On the hydrogen symmetry
We construct O(4)-invariant hydrogen wave function in coordinate representation.
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| Published in: | Вопросы атомной науки и техники |
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| Date: | 2001 |
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| Format: | Article |
| Language: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/79482 |
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| Cite this: | On the hydrogen symmetry / Ya.I. Granovskii // Вопросы атомной науки и техники. — 2001. — № 6. — С. 185-186. — Бібліогр.: 5 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860126065684381696 |
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| author | Granovskii, Ya.I. |
| author_facet | Granovskii, Ya.I. |
| citation_txt | On the hydrogen symmetry / Ya.I. Granovskii // Вопросы атомной науки и техники. — 2001. — № 6. — С. 185-186. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | We construct O(4)-invariant hydrogen wave function in coordinate representation.
|
| first_indexed | 2025-12-07T17:42:19Z |
| format | Article |
| fulltext |
ON THE HYDROGEN SYMMETRY
Ya.I. Granovskii
Institute of Physics, Szczecin University, Szczecin, Poland
e-mail: gran@wmf.univ.szczecin.pl
We construct O(4)-invariant hydrogen wave function in coordinate representation.
PACS: 03.65Ge, 11.30Ly
The symmetry mentioned above is the famous O(4)-
symmetry of the nonrelativistic hydrogen atom, which
was transformed by V.A. Fock from the hidden form to
the explicit one [1]. Fock considered the problem at the
wave-function level: he pointed out that Schroedinger
equation in omentum representation is nothing else as
the defining property of the 4-dimensional spherical
harmonics.
At the operator level this symmetry was well known
long ago: 10 years before Fock the hydrogen energy
was quantized by W. Pauli [2] just using two conserved
operators - orbital momentum L
and Laplace vector K
.
As was shown by V. Bargmann [3] both are generators
of the O(4)-symmetry.
For more than 60 years still remains without answer
the question - how to see this symmetry immediately in
coordinate representation? Aside of a simple curiosity
this question is also of practical meaning because it is
more easy to work with symmetric quantities.
Indeed, what is the symmetry? In a simplest case it
is an independence of the function on the corresponding
variable. For example, an axial symmetry means that
function has not angle ϕ among its arguments. Another
example, isotropy, means that the mentioned axis may
be rotated in an arbitrary mode -- now the angle θ drops
out of consideration. The isotropic function depends not
on three variables x,y,z but only on the one combination
of
them 222 zyxr ++= . Just this contraction is the
main cause of most simplifications when working with
symmetric entities.
Let us turn to the hydrogen problem. Its Hamiltonian
rZempH /2/ 22 −= (1)
is explicitly isotropic. This property is usually used to
separate the angles from radius, factorizing the spherical
function Ylm(θ,ϕ) from the radial Rnl(r) one.
But then the Fock symmetry becomes completely
elusive: indeed, O(4)--symmetry mixes the states with
different values of the quantum number while during
separation of the angles this number is kept fixed.
Therefore our goal may be achieved only in the one
way: to find the wave function of the given energy state
without expanding in the partial waves -- it must be
some linear combination of the ''standard'' radial-angular
solutions.
First we separate the asymptotic factor exp(-λr)
bearing in mind that state is bound, E<0, and λ=(2mE/
2 )1/2. Schroedinger equation takes on the form
0)()1(222 =Φ
−−∇−∇ Ri
RR
R ξ
, (2)
where we have introduced the dimensionless radius
rR
λ= and insert the notation vZe /2=ξ for the
dimensionless Coulomb parameter.
Axially symmetric wave function )(RΦ must not
depend on the angle ϕ. Thus, two possible arguments R-
z and R+z remain. The wave function, which depends
only on one of them u=R+z1,
)()( ufR =Φ , (3)
has a property being constant on the paraboloid u=R+z
-- displacements along its surface do not change it.
One of these displacements is motion along the
''parallel'' and is simply a rotation around the z-axis; it
generates the axial symmetry mentioned above. But the
other one -- displacement along the ''meridian'' -- is
something new. Parametrizing the paraboloid as
R=ucos2τ, z=usin2τ (4)
we can change the angle τ by arbitrary additive: τ→τ+δ.
The τ is that same third angle needed for the 4-dim
symmetry!
To ''liberate'' the z -axis it is sufficient to write the
argument as RRu
µ+= using unit vector µ. No
property will be lost, only the axis of the paraboloid will
change its direction, now pointing along that vector.
It is not difficult to treat the positive energies: the
asymptotic factor will be exp(ikr) with k=(+2mE/ 2 )1/2
dimensionless radius becomes to be rkR
= and u=ik(r+
µr).
1 The choice v=R-z is also possible but gives nothing
new because of parity conservation.
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 185-186. 185
Unlike to momentum representation, where Fock
symmetry is realized differently: on the sphere when
E<0 and on the hyperboloid when E>0 (see [3,4]), in
our case all the wave functions are living on the same
paraboloid
independently on the sign of the energy.
It must be mentioned that in classical limit this
paraboloid obtains a simple visual sense: it represents
the surface, whose interior is unattainable for positively
charged particles falling on the nucleus.
Another interesting and important property is that
homogeneous flux of particles falling inside of
paraboloid, after being gathered in its focus is diverging
in space according to Rutherford law:
dN/dΩ∼[sin(θ/2)]-4 . (5)
This property explains a mysterious coincidence of the
quantum and classical cross-sections for Coulomb
scattering (for more details see [5]).
APPENDIX: SOLUTIONS IN EXPLICIT FORM
Here some technical questions are sampled together.
Equation (2) under assumption (3) takes on the form
0)()()1()( ''' =Φ−Φ−+Φ uuuuu η , η=1-iξ.
(a.1)
Its solution for the bound states ( iξ=n=1,2,…) is simply
a Laguerre polynomial
)1;1()()( 111 unFuLu nn −==Φ − (a.2)
where F11(a;bu) is degenerate hyper-geometric
function.
Despite of its simplicity, this expression contains
much information. Its expansion in Legendre
polynomials
∑
−
=
=Φ
1
0
)()(cos)(
n
l
nllnln RRPCu θ ,
)!2()!1(
)!1(
lln
nCnl −−
−= (a.3)
generates all the radial functions
)222;1()2()( 11 RlnlFRRR l
nl +−+= belonging to the n-th
energy level. For example, the n=3 wave function
2/21)()( 2
23 uuuLu +−==Φ (a.4)
contain all the three angular states S,P,D:
)(cos)(~)(cos)(~)(~
221103 θθ PRRPRRRR ++=Φ with
well-known radial functions:
3/221)(~ 2
0 RRRR +−= ;
)2()(~
1 RRRR −= ;
3/R(R)R~ 2
2 = . (a.5)
Fourier-transformation of the considered solutions
[ ]∫ +Φ= − rdzreepf n
rrpi
n
3)()( λλ (a.6)
after evaluation
1
223 1
1
)1(
8)(
−
−
+
+
=
n
n ip
ip
p
npf
λ
π (a.7)
coincides with appropriately transformed expression of
V. Fock.
REFERENCES
1. V Fock. Zur Theorie des Wasserstoffsatoms // Zs.
f. Phys. 1935, v. 98, S. 145-154.
2. W. Pauli jr. Ueber des Wasserstoffspektrum vom
Standpunkt der neuen Quantenmechanik // Zs. f. Phys.
1926, v. 36, S. 336-363.
3. V. Bargmann. Zur Theorie des Wasserstoffsatoms
// Zs. f. Phys. 1936, v. 99, S. 576-582 (1936).
4. M. Bander, C. Itzykson. Group theory and the
hydrogen atom // Rev. Mod. Phys. 1966, v. 38, p. 330-345.
5. Ya. Granovskii, A.S. Zhedanov. ''Hidden'' sym-
metry of Coulomb problem as a cause of coincidence of
the quantum and classical Rutherford formulae //
Ukranian Phys. J. 1987, v. 32, p. 1290-1293 (in Russian).
186
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| id | nasplib_isofts_kiev_ua-123456789-79482 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:42:19Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Granovskii, Ya.I. 2015-04-02T15:31:39Z 2015-04-02T15:31:39Z 2001 On the hydrogen symmetry / Ya.I. Granovskii // Вопросы атомной науки и техники. — 2001. — № 6. — С. 185-186. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 03.65Ge, 11.30Ly https://nasplib.isofts.kiev.ua/handle/123456789/79482 We construct O(4)-invariant hydrogen wave function in coordinate representation. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Electrodynamics of high energies in matter and strong fields On the hydrogen symmetry О симметрии водорода Article published earlier |
| spellingShingle | On the hydrogen symmetry Granovskii, Ya.I. Electrodynamics of high energies in matter and strong fields |
| title | On the hydrogen symmetry |
| title_alt | О симметрии водорода |
| title_full | On the hydrogen symmetry |
| title_fullStr | On the hydrogen symmetry |
| title_full_unstemmed | On the hydrogen symmetry |
| title_short | On the hydrogen symmetry |
| title_sort | on the hydrogen symmetry |
| topic | Electrodynamics of high energies in matter and strong fields |
| topic_facet | Electrodynamics of high energies in matter and strong fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79482 |
| work_keys_str_mv | AT granovskiiyai onthehydrogensymmetry AT granovskiiyai osimmetriivodoroda |