APPLICATION OF GENERALIZED HAMILTONIAN DYNAMICS
TO MODIFIED COULOMB POTENTIAL
Except where reference is made to the work of others, the work described in this thesis
is my own or was done in collaboration with my advisory committee.
This thesis does not include proprietary or classified information.
__________________________
Julian Antolin Camarena
Certificate of Approval:
__________________________ __________________________
Michael S. Pindzola Eugene A. Oks, Chair
Professor Professor
Physics Physics
__________________________ __________________________
Joseph D. Perez George T. Flowers
Professor Dean
Physics Graduate School
APPLICATION OF GENERALIZED HAMILTONIAN DYNAMICS
TO MODIFIED COULOMB POTENTIAL
Julian Antolin Camarena
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
December 19, 2008?
iii?
APPLICATION OF GENERALIZED HAMILTONIAN DYNAMICS
TO MODIFIED COULOMB POTENTIAL
Julian Antolin Camarena
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon request of individuals or institutions and at their expense.
The author reserves all publication rights.
________________________
Signature of Author
________________________
Date of Graduation
iv?
THESIS ABSTRACT
APPLICATION OF GENERALIZED HAMILTONIAN DYNAMICS
TO MODIFIED COULOMB POTENTIAL
Julian Antolin Camarena
Master of Science, December 19, 2008
(B.S. University of Texas at El Paso, 2006)
53 Typed Pages
Directed by Eugene Oks
We apply Dirac?s generalized Hamiltonian dynamics (GHD), a purely classical
formalism, to spinless particles under the influence of a binomial potential. The integrals
of the motion for this potential were chosen as the constraints of GHD, and use Fradkin?s
unit Runge vector in place of the Laplace-Runge-Lenz vector.
A functional form of the unit Runge vector is derived for the binomial potential. It
is shown in accordance with Oks and Uzer (2002) that a new kind of time dilation occurs
for stable, nonradiating states. The primary result which is derived is that the energy of
these classical stable states agrees exactly with the quantal results for the ground state and
all states of odd values of the radial and angular harmonic numbers.
v?
ACKNOWLEDGEMENTS
I would like to thank the Auburn University Department of Physics for having me
as a graduate student for the past two years and for the education I have received under
them.
I wish to extend my heartfelt thanks to Prof. Eugene A. Oks for having been such
a wonderful teacher and mentor to. You have been a very important person in my
educational refinement, I thank you for this.
I dedicate this thesis to my parents, Cecilia and Antonio, and Cadmiel for all their
love and support throughout the time it has taken me to get this far. I thank you most of
all. I never could have done this without you.
Last, but not least, I thank all my friends. Whatever it is you have done, rest
assured it means a lot to me.
vi?
This thesis has been prepared in accordance with Auburn University Graduate School?s
Guide to Preparation and Submission of Thesis and Dissertations 2005, using Microsoft
Word 2007.
vii?
?
TABLE OF CONTENTS
Introduction ??????????????????????????.?......... 1
Dirac?s Generalized Hamiltonian Dynamics ?..????????????.??..... 4
Applications of the binomial potential ????????????????.??..... 8
Dynamical symmetries of Fradkin ????????????????????.. 10
Further properties of the generalized Runge-Lenz vector ???????????.. 14
Application of GHD to the binomial potential ???????????????... 15
Conclusions ?????????????????????????????. 27
Bibliography ????????????????????????????... 29
Appendix A:
Derivation of the functional form of the unit Runge vector ......??????. 31
Appendix B:
Derivation of the frequency of precession of the Laplace-Runge-Lenz
vector ????????????????????????...??...?. 36
Appendix C:
Derivation of the equations of motion via the Poisson bracket formalism ??. 43
1
?
1. INTRODUCTION
In 1950, Dirac developed a generalized Hamiltonian dynamics (hereafter GHD)
[1-3]. The conventional Hamiltonian dynamics is based on the assumption that the
momenta are independent functions of velocities. Dirac analyzed a more general situation
where momenta are not independent functions of velocities [1-3]. Physically, the GHD is
a purely classical formalism for constrained systems; it incorporates the constraints into
the Hamiltonian. Dirac designed the GHD with applications to quantum field theory in
mind [3].
The present work, where GHD is applied to atomic and molecular systems by
choosing integrals of the motion as the constraints of the system, stems from a paper in
which this idea was applied to hydrogenic atoms treated non-relativistically on the basis
of the Coulomb potential [4]. Using this purely classical formalism, Oks and Uzer
demonstrated the existence of non-radiating states and found their energy to be in exact
agreement with the corresponding results of quantum mechanics. They employed two
fundamental experimental facts, but did not ?forcefully? quantize any physical quantity
describing the atom. In particular, this amounted to classically deriving Bohr?s postulate
on the quantization of the angular momentum rather than accepting it on an axiomatic
basis.
2
?
It important to point out that the physics behind classical non-radiating states is a
new kind of time-dilation found by Oks and Uzer.
The content of this thesis differs from the above mentioned paper by Oks and
Uzer in that the dynamics analyzed are of a more general nature: a term proportional to
1/r
2
is added to the Coulomb
potential. This more complicated potential we call here the
binomial potential. Then the generalized unit Laplace-Runge-Lenz vector [5,6], or as
named by Fradkin, the unit Runge vector [5], is utilized instead of the classical Laplace-
Runge-Lenz vector.
This binomial potential has interesting applications. The primary application
considered here is to pionic atoms. We will classically obtain results corresponding to the
solution of the quantal (relativistic) Klein-Gordon equation, which is appropriate because
pions are spinless particles. Another application concerns the precession of planetary
orbits: for this phenomenon Einstein?s equations of general relativity are equivalent to
non-relativistic equations for the motion in the binomial potential [7]. We shall also
briefly mention an application furnished by the description of the energy of nonradiating
states of the so-called nanoplasmas [14].
An outline of the remainder of the thesis is in order:
In section 2, we briefly outline Dirac?s generalized Hamiltonian dynamics.
Section 3 serves to describe with more detail the applications of the binomial potential
given in the above paragraph. In sections 4 and 5 we discuss the dynamical symmetries or
Fradkin and the generalization of the Laplace-Runge-Lenz vector.
3
?
We present our new results in section 6 and appendices A, B, and C. Section 7,
contains the conclusions.
2. DIRAC?S GENERALIZED HAMILTONIAN DYNAMICS.
Dirac [1-3] considered a dynamical system of N degrees of freedom characterized
by generalized coordinates q
n
and velocities
dt
dq
v
n
n
= , where n = 1, 2, ..., N. If the
Lagrangian of the system is
( )vqLL ,= , (2.1)
then momenta are defined as
n
n
v
L
p
?
?
= . (2.2)
Each of the quantities q
n
, v
n
, p
n
can be varied by ?q
n
, ?v
n
, ?p
n
, respectively. The latter
small quantities are of the order of ?, the variation being worked to the accuracy of ?. As
a result of the variation, eq. (2.2) would not be satisfied any more, since their right-hand
side would differ from the corresponding left side by a quantity of the order of ? as can be
seen from:
()0=?=
?
?
?
?
?
?
?
?
?
?
?=
nnnn
n
n
pvvp
v
H
vL ???
for an arbitrary variation in the momenta. In the above, Hamilton?s canonical equations
of motion were invoked. Further, Dirac distinguished between two types of equations. To
one type belong equations such as eqs. (2.2), which does not hold after the variation (he
4
?
called them "weak" equations). In what follows, for weak equations, adopting Dirac?s
nomenclature, we use a different equality sign ? from the usual. Another type constitute
equations such as eq. (2.1), which holds exactly even after the variation (he called them
"strong" equations).
If quantities ?L/?v
n
are not independent functions of velocities, one can exclude
velocities v
n
from Eqs. (2.2) and obtain one or several weak equations
( ) 0, ?pq? , (2.3)
containing only q and p. In his formalism, Dirac [1-3] used the following complete
system of independent equations of the type (3):
( ) 0, ?pq
m
? , ( )Mm ,...2,1= . (2.4)
Here the word "independent" means that neither of the ??s can be expressed as a linear
combination of the other ??s with coefficient depending on q and p. The word "complete"
means that any function of q and p, which would become zero allowing for eqs. (2.2) and
which would change by ? under the variation, should be a linear combination of the
functions ?
m
(q, p) from (4) with coefficients depending on q and p.
Finally, proceeding from the Lagrangian to a Hamiltonian, Dirac [1-3] obtained
the following central result:
( ) ( )pqupqHH
mmg
,, ?+= (2.5)
(here and below, the summation over a twice repeated suffix is understood). Equation
(2.5) is a strong equation expressing a relation between the generalized Hamiltonian H
g
and the conventional Hamiltonian H(q, p). Quantities u
m
are coefficients to be
determined. Generally, they are functions of q, v, and p; by using Eqs. (2.2), they could
5
?
be made functions of q and p. It should be emphasized that H
g
? H(q, p) would be only a
weak equation - in distinction to Eq. (2.5).
Equation (2.5) shows that the Hamiltonian is not uniquely determined, because a
linear combination of ??s may be added to it. Equations (2.4) are called constraints. The
above distinction between constraints (i.e., weak equations) and strong equations can be
reformulated as follows.
Constraints must be employed in accordance to certain rules. Constraints can be
added. Constraints can be multiplied by factors (depending on q and p), but only on the
left side, so that these factors must not be used inside Poisson brackets.
If f is some function of q and p, then
dt
df
(i.e., a general equation of motion) in the Dirac's
GHD is
[][
mm
fuHf
dt
df
?,, += ], (2.6)
where [f, g] is the Poisson bracket defined for two functions f and g of the canonical
variables p and q as:
[]
rrrr
q
g
p
f
p
g
q
f
gf
?
?
?
?
?
?
?
?
?
=, . (2.7)
where r is an index put to stress the fact that in general there will be several generalized
coordinates and momenta. Here and throughout we adopt the summation convention so
that a sum is understood over any repeated index unless it is explicitly stated otherwise.
Substituting ?
m'
in (2.6) instead of f and taking into account eqs. (2.4), one obtains:
[][ ] 0,,
mmm
?+
??
???
m
uH. (m? = 1, 2, ?, M). (2.8)
6
?
7
?
These consistency conditions allow determining the coefficients u
m
.
Last of all, we note that the GHD was designed by Dirac specifically for
applications to quantum field theory [3], that is, for the purpose totally different from our
purpose.
3. APPLICATIONS OF THE BINOMIAL POTENTIAL
A. Pionic atoms described by the Klein-Gordon equation of relativistic quantum
mechanics.
Relativistic treatments of the hydrogenic atoms are typically presented working
with the Dirac equation, which is a relativistic wave equation that is particularly suited
well for spin-1/2 particles. However, in the literature one may also find a treatment of
hydrogen and hydrogenlike atoms ignoring spin; that is, working with the Klein-Gordon
equation (hereafter, the KG equation) [8,10-13].
The radial KG equation for the problem of the hydrogenic atom is given by:
( )
0
)1(
4
12
2
2
2
2
=
?
?
?
?
?
?
?+
??++ R
Zll
d
dR
d
Rd
?
?
?
?
???
. (3.1)
where Z is the atomic number and
137
1
2
?=
hc
e
? is the fine structure constant.
Thus, the radial KG equation for the Coulomb potential is equivalent to the radial
Schr?dinger equation for the binomial potential - ?/? - ?
2
/?
2
.
For usual hydrogenic atoms, the fine structure splitting predicted by the KG
equation is greater than what is observed experimentally [8]. However, for pionic atoms,
the KG equation becomes exact. Indeed, the pionic atom is an exotic hydrogenic atom,
8
?
where the atomic electron is substituted by a negative pion. Negative pions are spinless
particles of the same charge as electrons, but 273 times heavier than electrons. Due to the
spinless nature of pions, the KG equation for pionic atoms becomes exact.
B. Precession of planetary orbits
In his seminal paper, Die Grundlange der allgemeinen Relativit?sthoerie [7],
Einstein showed that general relativistic effects perturb the Kepler potential by an
additive term proportional to 1/r
2
and used it to calculate the precession of Mercury?s
orbit around the sun. His calculations for the precession yielded 43??/century, which was
later confirmed by observations. There are many good textbooks on general relativity that
derive this result [15-17].
C. Radiation of nonrelativistic particles in a central field
Karnakov et al. [14] derive the spectrum and expressions for the intensity of
dipole radiation for a classical nonrelativistic particle executing nonperiodic motion. The
potential in which the particles under consideration move is of the form ()
2
rr
rU
??
+?= .
The authors of this paper apply their results to the description of the radiation and the
absorption of a classical collisionless electron plasma in nanoparticles irradiated by an
intense laser field. Also, they find the rate of collisionless absorption of electromagnetic
wave energy in equilibrium isotropic nanoplasma.
9
?
4. DYNAMICAL SYMMETRIES OF FRADKIN
Fradkin [5] has shown that all classical dynamical problems of both the
relativistic and non-relativistic type, dealing with a central potential, necessarily possess
O(4) and SU(3). This led him to a generalization of the Runge-Lenz vector in the Kepler
problem. Fradkin also found a generalization of the conserved symmetric tensor for the
harmonic oscillator problem, and constructs a systematic way of imbedding the Lorentz
and the SU(3) group in and infinite-dimensional Lie algebra. Here we will only be
concerned with the results relating to the generalization of the Runge-Lenz vector and the
construction of the elements of the Lie algebra of O(4) and SU(3) in terms of canonical
variables.
In the non-relativistic Kepler problem the force on the affected particle is an
inverse square force given by:
r
r
?
2
?
?=
?
p ; ,
?
= rp m
r
r
r
=? (4.1)
and the overdot denotes total differentiation with respect to time. In the Kepler problem,
the Hamiltonian and the angular momentum (vector L and magnitude L
2
) are the
conserved quantities. There also exists another conserved vector quantity, namely the
Laplace-Runge-Lenz vector, or simply the Runge-Lenz vector. It is defined to be:
()(rmmE
?
2 2
1
????=
?
LpA ) (4.2)
10
?
For negative energies (E<0) A is a real vector. This vector, which is a constant of the
motion, lies in the plane of the orbit and points from the center of motion to perihelion
(that is, along the major axis from one focus to the closest point of the orbit); some
authors refer to it as the eccentricity vector [10].
Fradkin found, by differentiation via the standard Poisson bracket formalism, that
for the Kepler problem, and indeed for all central potential problems, that A, L, and H
satisfy the following closed Lie algebra:
[ ] [ ]
[]
[]
[]
kijkji
kijkji
kijkji
ii
LAA
AAL
LLL
HLHA
?
?
?
=
=
=
==
,
,
,
0,,
(4.3)
It is seen that the Lie algebra given above is isomorphic to that of the generator of the
O(4) symmetry group, which is the group of orthogonal transformations representing
rotations in four dimensions. Fradkin also concluded that if the existence of the Runge-
Lenz vector is simply to ensure that the plane of the motion is conserved, then it should
always be possible to find a vector analogous to the Runge-Lenz vector for all central
potentials.
Fradkin proposed a generalization for the Runge-Lenz vector choosing
as a mutually orthogonal triad of unit vectors. This unit Runge vector is LrLr
?
? and ,
?
,? ?
LrLrkLLkrrkk
?
?)
?
?
?
(
?
)
??
(?)?
?
(
?
???+?+?= , (4.4)
11
?
but since the unit Runge vector is in the plane of the orbit and the angular momentum
vector is perpendicular to the plane of motion, then the second term is identically zero
( ). may be chosen to be the direction from which the azimuthal angle ? is L?k
?
k
?
measured (with the positive sense given by a right-handed rotation about L
?
), then we
have:
?? sin
?
?
?
and cos
?
? =??=? Lrkkr
(4.5)
thus
Lrrk
?
?)(sin?)(cos
?
?+= ?? (4.6)
Defining u=1/r, we may write the following differential equation for u and the azimuthal
angle ? in terms of the energy E, potential V and angular momentum L:
()
2
2
2
2
uVE
L
m
d
du
??
?
?
?
?
?
?
=
?
?
?
?
?
?
?
(4.7)
At this point we note the following relations and definition:
()
L
r
u
f
ELuf
p?
?
?
?
?
?
?
?
?
=
=
?
sin
),,(cos
2
?
?
. (4.8)
Further, the putting V=-?u for the potential of the Kepler problem, the orbit equation
becomes:
[ ] ( )muLmmELf ??? ?+==
?
2
2
1
22
)(2cos . (4.9)
The unit Runge vector may be expressed as:
Lp?
?
?
+
?
?
?
?
?
?
?
?
?=
?
u
f
Lr
u
f
ufk
2
?
?
(4.10)
and it?s Poisson bracket with the total energy E (or, more importantly, the Hamiltonian)
vanishes.
12
?
Lastly, all of its entries have mutually vanishing Poisson brackets and it satisfies
the following relation with the angular momentum:
[ ] ;0,
?
=Hk
i
[ ] ;0
?
,
?
=
ji
kk
[ ] 3,2,1,,for ;
??
, == kjikkL
kijkji
? . (4.11)
13
?
5. FURTHER TOPICS ON THE GENERALIZATION OF THE LAPLACE-
RUNGE-LENZ VECTOR
We now turn to a brief discussion of further results that were utilized in our work.
They are the results of Holas and March [6] on a further treatment of the unit Runge
vector of Fradkin discussed in the previous section. These results, however, are centered
on the construction and time dependence of the vector itself rather than on the dynamical
symmetries of central potentials or the algebras satisfied by the unit Runge vector.
Holas and March using
( )
rL
rpr
Lp ?
?
?=?
Lrr
L
22
2
(5.1)
they rewrite the unit Runge vector, eq. (4.10), as:
( )
rL
u
f
Lr
rfk
?
?
?
?
?
?
??
?=
rp
(5.2)
where the function f is specified in the next section. This is the form of the unit Runge
vector with which we shall work.
14
?
6. APPLICATION OF GENERALIZED HAMILTONIAN DYNAMICS
TO THE BINOMIAL POTENTIAL
In our work, the angular momentum vector and the unit Runge vector are
constants of the motion for a centrally symmetric potential and consequently have
vanishing Poisson brackets with the Hamiltonian for the system and are thus suitable
constraints for the application of GHD. Following Oks and Uzer [4], the Hamiltonian for
this system is:
()(
00
2
22
??
22
kk
rr
Zep
H
g
??+??+
?
+?= wLLu
??
), (6.1)
where ? is the strength of the binomial potential, Ze is the nuclear charge, e is the charge
of an electron, ? is the reduced mass, u and w are the yet unknown constant vectors (to be
determined later) of the GHD formalism, L
0
and are the values of the angular
momentum and unit Runge vector in a particular state of the motion so that in those states
0
?
k
0
LL ? (6.2)
and
. (6.3)
0
??
kk ?
15
?
We may define the following quantities:
2
0
22
0
2
2
r
HH
r
Zep
H
B
?
?
?
+=
?=
(6.4)
where the subscript B is for binomial. The consistency conditions for this system are:
[ ]
[]0,
?
0,
?
?
g
g
Hk
HL
. (6.5)
First we must derive the form of the unit Runge vector in this problem. It is derived in
Appendix A. We arrive at the result:
( )
()
rL
uu
gg
f
gg
gg
gg
fg
Lr
r
gggg
gg
k
?
?
1
1
1
?
1
1
?
3
3
3
2
0
2
0
0
0
2
0
22
0
2
0
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
+
?
?
+++
+
=
rp
. (6.6)
where
2
0
22
0
2
0
1
1
gggg
gg
f
+++
+
=
. (6.7)
and
()( )
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+++
++
?
+++
=
?
?
u
g
gggg
ggggg
gggg
g
u
f
2
3
2
0
22
0
2
2
00
2
0
22
0
2
0
1
1
1
, (6.8)
16
?
The functions and are defined in Appendix A. The unit Runge vector as appears in
eq. (6.6) is a general result, valid for any value of the parameter
g
0
g
?. Hereafter, however,
we only consider a small perturbation in the binomial potential such that
. We
therefore perform a Taylor series expansion about
2
L<> . ?
At this point, in keeping with the development of the problem as in [4], we should
introduce Planck?s hypothesis, whereby we assume that the smallest possible change in
energy is proportional to the frequency of the motion, and the proportionality constant is
the reduced Planck?s constant Jsh
34
1005.1
2
?
???
?
h
in SI units. In our particular
problem, however, this is not so simple because, as is established in Holas and March [6],
the unit Runge vector is only piecewise continuous reflecting the well-known fact that the
motion in the modified Coulomb potential is only conditionally periodic (as opposed to
periodic). Given this fact, the relation between changes of the energy and of the angular
momentum should be refined as follows:
???
?=?=? ?? LdtLdEdt
00
?
TT
r
(6.36)
where T
r
is the period of radial motion and T
?
is the period angular motion. Eq. is
justified as the change in energy correlates with the change, in this case a decrease, of the
size of the orbit. Therefore, the integral containing the energy is over the period of radial
motion. On the right-hand side of eq. (6.36), the integral contains the angular momentum
which is the variable canonically conjugate to the angular variable ?, therefore the
integration is performed over the period of angular motion.
Combining eq. (6.36) with Planck?s hypothesis we get:
????
?
?
hLE
T
T
LdtLdEdt
r
TT
r
=?=???=?=?
???
00
(6.37)
In eq. (6.37), the change in energy must, of course, satisfy
(
SSgSS
LH
h
hHHHE ,
2
00
?? ==??=? )?????????????????????????????(6.38)
or
(
SSS
LH
h
H ,
2
0
?? ). (6.39)
We note that on both sides of the eq. (6.39) only physical quantities pertaining to the
stable states are present. Also, in eq. (6.37) we have
24
?
2
1 ;
1
LT
T
r
r
?
+=== ?
??
?
?
?
(6.40)
(note that as ,0?? 11
2
?
?
+=
L
? , which implies that
?
TT
r
= , as known from the
Coulomb potential)
and therefore
()
( )
2
3
42
0
2
1
2222
,
2
1
S
rr
SSS
H
eZ
m
nh
m
n
hmnh
LH
h
HE
??
?
?
??
?
?
?
?
?
?
?
?
?
?
?
+=
?
?
?
?
?
?
?
?
+=
+
=?=?
(6.41)
where n,m = 1,2,3? . In the third step of eq. (6.41) we used the relation between the
frequencies given in eq. (6.40) and we have substituted
( )
SS
LH ,
0
?
for the term
()
2
?
?? mn
r
+
, which is the average of the two frequencies throughout the motion
(hence the 1/2); and, further, the expression must be valid not only for the first harmonic,
but for all occurring harmonics of the radial and angular frequencies, hence the integer
factors n and m. We have also used:
2
2
3
42
0
0
2
1
8
L
Ze
H
eZ
S
r
???
?
?
?
+
== . (6.42)
We now take notice that from eq. (6.41) we may obtain an expression for the
Hamiltonian in the radiationless state of motion in terms of the integers n and m, we find:
()
...2,1
2
22
42
=
+
= ; n,m
mnh
eZ
H
S
?
?
. (6.43)
25
?
We compare this classically-derived result with the known quantal result as may be found
in, say, Quantum Mechanics: Nonrelativistic Theory of Landau and Lifschitz [18] in
problem 3 after section 36:
()( )
...2,1,0
1212
2
2
2
42
=
+++
= l
l
,; n
nh
eZ
H
r
r
quantal
?
?
. (6.44)
where n
r
and ? are the radial and angular momentum quantum numbers, respectively. We
see that in the quantal result, the ground state ( 0, =
r
nl ), agrees exactly with our derived
expression for n,m=1. Furthermore, the correspondence between the quantal result and
ours agrees for all odd n and m, i.e. when these integers are of the form n=2k+1 and
m=2q+1, q,k=0,1,2?. We may identify n and m as the radial and angular harmonic
numbers.
26
?
27
?
7. CONCLUSIONS
We close with a brief recapitulation of the work put forth in the preceding.
In the section of application, motivation was given for the use and interest of the
binomial potential. The well-known and interesting applications mentioned were that of
the solution to the Klein-Gordon equation governing the dynamics of pionic atoms;
radiation of particles in nanoplasmas; and the advance of the perihelion of planets
orbiting in a central potential as can be shown by means of general relativity. The main
new results obtained for the binomial potential are as follows.
1. We obtained an explicit expression for the additional (to the angular momentum)
vector integral of the motion: the unit Runge-Lenz vector.
2. Beginning with Dirac?s generalized Hamiltonian dynamics, a purely classical
formalism, a (generalized) Hamiltonian was set up that described the dynamics of
a spinless particle in a Coulomb potential perturbed by the presence of a binomial
potential, i.e. one that varies inversely with the square of the distance from the
center of force. With this Hamiltonian and the use of consistency conditions, in
this case the necessity that the angular momentum, energy (Hamiltonian), and the
unit Runge-Lenz vector be the seven conserved quantities of the central potential
it was shown that the use of GHD leads to an effective time dilation.
3. We derived classical energies of radiationless states in the system of bound
spinless particles and found that they agree with quantum theory for the ground
state and with all states of odd principle and angular momentum quantum
numbers.
4. We derived the explicit expression for the generalized Hamiltonian. It leads to a
dynamics that is much richer than the usual classical dynamics. This can be seen
from the many additional terms in the equations of motion derived in Appendix
C.
It is worth emphasizing some interesting physics of classical nonradiating stable
states following Oks and Uzer [4]: In those states, 0==
dt
d
dt
d pr
, so that
0
)( rr =t and
, where and are some constant vectors. Thus, the particle (for example,
the pion) is motionless, but its momentum is nonzero. This is not surprising: for example,
for a charge in an electromagnetic field characterized by a vector potential A, it is also
possible to have
0
)( pp =t
0
r
0
p
0=
?
=v
m
mc
e
Ap
, while 0?= Ap
mc
e
.
28
?
29
?
BIBLIOGRAPHY
1. Dirac, P. A. M 1950 Canad. J. Math. 2, 129
2. Dirac, P. A. M. 1958 Proc. R. Soc. A 246 326
3. Dirac, P. A. M 1964 Lectures on Quantum Mechanics (New York: Academic).
Reprinted by Dover Publications, 2001.
4. Oks E. and Uzer T. 2002 J. Phys. B: At. Mol. Opt. Phys 35 165
5. Fradkin D. M. 1967 Prog. Theor. Phys. 37 798
6. Holas A. and March N. H. 1990 J. Phys. A: Math. Gen. 23 735
7. Einstein A. 1916 Annalen der Physik 49. It is reprinted in The Principle of Relativity
(Dover 1952) with other landmark papers by Weyl H., Lorentz H., and Minkowski H.
8. Josephson J. 1980 Found. of Phys. 10 243
9. Landau L. D. and Lifschitz E. M. 1982 Mechanics 3
rd
Edition Butterworth-
Heinneman
10. Sokolov A. A., Ternov I. M., and Zhukovskii V. Ch. 1984 1
st
edition Quantum
Mechanics Mir Publishers.
11. Greiner W. 1990 Relativistic Quantum Mechanics: Wave Equations
12. Schiff L. I. 1968 Quantum Mechanics (International Pure and Applied Physics
Series) 3rd edition McGraw-Hill Companies.
30
?
13. Capri A. 2002 Relativistic Quantum Mechanics and Introduction to Quantum Field
Theory 1
st
edition World Scientific Publishing Company.
14. B. M. Karnakov, Ph. A. Korneev, and S. V. Popruzhenko 2008 J. of Exp. and Theor.
Phys., 106, No. 4, 650
15. Landau L. D. and Lifschitz E. M. 1980 Classical Theory of Fields 2
nd
Edition
Butterworth-Heinneman
16. Walecka J. D. 2007 Introduction to General Relativity 1
st
edition World Scientific
Publishing Company
17. Schutz S. 1985 A First Course in General Relativity Cambridge University Press.
18. Landau L. D. and Lifschitz E. M. 1981 Quantum Mechanics: Nonrelativistic Theory
3
rd
Edition Butterworth-Heinneman?
?
?
?
?
?
?
?
?
APPENDIX A
DERIVATION OF THE FUNCTIONAL FORM
OF THE UNIT RUNGE VECTOR
The function f, given by
()()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?==
?
u
u
eff
ud
uL
u
VE
L
f
0
2
1
2
1
2
;cos
???
?? (A.1)
where
??=
22
LL
eff
(A.2)
is the effective angular momentum and shows a correction due to the binomial potential.
The integral in eq. (A.1), upon the substitution of the Coulomb potential, may be
rewritten as:
()()
()
??
?
?
?
?
?
?
?
? ?
??+
?
?
?
?
?
?
?
?
?
=?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
u
u
u
u
eff
uL
uZeE
ud
ud
uL
u
VE
00
2
2
2
1
2
2
1
2
??
??
. (A.3)
If we now introduce the substitutions
31
?
?
?
?
?
?
?
?
?
?=
?
?
?
?
?
?
?
?
+=
?
?
?
?
42
2
2
2
2
42
2
2
2
1
2
1
2
1
eZ
EL
L
Ze
u
eZ
EL
L
Ze
u
, (A.4)
then the left-hand side of eq. (A.3), in the indefinite form of the integral, becomes:
()() ()()
?
?
?
?
?
?
?
?
?
?
?
?
??
+
?
=
??
?
?
?
21
21
1
21
2
tan
uuuu
uu
u
uuuu
ud
(A.5)
after some simplifications. It is convenient to define
2
2
21
3
2 L
Zeuu
u
?
=
+
= (A.6)
and thus eq. (A.5) reduces to:
()() ()()
?
?
?
?
?
?
?
?
??
?
=
?
?
?
?
?
?
?
?
?
?
?
?
??
+
?
??
21
31
21
21
1
tan
2
tan
uuuu
uu
uuuu
uu
u
. (A.7)
Putting in the limits of integration yields:
()() ()()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
??
?
=
??
2001
301
21
31
tantancos
uuuu
uu
uuuu
uu
f . (A.8)
It is convenient to define:
()()
00
21
3
)( ;)( gug
uuuu
uu
ugg ?
??
?
?= . (A.9)
Using the identity
()
2222
11
1
1
)(tan)(tancos
BABA
AB
BA
+++
+
=?
??
, (A.10)
32
?
we may then write
2
0
22
0
2
0
1
1
gggg
gg
f
+++
+
= . (A.11)
Consequently, the partial derivative in the unit Runge vector becomes:
()( )
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+++
++
?
+++
=
?
?
u
g
gggg
ggggg
gggg
g
u
f
2
3
2
0
22
0
2
2
00
2
0
22
0
2
0
1
1
1
, (A.12)
where
()()
()
()()
2
3
21
21
21
21
2
2
2
11
uuuu
uuu
uu
u
uuuuu
g
??
++??
?
?
?
?
? +
?
?
??
=
?
?
. (A.13)
We may use the definitions (A.9) and (A.11) to rewrite eq. (A.13) and put it into
eq. (A.12) to get the following compact form:
( )
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
+
=
?
?
3
3
3
2
0
2
0
0
0
1
1
1 uu
gg
f
gg
gg
gg
fg
u
f
. (A.14)
where the term in the second set of parenthesis is the simplification of
u
g
?
?
. We thus
arrive at:
( )
()
rL
uu
gg
f
gg
gg
gg
fg
Lr
r
gggg
gg
k ?
?
1
1
1
?
1
1
?
3
3
3
2
0
2
0
0
0
2
0
22
0
2
0
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
+
?
?
+++
+
=
rp
. (A.15)
This is a general result valid for any value of ?. However, since we are
considering a small perturbation in the binomial potential, such that , then we
may perform a Taylor series expansion of the unit Runge vector with respect to
2
L<