Download The two-dimensional hydrogen atom revisited

JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 43, NUMBER 10
OCTOBER 2002
The two-dimensional hydrogen atom revisited
D. G. W. Parfitt and M. E. Portnoia)
School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom
共Received 20 May 2002; accepted for publication 2 June 2002兲
The bound-state energy eigenvalues for the two-dimensional Kepler problem are
found to be degenerate. This ‘‘accidental’’ degeneracy is due to the existence of a
two-dimensional analog of the quantum-mechanical Runge–Lenz vector. Reformulating the problem in momentum space leads to an integral form of the Schrödinger
equation. This equation is solved by projecting the two-dimensional momentum
space onto the surface of a three-dimensional sphere. The eigenfunctions are then
expanded in terms of spherical harmonics, and this leads to an integral relation in
terms of special functions which has not previously been tabulated. The dynamical
symmetry of the problem is also considered, and it is shown that the two components of the Runge–Lenz vector in real space correspond to the generators of
infinitesimal rotations about the respective coordinate axes in momentum space.
© 2002 American Institute of Physics. 关DOI: 10.1063/1.1503868兴
I. INTRODUCTION
A semiconductor quantum well under illumination is a quasi-two-dimensional system, in
which photoexcited electrons and holes are essentially confined to a plane. The mutual Coulomb
interaction leads to electron–hole bound states known as excitons, which are extremely important
for the optical properties of the quantum well. The relative in-plane motion of the electron and
hole can be described by a two-dimensional Schrödinger equation for a single particle with a
reduced mass. This is a physical realization of the two-dimensional hydrogenic problem, which
originated as a purely theoretical construction.1 An important similarity with the three-dimensional
hydrogen atom is the ‘‘accidental’’ degeneracy of the bound-state energy levels. This degeneracy
is due to the existence of the quantum-mechanical Runge–Lenz vector, first introduced by Pauli2
in three dimensions, and indicates the presence of a dynamical symmetry of the system.
The most important study relating to the hidden symmetry of the hydrogen atom was that by
Fock in 1935.3 He considered the Schrödinger equation in momentum space, which led to an
integral equation. Considering negative-energy 共bound-state兲 solutions, he projected the threedimensional momentum space onto the surface of a four-dimensional hypersphere. After a suitable
transformation of the wavefunction, the resulting integral equation was seen to be invariant under
rotations in four-dimensional momentum space. Fock deduced that the dynamical symmetry of the
hydrogen atom is described by the four-dimensional rotation group SO共4兲, which contains the
geometrical symmetry SO共3兲 as a subgroup. He related this hidden symmetry to the observed
degeneracy of the energy eigenvalues.
Shortly afterwards, Bargmann4 made the connection between Pauli’s quantum mechanical
Runge–Lenz vector and Fock’s discovery of invariance under rotations in four-dimensional momentum space. Fock’s method was also extended by Alliluev5 to the case of d dimensions (d
⭓2). A comprehensive review concerning the symmetry of the hydrogen atom was later given by
Bander and Itzykson,6,7 including a detailed group-theoretical treatment and extension to scattering states.
Improvements in semiconductor growth techniques over the subsequent decades, which enabled the manufacture of effectively two-dimensional structures, led to a resurgence of interest in
a兲
Also at A.F. Ioffe Physico-Technical Institute, St. Petersburg, Russia. Electronic mail: [email protected]
0022-2488/2002/43(10)/4681/11/$19.00
4681
© 2002 American Institute of Physics
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4682
J. Math. Phys., Vol. 43, No. 10, October 2002
D. G. Parfitt and M. E. Portnoi
the two-dimensional hydrogen atom. The Runge–Lenz vector for this case was defined for the first
time,8 and real-space solutions of the Schrödinger equation were applied to problems of atomic
physics in two dimensions.9
Recent studies have focused on diverse aspects of the hydrogenic problem. The d-dimensional
case has been reconsidered, leading to a generalized Runge–Lenz vector 共see Ref. 10 and references therein兲. The algebraic basis of the dynamical symmetry has also been given a thorough
mathematical treatment.11,12
In the present work we return to the two-dimensional problem, and use the method of Fock to
obtain a new integral relation in terms of special functions. The dynamical symmetry of the system
is also considered, and a new interpretation of the two-dimensional Runge–Lenz vector is presented.
II. PROBLEM FORMULATION
A. Preliminaries
The relative in-plane motion of an electron and hole, with effective masses m e and m h ,
respectively, may be treated as that of a single particle with reduced mass ␮ ⫽m e m h /(m e ⫹m h )
and energy E, moving in a Coulomb potential V( ␳ ). The wavefunction of the particle satisfies the
stationary Schrödinger equation
冋
Ĥ⌿ 共 ␳兲 ⫽ ⫺
冉 冊
册
1 ⳵
⳵
1 ⳵2
␳
⫺ 2 2 ⫹V 共 ␳ 兲 ⌿ 共 ␳兲 ⫽E⌿ 共 ␳兲 ,
␳ ⳵␳
⳵␳
␳ ⳵␾
共1兲
where 共␳,␾兲 are plane polar coordinates. Note that excitonic Rydberg units are used throughout
this article, which leads to a potential of the form V( ␳ )⫽⫺2/␳ .
The eigenfunctions of Eq. 共1兲 are derived in Appendix A. It is well known that the bound-state
energy levels are of the form1
E⫽⫺
1
共 n⫹ 21 兲 2
,
n⫽0,1,2,...,
共2兲
where n is the principal quantum number. Notably, Eq. 共2兲 does not contain explicitly the azimuthal quantum number m, which enters the radial equation 关see Appendix A, Eq. 共A4兲兴. Each
energy level is (2n⫹1)-fold degenerate, the so-called accidental degeneracy.
It is convenient to introduce a vector operator corresponding to the z-projection of the angular
momentum, L̂z ⫽ez L̂ z , where ez is a unit vector normal to the plane of motion of the electron and
hole. We now introduce the two-dimensional analog of the quantum-mechanical Runge–Lenz
vector as the dimensionless operator
2
Â⫽ 共 q̂⫻L̂z ⫺L̂z ⫻q̂兲 ⫺ ␳,
␳
共3兲
where q̂⫽⫺iⵜ is the momentum operator. Note that  lies in the plane and has Cartesian
components  x and  y .
L̂ z ,  x , and  y represent conserved quantities and therefore commute with the Hamiltonian:
关 Ĥ,L̂ z 兴 ⫽ 关 Ĥ, x 兴 ⫽ 关 Ĥ, y 兴 ⫽0.
共4兲
They also satisfy the following commutation relations:
关 L̂ z , x 兴 ⫽i y ,
共5兲
关 L̂ z , y 兴 ⫽⫺i x ,
共6兲
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J. Math. Phys., Vol. 43, No. 10, October 2002
The two-dimensional hydrogen atom revisited
关  x , y 兴 ⫽⫺4iL̂ z Ĥ.
4683
共7兲
B. Derivation of energy eigenvalues from Â
The existence of the noncommuting operators  x and  y , representing conserved physical
quantities, implies that the Runge–Lenz vector is related to the accidental degeneracy of the
energy levels in two dimensions.13 We now present a simple interpretation of the hidden symmetry
underlying this degeneracy.
For eigenfunctions of the Hamiltonian we can replace Ĥ by the energy E, and defining
Â⬘ ⫽
Â
2 冑⫺E
,
共8兲
we obtain the new commutation relations:
关 L̂ z , x⬘ 兴 ⫽i ⬘y ,
共9兲
关 L̂ z , ⬘y 兴 ⫽⫺i x⬘ ,
共10兲
关  ⬘x , ⬘y 兴 ⫽iL̂ z .
共11兲
If we now construct a three-dimensional vector operator
Ĵ⫽Â⬘ ⫹L̂z ,
共12兲
then the components of Ĵ satisfy the commutation rules of ordinary angular momentum:
关 Ĵ j ,Ĵ k 兴 ⫽i ⑀ jkl Ĵ l ,
共13兲
where ⑀ jkl is the Levi-Civita symbol.
Noting that Â⬘ •L̂z ⫽L̂z •Â⬘ ⫽0, we have
Ĵ2 ⫽ 共 Â⬘ ⫹L̂z 兲 2 ⫽Â⬘ 2 ⫹L̂z2 ,
共14兲
where the operator Ĵ2 has eigenvalues j( j⫹1) and commutes with the Hamiltonian.
We now make use of a special expression relating Â2 and L̂z2 , the derivation of which is given
in Appendix B:
Â2 ⫽Ĥ 共 4L̂z2 ⫹1 兲 ⫹4.
共15兲
Substituting in Eq. 共14兲 and again replacing Ĥ with E, we obtain
Ĵ2 ⫽⫺
1
关 E 共 4L̂z2 ⫹1 兲 ⫹4 兴 ⫹L̂z2 .
4E
共16兲
Because 关 Ĥ,Ĵ2 兴 ⫽0, an eigenfunction of the Hamiltonian will also be an eigenfunction of Ĵ2 .
Operating with both sides of Eq. 共16兲 on an eigenfunction of the Hamiltonian, we obtain for the
eigenvalues of Ĵ2 :
j 共 j⫹1 兲 ⫽⫺
冉 冊
1 1
⫹ .
4 E
共17兲
Rearranging, and identifying j with the principal quantum number n, we obtain the correct expression for the energy eigenvalues:
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J. Math. Phys., Vol. 43, No. 10, October 2002
E⫽⫺
1
共 n⫹ 21 兲 2
D. G. Parfitt and M. E. Portnoi
,
n⫽0,1,2,... .
共18兲
Note that the z-component of Ĵ is simply L̂ z . Recalling that the eigenvalues of L̂ z are denoted
by m, there are (2 j⫹1) values of m for a given j. However, as j⫽n, we see that there are (2n
⫹1) values of m for a given energy, which corresponds to the observed (2n⫹1)-fold degeneracy.
III. FOCK’S METHOD IN TWO DIMENSIONS
A. Stereographic projection
The method of Fock,3 in which a three-dimensional momentum space is projected onto the
surface of a four-dimensional hypersphere, may be applied to our two-dimensional problem. We
begin by defining a pair of two-dimensional Fourier transforms between real space and momentum
space:
⌽ 共 q兲 ⫽
⌿ 共 ␳兲 ⫽
冕
1
共 2␲ 兲2
⌿ 共 ␳兲 e iq"␳ d ␳,
共19兲
冕
共20兲
⌽ 共 q兲 e ⫺iq"␳ dq.
We shall restrict our interest to bound states, and hence the energy E⫽⫺q 20 will be negative.
Substitution of Eq. 共20兲 in Eq. 共1兲 yields the following integral equation for ⌽共q兲:
共 q 2 ⫹q 20 兲 ⌽ 共 q兲 ⫽
冕
1
␲
⌽ 共 q⬘ 兲 dq⬘
.
兩 q⫺q⬘ 兩
共21兲
The two-dimensional momentum space is now projected onto the surface of a threedimensional unit sphere centered at the origin, and so it is natural to scale the in-plane momentum
by q 0 . Each point on a unit sphere is completely defined by two polar angles, ␪ and ␾, and the
Cartesian coordinates of a point on the unit sphere are given by
u x ⫽sin ␪ cos ␾ ⫽
u y ⫽sin ␪ sin ␾ ⫽
u z ⫽cos ␪ ⫽
2q 0 q x
q 2 ⫹q 20
2q 0 q y
q 2 ⫹q 20
q 2 ⫺q 20
q 2 ⫹q 20
,
共22兲
,
共23兲
共24兲
.
An element of surface area on the unit sphere is given by
d⍀⫽sin ␪ d ␪ d ␾ ⫽
冉
2q 0
q
2
⫹q 20
冊
2
dq,
共25兲
and the distance between two points transforms as
兩 u⫺u⬘ 兩 ⫽
共q
2
2q 0
兩 q⫺q⬘ 兩 .
2 1/2
⫹q 0 兲 共 q ⬘ 2 ⫹q 20 兲 1/2
共26兲
If the wavefunction on the sphere is expressed as
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J. Math. Phys., Vol. 43, No. 10, October 2002
The two-dimensional hydrogen atom revisited
1
␹ 共 u兲 ⫽
冑q 0
冉
q 2 ⫹q 20
冊
4685
3/2
⌽ 共 q兲 ,
共27兲
共 u⬘ 兲 d⍀ ⬘
.
兩 u⫺u⬘ 兩
共28兲
2q 0
then Eq. 共21兲 reduces to the simple form
␹ 共 u兲 ⫽
1
2␲q0
冕␹
B. Expansion in spherical harmonics
Any function on a sphere can be expressed in terms of spherical harmonics, so for ␹共u兲 we
have
⬁
␹ 共 u兲 ⫽
l
兺 兺
l⫽0 m⫽⫺l
A lm Y m
l 共 ␪,␾ 兲,
共29兲
where Y m
l ( ␪ , ␾ ) are basically defined as in Ref. 14:
Ym
l 共 ␪ , ␾ 兲 ⫽c lm
冑
2l⫹1 共 l⫺ 兩 m 兩 兲 ! 兩 m 兩
P 共 cos ␪ 兲 e im ␾ ,
4 ␲ 共 l⫹ 兩 m 兩 兲 ! l
共30兲
where P 兩nm 兩 (cos ␪) is an associated Legendre function as defined in Ref. 15. The constant c lm is an
arbitrary ‘‘phase factor.’’ As long as 兩 c lm 兩 2 ⫽1 we are free to choose c lm , and for reasons which
will become clear we set
c lm ⫽ 共 ⫺i 兲 兩 m 兩 .
共31兲
The kernel of the integral in Eq. 共28兲 can also be expanded in this basis as16
⬁
␭
1
⫽
兩 u⫺u⬘ 兩 ␭⫽0
4␲
兺 ␮ ⫽⫺␭
兺 2␭⫹1 Y ␭␮共 ␪ , ␾ 兲 Y ␭␮ *共 ␪ ⬘ , ␾ ⬘ 兲 .
共32兲
Substituting Eqs. 共29兲 and 共32兲 into Eq. 共28兲 we have
⬁
l
兺 兺
l⫽0 m⫽⫺l
A lm Y m
l 共 ␪,␾ 兲⫽
2
q0
⬁
⬁
l1
l2
兺 兺 兺 兺
l ⫽0 l ⫽0 m ⫽⫺l m ⫽⫺l
1
2
1
1
2
m
m
⫻Y l 2 共 ␪ , ␾ 兲 Y l 2 *共 ␪ ⬘ , ␾ ⬘ 兲
2
2
2
冕
1
m
Y 1共 ␪ ⬘ , ␾ ⬘ 兲
A
2l 2 ⫹1 l 1 m 1 l 1
d⍀ ⬘ .
共33兲
We now make use of the orthogonality property of spherical harmonics to reduce Eq. 共33兲 to
the following:
⬁
l
兺 兺
l⫽0 m⫽⫺l
A lm Y m
l 共 ␪,␾ 兲⫽
2
q0
⬁
l1
兺 兺
l ⫽0 m ⫽⫺l
1
1
1
1
m
A
Y 1共 ␪ , ␾ 兲 .
2l 1 ⫹1 l 1 m 1 l 1
共34兲
⬘ * ( ␪ , ␾ ) and integrating over d⍀ gives
Multiplying both sides of Eq. 共34兲 by Y m
n
A nm ⬘ ⫽
2
A
,
q 0 共 2n⫹1 兲 nm ⬘
共35兲
where we have again used the orthogonality relation for spherical harmonics. The final step is to
rearrange for q 0 and identify the index n with the principal quantum number. This enables us to
find an expression for the energy in excitonic Rydbergs:
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4686
J. Math. Phys., Vol. 43, No. 10, October 2002
E⫽⫺q 20 ⫽⫺
D. G. Parfitt and M. E. Portnoi
1
共 n⫹ 21 兲 2
,
共36兲
n⫽0,1,2,... .
This is seen to be identical to Eq. 共2兲.
For a particular value of n, the general solution of Eq. 共28兲 can be expressed as
n
␹ n 共 u兲 ⫽
兺
m⫽⫺n
A nm Y m
n 共 ␪,␾ 兲.
共37兲
Each of the functions entering the sum in Eq. 共37兲 satisfies Eq. 共28兲 separately. So, for each value
of n we have (2n⫹1) linearly independent solutions, and this explains the observed (2n⫹1)-fold
degeneracy.
We are free to choose any linear combination of spherical harmonics for our eigenfunctions,
but for convenience we simply choose
␹ nm 共 u兲 ⫽A nm Y mn 共 ␪ , ␾ 兲 .
共38兲
If we also require our eigenfunctions to be normalized as follows:
1
共 2␲ 兲2
冕
1
共 2␲ 兲2
兩 ␹ 共 u兲 兩 2 d⍀⫽
冕
q 2 ⫹q 20
2q 20
兩 ⌽ 共 q兲 兩 2 dq⫽
冕
兩 ⌿ 共 ␳兲 兩 2 d ␳⫽1,
共39兲
then Eq. 共38兲 reduces to
␹ nm 共 u兲 ⫽2 ␲ Y mn 共 ␪ , ␾ 兲 .
共40兲
Applying the transformation in Eq. 共27兲, we can obtain an explicit expression for the orthonormal eigenfunctions of Eq. 共21兲:
⌽ nm 共 q兲 ⫽c nm
冑
2␲
冉
2q 0
共 n⫺ 兩 m 兩 兲 !
2
共 n⫹ 兩 m 兩 兲 ! q ⫹q 20
冊
3/2
P 兩nm 兩 共 cos ␪ 兲 e im ␾ ,
共41兲
where we have used the fact that q 0 ⫽(n⫹ 21 ) ⫺1 , and ␪ and ␾ are defined by Eqs. 共22兲–共24兲.
C. New integral relations
To obtain the real-space eigenfunctions ⌿共␳兲 we make an inverse Fourier transform:
⌿ 共 ␳兲 ⫽
1
共 2␲ 兲2
冕
⌽ 共 q兲 e ⫺iq"␳ dq⫽
1
共 2␲ 兲2
冕 冕
2␲
0
⬁
0
⌽ 共 q兲 e ⫺iq ␳ cos ␾ ⬘ q dq d ␾ ⬘ ,
共42兲
where ␾⬘ is the azimuthal angle between the vectors ␳ and q. However, if we now substitute Eq.
共41兲 into this expression, we have to be careful with our notation. The angle labeled ␾ in Eq. 共41兲
is actually related to ␾⬘ via
␾ ⫽ ␾ ⬘⫹ ␾ ␳ ,
共43兲
where ␾ ␳ is the azimuthal angle of the vector ␳, which can be treated as constant for the purposes
of our integration.
Taking this into account, the substitution of Eq. 共41兲 into Eq. 共42兲 yields
⌿ 共 ␳兲 ⫽
c nm
共 2 ␲ 兲 3/2
冑
共 n⫺ 兩 m 兩 兲 ! im ␾
e ␳
共 n⫹ 兩 m 兩 兲 !
冕 冕冉
2␲
0
⬁
0
2q 0
q
2
⫹q 20
冊
3/2
P 兩nm 兩 共 cos ␪ 兲 e i 共 m ␾ ⬘ ⫺q ␳ cos ␾ ⬘ 兲 q dq d ␾ ⬘ .
共44兲
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J. Math. Phys., Vol. 43, No. 10, October 2002
The two-dimensional hydrogen atom revisited
From Eq. 共24兲 we obtain
P 兩nm 兩 共 cos ␪ 兲 ⫽ P 兩nm 兩
冉
q 2 ⫺q 20
q 2 ⫹q 20
冊
4687
共45兲
,
and we use the following form of Bessel’s integral:16
冕
2␲
0
e i 共 m ␾ ⬘ ⫺q ␳ cos ␾ ⬘ 兲 d ␾ ⬘ ⫽2 ␲ 共 ⫺i 兲 m J m 共 q ␳ 兲 ,
共46兲
where J m (q ␳ ) is a Bessel function of the first kind of order m. Substituting Eqs. 共45兲 and 共46兲 into
Eq. 共44兲 leads to
⌿ 共 ␳兲 ⫽
c nm 共 ⫺i 兲 m
冑2 ␲
冑
共 n⫺ 兩 m 兩 兲 ! im ␾
e ␳
共 n⫹ 兩 m 兩 兲 !
冕冉
⬁
0
2q 0
q 2 ⫹q 20
冊 冉
3/2
P 兩nm 兩
q 2 ⫺q 20
q 2 ⫹q 20
冊
J m 共 q ␳ 兲 q dq.
共47兲
We now make a change of variables, x⫽q 0 ␳ and y⫽q 2 /q 20 , so that Eq. 共47兲 becomes
⌿ 共 ␳兲 ⫽c nm 共 ⫺1 兲 n⫹m 共 ⫺i 兲 m
冑
q 0 共 n⫺ 兩 m 兩 兲 ! im ␾
e ␳
␲ 共 n⫹ 兩 m 兩 兲 !
冕 冉 冊
⬁
0
P 兩nm 兩
1⫺y J m 共 x 冑y 兲
dy,
1⫹y 共 1⫹y 兲 3/2
共48兲
where we have used the fact that16
P 兩nm 兩
冉 冊
冉 冊
y⫺1
1⫺y
⫽ 共 ⫺1 兲 n⫹m P 兩nm 兩
.
y⫹1
1⫹y
共49兲
If we now equate the expression for ⌿共␳兲 in Eq. 共48兲 with that derived in Appendix A, we
obtain the following:
c nm 共 ⫺1 兲 n⫹m 共 ⫺i 兲 m
冕 冉 冊
⬁
0
P 兩nm 兩
1⫺y J m 共 x 冑y 兲
共 2x 兲 兩 m 兩 e ⫺x 2 兩 m 兩
dy⫽
L n⫺ 兩 m 兩 共 2x 兲 .
1⫹y 共 1⫹y 兲 3/2
n⫹1/2
共50兲
The value of c nm chosen earlier in Eq. 共31兲 ensures that both sides of Eq. 共50兲 are numerically
equal. If we restrict our interest to m⭓0, then the relation simplifies to
冕 冉 冊
⬁
0
Pm
n
1⫺y J m 共 x 冑y 兲
共 ⫺1 兲 n 共 2x 兲 m e ⫺x 2m
dy⫽
L n⫺m 共 2x 兲 ,
1⫹y 共 1⫹y 兲 3/2
n⫹1/2
n,m⫽0,1,2,...; m⭐n.
共51兲
As far as we can ascertain, this integral relation between special functions has not previously
been tabulated. For n,m⫽0 we recover the known integral relation15
冕
⬁
0
J 0 共 x 冑y 兲
dy⫽2e ⫺x .
共 1⫹y 兲 3/2
共52兲
IV. DYNAMICAL SYMMETRY
A. Infinitesimal generators
Consider now a vector u from the origin to a point on the three-dimensional unit sphere
defined in Sec. III A. If this vector is rotated through an infinitesimal angle ␣ in the (u x u z ) plane,
we have a new vector
u⬘ ⫽u⫹ ␦ u,
共53兲
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J. Math. Phys., Vol. 43, No. 10, October 2002
D. G. Parfitt and M. E. Portnoi
where the components of u are given in Eqs. 共22兲–共24兲, and
␦ u⫽ ␣ ey ⫻u.
共54兲
This rotation on the sphere corresponds to a change in the two-dimensional momentum from q to
q⬘. The Cartesian components of Eq. 共53兲 are then found to be
2q 0 q x⬘
u ⬘x ⫽
2⫽
q ⬘ 2 ⫹q 0
u ⬘y ⫽
u z⬘ ⫽
q 2 ⫺q 20
q 2 ⫹q 0
q 2 ⫹q 20
2 ⫹␣
2q 0 q ⬘y
q⬘
2
⫽
⫹q 20
q ⬘ 2 ⫺q 20
2⫽
q ⬘ ⫹q 0
2
2q 0 q x
2q 0 q y
q 2 ⫹q 20
q 2 ⫺q 20
q
2
⫺␣
⫹q 20
共55兲
,
共56兲
,
2q 0 q x
q 2 ⫹q 20
共57兲
,
where q 2 ⫽q 2x ⫹q 2y .
After some manipulation we can also find the components of ␦q⫽q⬘⫺q:
␦ q x⫽ ␣
q 2 ⫺q 20 ⫺2q 2x
2q 0
␦ q y ⫽⫺ ␣
共58兲
,
q xq y
.
q0
共59兲
The corresponding change in ⌽共q兲 is given by
␦ ⌽ 共 q兲 ⫽
冉
冊
q 2 ⫺q 20 ⫺2q 2x ⳵
␣
q xq y ⳵
⫺
关共 q 2 ⫹q 20 兲 3/2⌽ 共 q兲兴 .
2 3/2
2
2q 0
⳵qx
q0 ⳵qy
共 q ⫹q 0 兲
共60兲
We can write this as
␦ ⌽ 共 q兲 ⫽⫺
i
␣ Âx ⌽ 共 q兲 ,
2q 0
共61兲
where the infinitesimal generator is given by
Âx ⫽
i
共q
2
⫹q 20 兲 3/2
冋
共 q 2 ⫺q 20 ⫺2q 2x 兲
册
⳵
⳵
⫺2q x q y
共 q 2 ⫹q 20 兲 3/2.
⳵qx
⳵qy
共62兲
We now make use of the following operator expression in the momentum representation:
␳ˆ ⫽ex x̂⫹ey ŷ⫽iⵜq ,
共63兲
关 ␳ˆ , f 共 q兲兴 ⫽iⵜq f ,
共64兲
and the commutation relation
to derive a more compact expression for Âx :
Âx ⫽ 共 q 2 ⫺q 20 兲 x̂⫺2q x 共 q"␳ˆ 兲 ⫺3iq x .
共65兲
By considering an infinitesimal rotation in the (u y u z ) plane we can obtain a similar expression
for Ây :
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J. Math. Phys., Vol. 43, No. 10, October 2002
The two-dimensional hydrogen atom revisited
Ây ⫽ 共 q 2 ⫺q 20 兲 ŷ⫺2q y 共 q"␳ˆ 兲 ⫺3iq y .
4689
共66兲
These expressions operate on a particular energy eigenfunction with eigenvalue ⫺q 20 . If we
move the constant ⫺q 20 to the right and replace it with the Hamiltonian in momentum space, Ĥ,
Âx ⫽q 2 x̂⫹x̂Ĥ⫺2q x 共 q"␳ˆ 兲 ⫺3iq x ,
共67兲
Ây ⫽q 2 ŷ⫹ŷĤ⫺2q y 共 q"␳ˆ 兲 ⫺3iq y ,
共68兲
then Âx and Ây can operate on any linear combination of eigenfunctions.
B. Relation to Runge–Lenz vector
Recall the definition of the two-dimensional Runge–Lenz vector in real space:
2
Â⫽ 共 q̂⫻L̂z ⫺L̂z ⫻q̂兲 ⫺ ␳.
␳
共69兲
Using L̂z ⫽ ␳⫻q̂, and the following identity for the triple product of three vectors:
a⫻ 共 b⫻c兲 ⫽ 共 a"c兲 b⫺ 共 a"b兲 c,
共70兲
we can apply the commutation relation 关 ␳,q̂兴 ⫽i to rewrite Eq. 共69兲 in the form:
冉 冊
Â⫽q̂2 ␳⫹ ␳ q̂2 ⫺
2
⫺2q̂共 q̂"␳兲 ⫺3iq̂.
␳
共71兲
If we now return to the expression for the real-space Hamiltonian in Eq. 共1兲, it is apparent that
we may substitute
2
q̂2 ⫺ ⫽Ĥ
␳
共72兲
Â⫽q̂2 ␳⫹ ␳Ĥ⫺2q̂共 q̂"␳兲 ⫺3iq̂.
共73兲
in Eq. 共71兲 to yield
Comparing this with Eqs. 共67兲 and 共68兲, it is evident that the two components of the Runge–Lenz
vector in real space correspond to the generators of infinitesimal rotations in the (u x u z ) and (u y u z )
planes.
V. CONCLUSION
We have shown that the accidental degeneracy in the energy eigenvalues of the twodimensional Kepler problem may be explained by the existence of a planar analog of the familiar
three-dimensional Runge–Lenz vector. By moving into momentum space and making a stereographic projection onto a three-dimensional sphere, a new integral relation in terms of special
functions has been obtained, which to our knowledge has not previously been tabulated. We have
also demonstrated explicitly that the components of the two-dimensional Runge–Lenz vector in
real space are intimately related to infinitesimal rotations in three-dimensional momentum space.
APPENDIX A: SOLUTION OF REAL-SPACE SCHRÖDINGER EQUATION
We apply the method of separation of variables to Eq. 共1兲, making the substitution
⌿ 共 ␳兲 ⫽R 共 ␳ 兲 ⌽ 共 ␾ 兲 .
共A1兲
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4690
J. Math. Phys., Vol. 43, No. 10, October 2002
D. G. Parfitt and M. E. Portnoi
Introducing a separation constant m 2 , we can obtain the angular equation
d 2⌽
⫹m 2 ⌽⫽0,
d␾2
共A2兲
with the solution
⌽共 ␾ 兲⫽
1
冑2 ␲
e im ␾ .
共A3兲
The corresponding radial equation 共with E⫽⫺q 20 ) is
冉
冊
2
m2
d 2 R 1 dR
2
⫹
⫹
⫺q
⫺
R⫽0.
0
d␳2 ␳ d␳
␳
␳2
共A4兲
R 共 ␳ 兲 ⫽C ␳ 兩 m 兩 e ⫺q 0 ␳ w 共 ␳ 兲 ,
共A5兲
We make the substitution
where C is a normalization constant. This leads to the equation
␳
d 2w
dw
⫹ 共 2 兩 m 兩 ⫹1⫺2q 0 ␳ 兲
⫹ 共 2⫺2 兩 m 兩 q 0 ⫺q 0 兲 w⫽0.
d␳2
d␳
共A6兲
Making a final change of variables ␤ ⫽2q 0 ␳ , we obtain
␤
冉
冊
1
d 2w
dw
1
⫹
⫺ 兩 m 兩 ⫺ w⫽0.
2 ⫹ 共 2 兩 m 兩 ⫹1⫺ ␤ 兲
d␤
d␤
q0
2
共A7兲
This is the confluent hypergeometric equation,15 which has two linearly independent solutions. If
we choose the solution which is regular at the origin, then this becomes a polynomial of finite
degree if q 0 ⫽(n⫹ 12 ) ⫺1 with n⫽0,1,2,... . Equation 共A7兲 then becomes the associated Laguerre
equation,16 the solutions of which are the associated Laguerre polynomials:
2兩m兩
2兩m兩
w⫽L n⫺
兩 m 兩 共 ␤ 兲 ⫽L n⫺ 兩 m 兩 共 2q 0 ␳ 兲 .
共A8兲
We can now write the real-space wavefunction in the form
⌿ nm 共 ␳兲 ⫽
C 兩 m 兩 ⫺q ␳ 2 兩 m 兩
␳ e 0 L n⫺ 兩 m 兩 共 2q 0 ␳ 兲 e im ␾ ␳ ,
2␲
共A9兲
where the reason for the subscript on ␾ is explained in Sec. III C.
To normalize this wavefunction we need to make use of the integral16
冕
⬁
0
2兩m兩
2兩m兩
e ⫺2q 0 ␳ 共 2q 0 ␳ 兲 2 兩 m 兩 ⫹1 L n⫺
兩 m 兩 共 2q 0 ␳ 兲 L n⫺ 兩 m 兩 共 2q 0 ␳ 兲 d 共 2q 0 ␳ 兲 ⫽
共 n⫹ 兩 m 兩 兲 !
共 2n⫹1 兲 . 共A10兲
共 n⫺ 兩 m 兩 兲 !
The normalized wavefunctions are therefore
⌿ nm 共 ␳兲 ⫽
冑
q 30 共 n⫺ 兩 m 兩 兲 !
␲ 共 n⫹ 兩 m 兩 兲 !
2兩m兩
im ␾ ␳
,
共 2q 0 ␳ 兲 兩 m 兩 e ⫺q 0 ␳ L n⫺
兩 m 兩 共 2q 0 ␳ 兲 e
共A11兲
satisfying the following orthogonality condition:
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J. Math. Phys., Vol. 43, No. 10, October 2002
冕
The two-dimensional hydrogen atom revisited
⌿ n* m 共 ␳兲 ⌿ n 2 m 2 共 ␳兲 d ␳⫽ ␦ n 1 n 2 ␦ m 1 m 2 .
1 1
4691
共A12兲
APPENDIX B: DERIVATION OF EQ. „15…
From Eq. 共3兲 we have
冋
册
2 2
2
Â2 ⫽ 共 q̂⫻L̂z ⫺L̂z ⫻q̂兲 ⫺ ␳ ⫽ 关 2 共 q̂⫻L̂z 兲 ⫺iq̂兴 2 ⫺ ␳• 关 2 共 q̂⫻L̂z 兲 ⫺iq̂兴
␳
␳
2
⫺ 关 2 共 q̂⫻L̂z 兲 ⫺iq̂兴 • ␳⫹4.
␳
共B1兲
We further expand as follows:
关 2 共 q̂⫻L̂z 兲 ⫺iq̂兴 2 ⫽4 共 q̂⫻L̂z 兲 2 ⫺2iq̂• 共 q̂⫻L̂z 兲 ⫺2i 共 q̂⫻L̂z 兲 •q̂⫺q̂2
⫽4q̂2 L̂z2 ⫹2q̂2 ⫺q̂2 ⫽q̂2 共 4L̂z2 ⫹1 兲 ,
共B2兲
and
2
2
2
⫺ ␳• 关 2 共 q̂⫻L̂z 兲 ⫺iq̂兴 ⫺ 关 2 共 q̂⫻L̂z 兲 ⫺iq̂兴 • ␳⫽⫺ 共 4L̂z2 ⫹1 兲 .
␳
␳
␳
共B3兲
Substituting Eqs. 共B2兲 and 共B3兲 into Eq. 共B1兲 gives
2
Â2 ⫽q̂2 共 4L̂z2 ⫹1 兲 ⫺ 共 4L̂z2 ⫹1 兲 ⫹4,
␳
共B4兲
Â2 ⫽Ĥ 共 4L̂z2 ⫹1 兲 ⫹4.
共B5兲
which, from Eq. 共72兲, is just
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