward. We investigate the action of
 ⫹ on the spherical hydrogen eigenfunction 兩 nᐉm 典 . We indicate that this is a hydrogen eigenfunction by omitting the
subscript C that was used to designate an eigenfunction for
an arbitrary central potential. Using the relations contained in
Table I we find that
L̂ z 兵  ⫹ 兩 nᐉm 典 其 ⫽L̂ z 共  x ⫹i y 兲 兩 nᐉm 典
共9兲
⫹
D ᐉᐉ
is a constant that depends on ᐉ. This result is
where
stated in the textbook by Shankar,7 but the derivation there
employs tensor operators.
⫹
. We
To complete the task it is necessary to evaluate D ᐉᐉ
begin by following a procedure similar to that used to evaluate the constants when finding the action of L̂ ⫾ on the states
兩 nᐉm 典 C . We consider the matrix element
⫹
⫹
,
兲 * D ᐉᐉ
具 nᐉᐉ 兩 Â ⫺ Â ⫹ 兩 nᐉᐉ 典 ⫽ 共 D ᐉᐉ
共10兲
⫹
)*
(D ᐉᐉ
was obtained by operating to the
where the factor
left with  ⫺ and noting that the Hermitian conjugate of  ⫺
is  ⫹ . We specify for convenience, and by analogy with the
⫹
is
assumption employed in the L̂ ⫾ construction, that D ᐉᐉ
real. The operator  ⫺  ⫹ may be expanded giving
 ⫺  ⫹ ⫽ 2 ⫺ z2 ⫹2L̂ z Ĥ
⫽ 共 2L̂ 2 Ĥ⫹2Ĥ⫹1 兲 ⫺Â z2 ⫹2L̂ z Ĥ.
共11兲
Since the 兩 nᐉm 典 are eigenfunctions of all operators on the
right hand side except  z we obtain
⫹ 2
兲 ⫽
共 D ᐉᐉ
再
n 2 ⫺ 共 ᐉ⫹1 兲 2
n2
冎
⫺ 具 nᐉᐉ 兩 Â z2 兩 nᐉᐉ 典 ,
共12兲
where we have used
冉 冊
Ĥ 兩 nᐉᐉ 典 ⫽ ⫺
1
2n 2
L̂ z 兩 nᐉᐉ 典 ⫽ᐉ 兩 nᐉᐉ 典 ,
兩 nᐉᐉ 典 .
共13兲
⫹
we must determine the
To complete the evaluation of D ᐉᐉ
2
matrix element 具 nᐉᐉ 兩 Â z 兩 nᐉᐉ 典 . We operate on 兩 nᐉᐉ 典 with
 z , and using a result from Table I we find
 z 兩 nᐉᐉ 典 ⫽⫺ 关 21 L̂ ⫺  ⫹ ⫺ 21  ⫺ L̂ ⫹ ⫹ z L̂ z 兴 兩 nᐉᐉ 典
⫹
冑2 共 ᐉ⫹1 兲 兩 n 共 ᐉ⫹1 兲 ᐉ 典 ⫺ᐉÂ z 兩 nᐉᐉ 典 ,
⫽⫺ 12 D ᐉᐉ
共14兲
from which we obtain
⫽Â ⫹ 共 L̂ z ⫹1 兲 兩 nᐉm 典 ⫽ 共 m⫹1 兲 兵 Â ⫹ 兩 nᐉm 典 其 ,
共6兲
which shows that 兵 Â ⫹ 兩 nᐉm 典 其 is an eigenfunction of L̂ z with
eigenvalue (m⫹1). Similarly
Am. J. Phys., Vol. 72, No. 8, August 2004
⫹
兩 n; 共 ᐉ⫹1 兲 ; 共 ᐉ⫹1 兲 典 ,
 ⫹ 兩 nᐉᐉ 典 ⫽D ᐉᐉ
L̂ 2 兩 nᐉᐉ 典 ⫽ᐉ 共 ᐉ⫹1 兲 兩 nᐉᐉ 典 ,
III. THE ACTION OF Â ¿
1014
共8兲
Because 兵 Â ⫹ 兩 nᐉᐉ 典 其 is an eigenfunction of L̂ 2 and L̂ z with
eigenvalues 共ᐉ⫹1兲共ᐉ⫹2兲 and (ᐉ⫹1) respectively, we know
that
Note that the quantum mechanical operators in the square
brackets are not inverses of each other, i.e., 共p̂ÃL̂兲⫽⫺共L̂
Ãp̂兲. It is easily shown that  as defined in Eq. 共4兲 is indeed
Hermitian. It can also be shown, but with considerably more
labor, that b Â,Ĥ c ⫽0.
It is shown in elementary quantum mechanics that the actions of L̂ ⫾ on a spherical eigenfunction for any central potential, which we designate as 兩 nᐉm 典 C , cause the following
state-to-state conversions:
L̂ ⫾ 兩 nᐉm 典 C ⫽ 冑共 ᐉ⫿m 兲共 ᐉ⫾m⫹1 兲 兩 nᐉ; 共 m⫾1 兲 典 C ,
共7兲
Note that the 兩 nᐉm 典 are not eigenfunctions of  z , but if m
⫽ᐉ the term with  z vanishes and thus 兵  ⫹ 兩 nᐉᐉ 典 其 is an
eigenfunction of L̂ 2 with eigenvalue
where p is the linear momentum, L is the angular momentum, and r̂ is the unit vector in the r direction. The direction
of A, toward apocenter or pericenter, is a matter of choice.
We elect the definition in Eq. 共3兲.2
The quantum mechanical operator corresponding to A
cannot be constructed by merely replacing each quantity
with its corresponding quantum mechanical operator because
the result is a non-Hermitian operator. Pauli4 recognized that
 must be defined as
Â⫽ 21 关共 p̂ÃL̂兲 ⫺ 共 L̂Ãp̂兲兴 ⫺r̂.
⫽ 共 Â ⫹ L̂ 2 ⫺2Â z L̂ ⫹ ⫹2Â ⫹ ⫹2Â ⫹ L̂ z 兲 兩 nᐉm 典
⫽ 兵 ᐉ 共 ᐉ⫹1 兲 ⫹2 共 m⫹1 兲 其兵 Â ⫹ 兩 nᐉm 典 其
L̂"Â⫽Â"L̂⫽0
(Â ⫿ ) † ⫽Â ␮
(Â ⫿ Â ␮ ) † ⫽Â ⫿ Â ␮
b L̂ i ,L̂ j c ⫽iL̂ k ⑀ i jk
b L̂ ⫾ , z c ⫽ ␮  ⫿
b L̂ i , j c ⫽i k ⑀ i jk
b  i , j c ⫽⫺2iL̂ k Ĥ ⑀ i jk
 2 ⫽2(L̂ 2 ⫹1)Ĥ⫹1
1
 z ⫽⫺ 2 (L̂ ⫺  ⫹ ⫹ ⫺ L̂ ⫹ )⫺ z L̂ z
A⫽pÃL⫺r̂,
L̂ 2 兵 Â ⫹ 兩 nᐉm 典 其
 z 兩 nᐉᐉ 典 ⫽⫺
1
冑2 共 ᐉ⫹1 兲
⫹
D ᐉᐉ
兩 n 共 ᐉ⫹1 兲 ᐉ 典 .
共15兲
Thus, the matrix element 具 nᐉᐉ 兩 Â z2 兩 nᐉᐉ 典 is given by
C. E. Burkhardt and J. J. Leventhal
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Table II. The results of operations on the specified spherical eigenfunctions
of the hydrogen atom.
Operation
Result
m 兩 nᐉm 典
L̂ z 兩 nᐉm 典
L̂ ⫾ 兩 nᐉm 典
 z 兩 n;ᐉ;(⫾ᐉ) 典
冑(ᐉ⫿m)(ᐉ⫾m⫹1) 兩 nᐉ;(m⫾1) 典
1
n
 ⫹ 兩 nᐉᐉ 典
冑
冑
⫺
 ⫺ 兩 n;ᐉ;(⫺ᐉ) 典
1
n
1
n
具 nᐉᐉ 兩 Â z2 兩 nᐉᐉ 典 ⫽
冑
关 n 2 ⫺ 共 ᐉ⫹1 兲 2 兴
兩 n; 共 ᐉ⫹1 兲 ; 共 ⫾ᐉ 兲 典
共 2ᐉ⫹3 兲
2 共 ᐉ⫹1 兲 2
关 n ⫺ 共 ᐉ⫹1 兲 2 兴 兩 n; 共 ᐉ⫹1 兲 ; 共 ᐉ⫹1 兲 典
共 2ᐉ⫹3 兲
2 共 ᐉ⫹1 兲 2
关 n ⫺ 共 ᐉ⫹1 兲 2 兴 兩 n; 共 ᐉ⫹1 兲 ;⫺ 共 ᐉ⫹1 兲 典
共 2ᐉ⫹3 兲
⫹ 2
兲
共 D ᐉᐉ
2 共 ᐉ⫹1 兲
.
共16兲
⫹ 2
) and
Inserting Eq. 共16兲 into Eq. 共12兲, solving for (D ᐉᐉ
taking the square root, we have
⫹
⫽⫾
D ᐉᐉ
1
n
冑
2 共 ᐉ⫹1 兲 2
关 n ⫺ 共 ᐉ⫹1 兲 2 兴 .
共 2ᐉ⫹3 兲
共17兲
It is important to note that, although the operator  ⫺ was
present in our analysis of the action of  ⫹ on 兩 nᐉᐉ 典 , the
only property of  ⫺ that was used was that it is the Hermitian conjugate is  ⫹ . The action of  ⫺ on an eigenfunction
⫹
was not needed. Also, because D ᐉᐉ
was obtained by taking
⫹ 2
the square root of (D ᐉᐉ ) , there is no information on the sign
⫹
. In accordance with convention it is the minus sign
of D ᐉᐉ
that is retained.6
change only the m quantum number, but neither n nor ᐉ. This
is a consequence of the spatial symmetry that causes the
degeneracy in m for any central potential. L̂ z , of course, does
not change any quantum numbers because the spherical
eigenfunctions are also eigenfunctions of L̂ z . 共Recall that the
spherical eigenfunctions are obtained using the mutually
commuting operators Ĥ, L̂ 2 and L̂ z .) On the other hand, Â ⫾
change both ᐉ and m. We note that in this paper we consider
the actions of the  ⫾ on only eigenstates for which 兩 m 兩
⫽ᐉ. In general, however, the  ⫾ acting on 兩 nᐉm 典 produce
linear combinations of 兩 n(ᐉ⫹1)(m⫾1) 典 and 兩 n(ᐉ⫺1)(m
⫾1) 典 as a consequence of the Wigner–Eckart theorem.
While L̂ z can change neither ᐉ nor m, Â z changes ᐉ but not
m. Â z cannot change m because A z and L z commute so
兵 A z 兩 nᐉm 典 其 must be an eigenfunction of L z with eigenvalue
m. This does not, however, preclude A z changing ᐉ 共also a
consequence of the Wigner–Eckart theorem兲. The ability to
shift the angular momentum quantum number ᐉ using Lenz
vector operators in a manner analogous to the shift of m that
can be accomplished for any central potential using angular
momentum operators is a manifestation of the ‘‘extra’’ symmetry of the Coulomb/gravitational potential. This extra
symmetry leads to an additional operator, the Lenz vector
operator, that commutes with the Hamiltonian and accounts
for the accidental degeneracy of the hydrogen atom.
ACKNOWLEDGMENTS
The authors would like to thank their colleagues, Joseph F.
Baugh and Ta-Pei Cheng for helpful discussions. Thanks are
also due to a diligent referee for several helpful suggestions.
IV. THE ACTION OF Â À
It is tempting to assume that  ⫺ acts as a lowering operator on certain spherical eigenfunctions. In fact, for certain
state-to-state conversions it does lower the z-component of
the angular momentum, but it raises the total angular momentum to 共ᐉ⫹1兲. In a manner identical to that employed for
兵 Â ⫹ 兩 nᐉm 典 其 we find that 兵 Â ⫺ 兩 nᐉm 典 其 is an eigenfunction of
L̂ z with eigenvalue (m⫺1). Again in an analogous way we
find that 兵 Â ⫺ 兩 nᐉm 典 其 is an eigenfunction of L̂ 2 provided m
⫽⫺ᐉ, but with eigenvalue 共ᐉ⫹1兲共ᐉ⫹2兲. We conclude therefore that
⫺
 ⫺ 兩 n;ᐉ; 共 ⫺ᐉ 兲 典 ⫽D ᐉ⫺ᐉ
兩 n; 共 ᐉ⫹1 兲 ;⫺ 共 ᐉ⫹1 兲 典 ,
共18兲
⫺
D ᐉ⫺ᐉ
is a constant. Using a procedure analogous to
where
⫹
⫺
that employed to evaluate D ᐉᐉ
gives the magnitude of D ᐉ⫺ᐉ
;
6
determination of the sign requires additional work. The result is
⫺
⫽
D ᐉ⫺ᐉ
1
n
冑
2 共 ᐉ⫹1 兲 2
⫹
.
关 n ⫺ 共 ᐉ⫹1 兲 2 兴 ⫽⫺D ᐉᐉ
共 2ᐉ⫹3 兲
共19兲
V. SUMMARY
The properties of the relevant operators are summarized in
Table II. It is interesting to contrast the actions of L̂ ⫾ and L̂ z
with those of  ⫾ and  z . For any central potential L̂ ⫾
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Am. J. Phys., Vol. 72, No. 8, August 2004
APPENDIX: SUGGESTED PROBLEMS
共1兲 Using the relations in Table I show that the commutator b L̂ z , ⫺ c ⫽⫺ ⫺ .
Solution:
b L̂ z , ⫺ c ⫽ b L̂ z , x c ⫺i b L̂ z , y c
⫽i y ⫺i 兵 ⫺i x 其 ⫽⫺ x ⫹i y ⫽⫺ ⫺ .
共2兲 共a兲 Show that 兵 Â ⫺ 兩 nᐉm 典 其 is an eigenfunction of L̂ z
with eigenvalue (m⫺1).
共b兲 Show that 兵 Â ⫺ 兩 nᐉm 典 其 is an eigenfunction of L̂ 2 if and
only if m⫽⫺ᐉ with eigenvalue 共ᐉ⫹1兲共ᐉ⫹2兲 thus proving
that the action of  ⫺ on 兩 nᐉ;⫺ᐉ 典 is to lower the
z-component of angular momentum by one unit of angular
momentum while raising the total angular momentum by one
⫺
兩 n;(ᐉ
unit of angular momentum, i.e., 兵 Â ⫺ 兩 nᐉm 典 其 ⫽D ᐉᐉ
⫹1);⫺(ᐉ⫹1) 典 .
Solution:
共a兲
L̂ z 兵  ⫺ 兩 nᐉm 典 其 ⫽L̂ z 共  x ⫺i y 兲 兩 nᐉm 典
⫽Â x L̂ z ⫺ 关 Â x ,L̂ z 兴 ⫺i 兵 Â y L̂ z ⫺ 关 Â y L̂ z 兴 其 .
Using Table I:
C. E. Burkhardt and J. J. Leventhal
1015
L̂ z 兵  ⫺ 兩 nᐉm 典 其 ⫽ 兵  x L̂ z ⫹i y ⫺i 共  y L̂ z ⫺i x 兲 其 兩 nᐉm 典
⫺1 兲 典 ⫹2 共 1⫺m 兲 兵 Â ⫺ 兩 nᐉm 典 其
⫽ 兵 共  x ⫺i y 兲 L̂ z ⫺ 共  x ⫺i y 兲 其 兩 nᐉm 典
⫽ 兵 ᐉ 共 ᐉ⫹1 兲 ⫹2 共 1⫺m 兲 其兵 Â ⫺ 兩 nᐉm 典 其
⫽Â ⫺ 共 L̂ z ⫺1 兲 兩 nᐉm 典
⫹2Â z 冑共 ᐉ⫹m 兲共 ᐉ⫺m⫹1 兲 兩 n;ᐉ; 共 m
⫽ 共 m⫺1 兲 兵 Â ⫺ 兩 nᐉm 典 其 .
⫺1 兲 典 .
共b兲
L̂ 2 兵  ⫺ 兩 nᐉm 典 其 ⫽L̂ 2 共  x ⫺i y 兲 兩 nᐉm 典 .
Using b X̂Ŷ ,Ẑ c ⫽ b X̂,Ẑ c Ŷ ⫹X̂ b Ŷ ,Ẑ c and Table I we have
b L̂ 2 , x c ⫽⫺i z L̂ y ⫺iL̂ y  z ⫹i y L̂ z ⫹iL̂ z  y ,
b L̂ 2 , y c ⫽i z L̂ x ⫹iL̂ x  z ⫺i x L̂ z ⫺iL̂ z  x .
Then b L̂ 2 , ⫺ c ⫽ z L̂ ⫺ ⫹L̂ ⫺  z ⫺ ⫺ L̂ z ⫺L̂ z  ⫺ .
Using the result of problem 共1兲 we have
b L̂ 2 , ⫺ c ⫽2 z L̂ ⫺ ⫹2 ⫺ ⫺2 ⫺ L̂ z .
So
L̂ 2 兵 Â ⫺ 兩 nᐉm 典 其 ⫽ 共 Â ⫺ L̂ 2 ⫹2Â z L̂ ⫺ ⫹2Â ⫺
⫺2Â ⫺ L̂ z 兲 兩 nᐉm 典
⫽ᐉ 共 ᐉ⫹1 兲 Â ⫺ 兩 nᐉm 典
⫹2Â z 冑共 ᐉ⫹m 兲共 ᐉ⫺m⫹1 兲 兩 n;ᐉ; 共 m
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Am. J. Phys., Vol. 72, No. 8, August 2004
So if m⫽⫺ᐉ (m⫽ᐉ⫹1 is not allowed兲 then L̂ 2 兵 Â ⫺ 兩 nᐉ;
⫺ᐉ 典 其 ⫽(ᐉ⫹1)(ᐉ⫹2) 兵 Â ⫺ 兩 nᐉ;⫺ᐉ 典 其 .
a兲
Electronic mail: [email protected]
L. I. Schiff, Quantum Mechanics, 3rd ed. 共McGraw-Hill, New York,
1968兲, p. 234.
2
H. Goldstein, Classical Mechanics, 2nd ed. 共Addison-Wesley, Reading,
MA, 1980兲, p. 422.
3
T. P. Hezel, C. E. Burkhardt, M. Ciocca, and J. J. Leventhal, ‘‘Classical
view of the Stark effect in hydrogen atoms,’’ Am. J. Phys. 60, 324 共1992兲.
4
W. Pauli, ‘‘On the hydrogen spectrum from the standpoint of the new
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C. E. Burkhardt and J. J. Leventhal
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