Bachelor Thesis

The symmetriesof the Kepler problem

Author:Marcus Reitz

Supervisor:Prof. dr. Erik Verlinde

Second reader:Prof. dr. Kostas Skenderis

University of AmsterdamInstitute for Theoretical Physics

Science Park 9041090 GL, Amsterdam

The Netherlands

August 2011

Abstract

The properties of the Kepler Problem are well known in classical and quantum mechan-ics. These properties are usually derived in a direct calculation that gives no informationabout the origin of these properties. There exists a more fundamental method that givesa deeper insight in why a specific physical system behaves as it does. By looking at thesymmetries of a given physical system one can often derive the essential governing laws ofthat system. This approach gives an answer to the question of the origin of these laws. Inthis thesis the goal is to investigate how useful this method is and if it really gives a deeperunderstanding of the origin of the properties then other methods. The Kepler problem isvery suitable for this approach because it is an integrable system and its solutions consistof bound orbits for negative energies. Therefore it will function here as an example as tohow this method works in a specific case. It will be shown that in the classical as wellin the quantum mechanical picture the well known properties can be derived by lookingat the symmetries of the Kepler problem. This approach also gives an intuitive bridgebetween the classical and the quantum mechanical formalism.

Contents

1 Introduction 2

2 Central force problems and the LRL vector 32.1 Inverse square problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Integrability of inverse square problems . . . . . . . . . . . . . . . . . . . . . 5

3 The Noether theorem and the LRL vector 7

4 A higher order symmetry 104.1 The hodograph method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 An extension of the momentum space . . . . . . . . . . . . . . . . . . . . . . 11

5 From Poisson brackets to quantum commutators 14

6 The SU(2) ⊗ SU(2) symmetry of the Kepler problems 166.1 Representations of a symmetry group . . . . . . . . . . . . . . . . . . . . . . 166.2 The group SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.3 The hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7 Conclusions 21

8 Appendix A: Representations of SU(2) 23

1

1 Introduction

An important aspect of physics is to discover the properties of the different physical systemsthat exist. There are several approaches to find these properties for a given system. Meth-ods that rely on direct calculation often are efficient in finding these properties but keep thequestion open as to what the origin of these laws is. A more fundamental approach is neededto answer this question. There can be found certain symmetries of the physical systemsthemselves. These symmetries come, although sometimes hidden, with the definition of thephysical system and it would therefore not be surprising if they also are important for itsproperties. There exists a theorem that connects symmetries of a certain physical system tothe conservation laws governing that system. This is the Noether theorem. This theorem wasthe starting point for the discussion in this thesis of the connection between symmetries of asystem and its properties.

For this thesis the goal was to look if the connection between the symmetries of a systemand its properties can be found in the case of the Kepler problem. The Kepler problem thusfunctions as a study case to examine if starting from its symmetries all the known propertiescan be derived. This will be done for the classical as for the quantum mechanical case. TheKepler problem has been chosen for this task because it is a so called integrable system. Thiswill be discussed later but in essence it means that the solutions to the equations of motionare fully determined by the constants of motion. For the Kepler problem there exist thewell known laws of conservation of energy and angular momentum but there is also anotherconserved quantity. There exists a conserved vector for the Kepler problem less known thanthe angular momentum. This vector is called the Laplace Runge Lenz vector. It was namedafter its multiple discoverers because it was rediscovered several times. Wilhelm Lenz usedthe vector to calculate the energy levels of perturbed Kepler motions and referred his use ofthe vector to a text by Carl Runge, but the earliest mentioning of this vector was by PierreSimon Laplace in the Traite de mecanique celeste although there are different views of whowas the original discoverer [9]. The LRL vector plays an important role in this thesis becausewithout it the symmetries can’t be made complete.

The setup of the thesis will be as follows. First there will be given a treatment of theclassical Kepler problem to introduce all the physical concepts needed later. In that sectionthe LRL vector will be shown as an aid to derive the Kepler orbits. From then the discussionwill turn into the direction of symmetries. The direct application of the Noether theorem willbe shown in the case of the LRL vector. Then there will be shown a special symmetry of theenergy of the Kepler problem. In the following section there will be discussed an interestingbyproduct of the discussion. The discussion of symmetries gives an insightful bridge from thecontinuous classical formalism to the discrete quantum mechanical formalism. The results ofthis section together with the symmetries already found will then be used in the quantummechanical version of the Kepler problem. For this the theory of Lie algebras and Lie groupsis needed as well as representation theory. These theories will be covered to the extend theyare needed for the rest of the section. In the final conclusions it will be evaluated if the usedapproach leads to a deeper understanding of the Kepler problem.

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2 Central force problems and the LRL vector

2.1 Inverse square problems

The most common inverse square problems are the two body systems of gravity and of theCoulomb force. Both of them are reducible to a one body problem in a central potential bythe classic introduction of the reduced mass µ and by describing the motion of the particle ofinterest relative to the center of mass. The corresponding Lagrangian will then after droppingthe irrelevant kinetic energy of the center of mass be

L =1

2µr +

α

r(2.1)

where α is the constant of the respective force. Inverse square problems are also known asKepler problems.

The problem has the well known conservation of angular momentum L = r × p. Fromthe conservation of L it follows that r must lie in a plane orthogonal to L. The equations ofmotion can then be written in polar coordinates.

µr2θ = l (2.2)

µr − µrθ2 +α

r2= 0 (2.3)

With some substitutions these equations are easily integrated. An explicit solution for thetime dependence is not solvable because of the need of an inversion of a function of the formr = x− sin(x), but it is possible to find r as function of θ [1].

r =a(1− e2)

1 + e cos(θ − θ0)(2.4)

In this equation the parameter a is called the semi-major axis, the parameter e is the ec-centricity and θ0 is the initial angle (the angle at t = 0). Because every central force isconservative the total energy of the system is conserved. With this the eccentricity can beexpressed in terms of the energy, the angular momentum l, the constant of the force α and µthe reduced mass. The semi - major axis can also be expressed in the energy.

e =

√1 +

2El2

µα(2.5)

a = − α

2E(2.6)

From equation (2.4) it is visible that for e < 1 the relation between r and θ gives an ellipsewith eccentricity e and semi - major axis a with one of the focus points in the origin. InKepler problems there is another vector of interest. The Laplace - Runge - Lenz vector. Themathematical definition that will be used here is

A = p× L− µαrr

(2.7)

3

This vector can be used as the basis to calculate the solution of the Kepler problem in analternative way. With the identities

r · (r× p) = l2 (2.8)

A ·A = 2µEl2 + µ2α2 (2.9)

it is possible to acquire the same relations between r and the angle θ. In (2.9) the definitionof the Hamiltonian as the total energy is used to introduce E in the equation.

r ·A = Ar cos(θ) = l2 −mαr (2.10)

giving1

r=µα

l2(1 +

A

µαcos(θ)) (2.11)

which is identical to equation (2.4) after some rearranging and substitution of (2.9). The LRLvector also gives an alternative way to calculate the energy of the system (2.9).

The LRL vector is of practical use because it immediately gives the geometry of the solutionof a given Kepler problem for given energy and angular momentum. This interpretation ofA as defining the geometry of the solutions becomes even more apparent after a rescalinge = A

µα . The vector e is called the eccentricity vector. By comparing (2.5) to (2.9) it followsthat e has a length equal to the eccentricity. For a first guess as to the direction of the vectore it is helpful to compute A · L. This dot product vanishes because of the orthogonality ofL to both terms in the definition of A. From this result follows that e must lie in the planeof motion of the Kepler orbit. Actually the vector is always in the direction of the perihelionof the orbit. So the vector always lies parallel to the semi major axis. This follows from theconservation discussed below.

As the eccentricity is enough to determine the form of an ellipse and as the vector pointsin the direction of the semi-major axis, this shows that the LRL vector can be interpreted asdefining the geometry of a solution of the Kepler problem.

This interpretation of the LRL vector implies that A should be conserved over the orbit.Because the geometry of a bound orbit doesn’t change over time. It is not to hard to showthat A is conserved.

d

dtA = −µα

r3(r(r · dr

dt)− r2dr

dt)− µα d

dt(r

r) = 0 (2.12)

Where there has been only made use of the conservation of L, the definition of the force asthe time derivative of the momentum and some basic vector calculus.

4

Figure 1: A bound orbit of the Kepler problem and the construction of the LRL vector A.

In this section the LRL vector has been introduced as an aid to calculate the orbit of aKepler problem and was interpreted as defining the geometry of the orbit. The calculationabove confirms the conservation of A in a direct way but does not give more informationabout the nature of this conservation. There exist other methods that connect conservationlaws to deeper structures in a given system. In the following sections the conservation of theLRL vector will be connected to these deeper structures, which will also bring the in thissection purely classical description of the Kepler problem closer to its quantum mechanicalcounterpart.

2.2 Integrability of inverse square problems

There exists a definition that categorizes systems in terms of how determined the solutions oftheir equations of motion are. A system is called integrable if there exist as many indepen-dent constants of motion as there exist initial conditions. The Kepler problem is an integrablesystem.

With the LRL vector the Kepler problem has 7 (dependent) constants of motion. Theseare the energy, the 3 components of the angular momentum and the 3 components of theLRL vector. Because this system has 6 initial conditions, the 3 components of the positionvector and the three components of the momentum vector, there must exist a relation thatmakes these 7 dependent constants into 6 independent constants. This relation is equation(2.9) which after rearranging gives the energy expressed in terms of the 6 independent compo-nents of L and A. The integrability of the Kepler problem is the reason why it is so suitableto explore in the light of its symmetries. In the next section there will be made a connectionbetween the symmetries of a system and the constants of the motion. This will be doneby the Noether theorem. Also the property that one can switch from one complete set of

5

independent constants to another will be used in section 6 to show a rather exotic symmetryin the quantum mechanical version of the Kepler problem.

6

3 The Noether theorem and the LRL vector

There exists an insightful theorem by Emmy Noether that connects the principle of leastaction to conserved quantities. In short it says that every symmetry of the Lagrangian cor-responds to a specific conserved quantity. This theorem can be made more generalized if itis stated in terms of gauge - variant Lagrangians. Then this generalized Noether theoremstates that every variation of the coordinates of the Lagrangian for which the variation of theLagrangian can be written as the total derivative of some function it produces a conservationlaw. A total derivative will only contribute to the action integral at the endpoints. Becauseof this such a variation of the Lagrangian will leave the variational problems invariant andthus will not change the equations of motion [3].

To make the theorem explicit assume a general variation of a set of N variables q = {q1, q2, ..., qn}possibly as a function of q that brings forth a variation of the Lagrangian in the form of atotal derivative.

δq = εf(q, q, t) (3.1)

δL = εd

dtΛ(q, q, t) (3.2)

In general a variation of the Lagrangian can be written as follows.

δL =δL

δqδq +

δL

δqδq (3.3)

By using (3.1), the identity ddtδLδq = δL

δq and some rearranging this reduces to

δL = εd

dt(δL

δqf) (3.4)

After equating this to (3.2) the result follows.

εd

dt(δL

δqf − Λ) = 0 (3.5)

This is the result that gives the corresponding conservation law to the variation (3.1). Thefunction

A =δL

δqf − Λ (3.6)

must in the view of (3.5) be a constant of the orbit following from the solutions of this La-grangian. This object A is the conserved quantity meant in the Noether theorem.

In this way it is easy to get the conservation of the angular momentum by looking at arotation of the coordinates. Because this has been done often and is fairly straight forwardthis calculation won’t be done explicitly here. The symmetry of the LRL vector is more exoticand also gives the general scheme to get to a conserved quantity from a certain symmetry.So to use the Noether theorem as another way to show the conservation of the LRL vectorit is first necessary to find the right variation of the position coordinates. This can be doneby calculating the Poisson brackets of A with the position vector but this calculation will be

7

omitted here.

This variation is the following [3].

δkri = ε(µrirk − µ1

2rirk − µδik(

1

2r · r)) i, k ∈ 1, 2, 3 and k fixed (3.7)

This must be seen as a simultaneous variation of all three of the coordinates ri with respectto rk. From (3.7) follows the variation of ri. Where ri was eliminated by use of the EulerLagrange equation of the Lagrangian for the Kepler problem (2.1).

δkri = ε1

2(µrirk − δikµr2 −

αrirkr3

+ δikα

r) (3.8)

The variation of the Lagrangian (2.1) can be written in terms of (3.7) and (3.8).

δL = µr · δr− αr · δrr3

(3.9)

Substituting (3.7) and (3.8) this reduces to (3.10).

δkL = εd

dt(µα

rkr

) (3.10)

With use of the general result of the Noether theorem the equations (3.7) and (3.10) can besubstituted into (3.6).

Ak = µ2rkr2 − µ2rk(r · r)− µαrk

r(3.11)

This is exactly the k-th component of the LRL vector. Thus it follows that all components ofA are conserved. It is also now more clear where this conservation comes from. The separatecomponents of A are conserved because it is possible to transform all of the components ofthe position vector as r→ r + δrk with respect to one of the coordinates k without changingthe equations of motion. The meaning of this result gives a deeper insight in the structure ofthe Kepler problem.

The orbits given by the solutions of the equations of motion form a family of functions.The transformation of the coordinates that do not change the Euler Lagrange equations canbe applied to a specific solution. In this respect the found invariant transformation is alsoa transformation working on this family of orbits mapping them on themselves. Formally,writing G as the family of orbits, g as the transformation of one orbit into another and f asthe transformation r→ r + δr this can be written as follows. Here G is a subset of the set ofdifferentiable functions on R3.

g : G → G g(r) → g(f(r)) g ∈ G (3.12)

The conserved quantity corresponding to the specific transformation f divides the set G. Thefunction f transforms certain orbits with a distinct value of the conserved quantity to anotherorbit with the same value for the conserved quantity. So the set G can be divided in subsets Fof orbits with the same value for a conserved quantity. For this subset F the transformationg is also a map from the subset to itself. The Noether theorem showed the transformationcorresponding to the LRL vector but it isn’t easily visible what kind of transformation the

8

function in (3.7) actually is. The form of (3.7) is rather peculiar and not very insightful. Aremark that can be made is that by scaling of an orbit the value of the LRL vector would notchange as the eccentricity of an orbit is invariant under scaling. This would make it plausiblethat the transformation corresponding to the LRL vector should be some kind of scaling butthis isn’t immediately visible from 3.7.

As a final remark in this section it should be said that although here only the Kepler problemand especially the LRL vector was discussed all of these remarks can be generalized to allclassical problems that can be solved with the theory of the Lagrangian.

9

4 A higher order symmetry

4.1 The hodograph method

Normally after solving the equations of motion the solutions are analyzed by looking at theposition vector as a function of time. Another approach is to look at the velocity vector. Thevelocity of an object is also a function of time and thus follows some trajectory in momentumspace. Such a trajectory in momentum space is called a hodograph. This name was coinedby William Rowan Hamilton [10].

Hamilton showed geometrically that the hodographs belonging to the bound orbits of inversesquare forces are circles. This is also visible from a simple calculation using the definition ofA. By taking the dot product of both sides with itself, some rearranging and by assumingthat the orbit lies in the x-y plane, (2.7) turns into (4.1).

p2x + (py −A/L)2 = (µα/L)2 (4.1)

The Hodograph following from this equation is drawn below.

Figure 2: Hodograph of a Kepler orbit with given values for L and A.

In this figure it is possible to identify some of the relations that were already found in sec-tion 2. The two points where the circle intersects with the px axis lie at distance p0 =

√−2µE.

The geometry of the triangle made up of the blue, red and green line has the same relation as(2.9) in section 2. Also the cosine of the angle η equals the length of the eccentricity vector.This shows that there is an unique hodograph for every combination of the constants of themotion L and A.

The next observation that can be made is that there must exist multiple hodographs withthe same energy. Circles corresponding to different values of L and A but intersecting with

10

the x axis in the same point have the same value for p0. This means there is a whole familyof orbits with the same energy.

Figure 3: Multiple hodographs with the same energy E.

A transformation from hodograph to hodograph within such a family will be a symmetrytransformation of the Kepler problem. To find the nature of this symmetry it is first neededto find this transformation explicitly.

4.2 An extension of the momentum space

The space of this symmetry transformation was introduced by the Russian physicist VladimirFock [7]. An extended 4-dimensional ”momentum” space can be constructed by taking the 3Euclidean momentum vectors and adding a fourth dimension. A given hodograph can thenbe connected with this space by an inverse stereographic projection on this 3-sphere withradius p0 =

√−2µE. The coordinates on this 4-dimensional sphere that correspond to a

11

given hodograph are then given by the following relations1.

ζ = p02p0pxp20 + p2

= p0 sinα sin θ cosφ (4.2)

η = p02p02pyp20 + p2

= p0 sinα sin θ sinφ (4.3)

ξ = p02p0pzp20 + p2

= p0 sinα cos θ (4.4)

χ = p0p2 − p20p20 + p2

= p0 cosα (4.5)

If the relation of the hodograph (4.1) is inserted in these equations for px and py this corre-sponds uniquely to a great circle on a normal sphere by a stereographic projection [7]. Thisis correct only if the value of p0 is equal for both the hodograph and the sphere.

Figure 4: Connection between the hodograph and a great circle on the sphere.

Because the pz component was eliminated by symmetry, it can be brought back to geta more general statement. Every hodograph corresponds to a great circle on the 3-spherethrough an inverse stereographic projection.

Next it can be seen that the statement in the previous section about multiple hodographswith the same energy can be used here. A family of hodographs with the same value for p0corresponds to a family of great circles on the 3-sphere. The transformation that correspondsto the symmetry of the energy has now been found. A rotation of a great sphere around someaxis transforms one hodograph into another but will keep the energy invariant2. From this

1In the article by Vladimir Fock the coordinates are in a somewhat different form as he projects thehodograph divided by p0 on the unit sphere (and the projection was made to the ”south” pole of the spherewhich results in a minus sign in the definition of χ.)

2Under certain rotations the hodographs will not pass trough the exact same coordinate point p0 anymorebut they will still have the same energy

12

discussion can be concluded that the energy is invariant under rotations over a 4-dimensionalsphere for inverse square problems. So there exists a symmetry in the Kepler problem thatis identical to the rotation group of the 4 -dimensional sphere. Another way of saying this isthat the subspace of solutions of the Kepler problem with a given energy FE is closed underthe symmetry group SO(4) which is the rotation group of the 4-dimensional sphere.

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5 From Poisson brackets to quantum commutators

In introductory courses of quantum mechanics the formalism is often introduced as a given. Ifthe correspondence between the classical and the quantum mechanical formalism is discussedit is often only in an incomplete manner. The well known transition from the classicalidentities to quantum operators has a deeper lying reasoning.

x→ x, p→ −i~ δδx

(5.1)

The structure of the classical formalism is in essence the same as the quantum mechanicalversion and this structure becomes visible in the discussion of symmetries.

Classical mechanics can be described in different ways, of which the Euler Lagrange equationsand the Hamilton formalism are the most common. The Hamilton equations can be writtenin a compact form by use of Poisson brackets and by this arises an interesting connectionbetween the Hamiltonian and conserved quantities.

d

dtf(q, p) =

δf

δqq +

δf

δpp =

δf

δq

δH

δp− δf

δp

δH

δq= {f,H} (5.2)

So if f(q, p) is a conserved quantity its time derivative vanishes and the Poisson brackets of theconstant and the Hamiltonian do as well. This structure takes the form of the commutationrelations in quantum mechanics. That is why a constant of the motion in classical mechanicsbecomes an operator commuting with the Hamiltonian in quantum mechanics. Actually thisequivalence can be made more exact. In mathematics there exists the definition of the Liealgebra. A Lie algebra is a vector space V over some field on which there is defined a binaryoperation.

[·, ·] : V × V → V (5.3)

This operation is bilinear and the equations [a,a]=0 and [a,[b,c]]+[b,[c,a]]+[c,[a,b]]=0 musthold for all a, b, c ∈ V . Now the equivalence between the classical and the quantum formalismgoes by the following map.

{f, g} → i[f , g] (5.4)

Where f and g are the quantum operators corresponding to their classical counterparts. It isvisible by this map that the Lie algebras of the Poisson brackets and commutation bracketshave the same structure.

The correspondence between the two formalisms can be taken even further if one starts withthe Noether theorem. From a continuous symmetry in classical mechanics follows a constantof the motion. This becomes an operator in quantum mechanics that commutes with theHamiltonian by (5.2). A set of operators then form a so called Lie group 3 by exponenti-ation. This will be discussed to more extent in the next section but is already mentionedhere to make the picture complete. The commutation relations of the operators with eachother define another Lie algebra on a subspace F of the solution space. If only one operatorworks on this space then the space is defined by a given value of the operator that works onit. If multiple non commuting operators work on a space then this space will be defined by

3A Lie group is a group that is also a differentiable manifold

14

a value of a combined operator that commutes with all of the separate operators. Such ancombined operator is called a Casimir operator and by construction lies in the center of theLie algebra. The Lie group mentioned before will now be a symmetry group working on thisspace F. Every element of this symmetry group working on F will transform an orbit in Fwith a given value for the Casimir operator into another orbit with the same value. So thefound symmetry group is the same symmetry as was started with in the Noether theorem.An equivalent way of saying this is that the Lie algebra mentioned before corresponds to theLie group defined through exponentiation of the chosen operators.

The next step is to get to the quantization of the orbits as this is one of the defining propertiesof the quantum mechanical formalism. This will be done with representation theory. Theexact definition will be given in the next section but in essence it comes down to the following.A representation of a group connects a matrix to every element of that group such that thestructure of the group stays intact. In this way the representation transforms the continuoussymmetry group into a group working on a space with as many orthogonal states as the dimen-sion of the matrices in the representation. This completes the quantization of the subspace F.

Now to summarize, by the Noether theorem a symmetry generates a conserved quantityand this quantity becomes an operator. Next, a set of operators defines a symmetry groupthrough exponentiation that is equivalent to the combined symmetries of the conserved quan-tities that were the origin of the operators. The group works on a subspace F that is definedby a value for the Casimir operator of the group. F is the same space as the subspace ofsolutions in the classical case. In the classical case F was the subset of solutions definedas all the solutions with the same value for a conserved quantity. This conserved quantityis the classical version of the Casimir operator. Next, the representation of the continuousgroup will then quantize the space on which the group acts. This makes that there is only afinite number of orthogonal states. A transformation by the representation of the symmetrygroup will then mix states in a linear combination of the orthogonal base of solutions. Thisin contrast to the classical case where the symmetry transformation transforms one solutioncontinuously into another. With this the transition of the classical to the quantum mechanicalformalism is complete and the discussion can return to the Kepler problem but now in thequantum mechanical formalism.

15

6 The SU(2) ⊗ SU(2) symmetry of the Kepler problems

6.1 Representations of a symmetry group

Another interesting result can be obtained from the discussion of the symmetries in the Ke-pler problem. In the quantum mechanical treatment of the Kepler problem i.e. the hydrogenatom, the known energy levels of the hydrogen atom can be derived from a symmetry. Alsothis symmetry will readily give the partition of the energy space into the usual subspacesdefined by the azimuthal quantum number l.

To show this it is necessary to make use of representation theory. A representation is ahomomorphic map D of some group to the general linear group of dimension n over the fieldof the complex numbers.

D : G → GL(n,C) (6.1)

For the discussion of symmetries this group will be a symmetry group. With a representationit is possible to see how a certain symmetry group works on the solution space of a givensystem.

With this method every element of G is linked to some matrix and because of the homo-morphism the product rules of the group are preserved. In the definition the dimension ofthe representation is not specified. Also there has to be made a difference between reduciblerepresentations and irreducible representations. A representation D(G ) is called reducible ifit can be written as the direct sum of irreducible representations.

D(G ) =⊕α

bα Dα(G ) (6.2)

Where the summation runs over all lower dimensional irreducible representations Dα(G ) andbα is the multiplicity in the given decomposition. A representation is called irreducible if itisn’t possible to find a decomposition as in (6.2).

6.2 The group SU(2)

The symmetry group of interest is the special unitary group of dimension 2 or SU(2). Theelements of SU(2) consist of all 2 × 2 unitary matrices with determinant 1. There exists asurjective homomorphism with kernel {I, -I}4 from SU(2) to SO(3) the group of 3 dimensionalrotations. From this it can be concluded that the structure of SU(2) is essentially the sameas that of SO(3) except that for each real rotation there exist two ”rotations” in SU(2), onewith a plus and one with a minus sign.

To find all representations of SU(2) it is enough to find the representations of the infinitesimalgenerators of the group. The infinitesimal generators of SU(2) are the matrices Ji = 1

2σi.These matrices σi are known as the Pauli matrices.

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)(6.3)

4I is the identity.

16

This means that every element of SU(2) can be obtained by an exponentiation of thesematrices.

X(α) = exp(iαi1

2σi) (6.4)

Further the commutation relations of these generators are given by (6.5).

[Ji,Jj ] = iεijkJk (6.5)

To find a representation it is necessary to construct a vector space on which these matricescan work. Take a vector space with two complex coordinates ϕα with α = 1, 2. If one takesthe n-th term of the Taylor series of some analytical function of these two coordinates thistakes the form of a homogeneous symmetrical polynomial of ϕα of degree n. In short thispolynomial can be written as Pα1α2···αn .

Pα1α2···αn = ϕα1ϕα2 · · · ϕαn (6.6)

This polynomial now transforms as follows.

Pα1α2···αn → P β1β2···βn = Xβ1α1Xβ2α2· · ·Xβn

αnPα1α2···αn (6.7)

The number of independent polynomials of this kind must be n+1 because of the homogeneityof the polynomials. Now the generators given earlier can be written as differential operatorsfrom how they work on a polynomial Pα1α2···αn .

Ji = −1

2(σi)

αβϕ

β δ

δϕα(6.8)

The next step is to derive a certain Casimir operator. A Casimir operator for this group is amatrix that commutes with all the generators of the group. For SU(2) the Casimir operatoris (Ji)

2.5 Using (6.8) this Casimir operator can be expressed as a differential operator.

(Ji)2 =

1

4(ϕα

δ

δϕα)2 +

1

2(ϕα

δ

δϕα) (6.9)

Where there has been made use of the identity∑i

(σi)αβ(σi)

γδ = −δαβ δ

γδ + 2δαδ δ

γβ .

Working on the constructed polynomial Pα1α2···αn this gives the following result.

(Ji)2Pα1α2···αn =

1

2n(

1

2n+ 1)Pα1α2···αn (6.10)

Or by substituting 12n = l.

(Ji)2Pα1α2···αn = l(l + 1)Pα1α2···αn (6.11)

This equation and the commutation relations of the generators of SU(2) are identical to thewell known relations of the quantum mechanical treatment of the angular momentum oper-ator L. Actually the rest of the derivation to find the representations of the generators ofSU(2) is fully analogue to the method of ladder operators in quantum mechanics and will beomitted here. The result of this derivation is a set of representations for the generators of

5Summation over i is implied

17

SU(2) D(l)i (Ji). These are determined by a number l and are of dimension n = 2l + 1 where

the parameter l is half or whole valued6. In all of these representations there exists a Casimir

operator (D(l)i (Ji))

2 with eigenvalue l(l + 1). From the relations of the ladder operators theform of the generators is derivable. The explicit form has been put in appendix A because itisn’t needed here.

With the use of the ladder operators it is possible to show that there are no other repre-sentations of SU(2). This follows from that by raising and lowering with the correspondingoperator the states will stay in the same ”order” and from that there aren’t multiple stateswith the same eigenvalue.

With the exponentiation in (6.4) but with the matrices D(l)i (Ji) instead of Ji all the rep-

resentations of SU(2) are obtained7.

These representations are irreducible but a tensor product of two such representations willnot be irreducible. The general result for the decomposition in irreducible components of thetensor product of two representations of SU(2) is known [5].

D(l′)i ⊗D

(m)i =

l′+m⊕l=|l′−m|

D(l)i (6.12)

This decomposition in irreducible components is nothing else than finding the independentsubspaces of the product space.

Before all of these results can be used for an actual symmetry of the Kepler problem an-other mathematical concept has to be introduced. The commutation relations (6.5) togetherwith the three matrices Ji as generators of their group define a so called Lie algebra. If onefinds other matrices Ki that have the same commutation relations as Ji and also are thegenerators of a group these matrices Ki form a representation of this Lie algebra throughexponentiation. Also the Lie algebra brought forth by these Ki is essentially the same as thatof the Ji and there exists an isomorphism between these Lie algebras. This means that ifone finds a set of operators that brings forth a group through exponentiation with the samecommutation relations as of the group SU(2) there exists an isomorphism of the correspond-ing Lie algebras and thus the two groups share their representations. Also the notion of aCasimir operator is actually an important aspect of the theory of the Lie algebras. The onlyproperty of the Casimir operator used here is that in every representation of the Lie algebrathe commutativity of the Casimir operator will be conserved in its representations. All ofthis can be used to identify operators with a certain symmetry group as it is enough to checktheir commutation relations. This is exactly what will be done in the next subsection.

6.3 The hydrogen atom

All the work in the last subsection has been done as a preparation for this section. One ofthe most important results of the quantum mechanical treatment of the Kepler problem i.e.

6(l = 0, 12, 1, 3

2, ...)

7To be complete it must be added that for the parameters αi in (6.4) the inequality |α| ≤ 2π holds. Herethe three parameters αi together form a vector α.

18

the hydrogen atom is the quantization of the energy levels. The formula for the energy levels(6.13) is well known and can be calculated directly by the Schrodinger equation.

En = −Rn2

(6.13)

With the method of the Shrodinger equation the n2 degeneration and the decomposition intosubspaces defined by the quantum number l are also obtained. If it is possible to connectthese characteristics to a symmetry group, this group could be a tool to find the underlyingstructure of these characteristics.

First it is necessary to define the LRL vector as a quantum mechanical operator. As theangular momentum operator L and the linear momentum operator p do not commute thedefinition as in (2.7) is not suitable. Because of the non-commutativity of L and p it isnot obvious in which order L and p should be placed in the cross product. The quantummechanical definition is thus defined as follows 8.

A =1

2(p× L− L× p)− µαr

r(6.14)

This operator commutes with the Hamiltonian H. Which means that an eigenfunction of theenergy is also an eigenfunction of this operator A. In this definition the LRL vector is also aconstant of the orbit. Now with definition (6.14) there can be made two new operators withH, L and A [2].

K± =1

2(L∓

√H−1

−2µA) (6.15)

This redefinition is done to find commutation relations that are more easily interpretable thenif one would look only at L and A. This would bring the same results but is far more work.The two operators are made up of elements that commute with the Hamiltonian so they com-mute as well. Furthermore they are 3 dimensional vectors and their vector components canbe seen as the infinitesimal generators of a group. As was discussed in the previous sectionthe structure of a group is defined by the commutation relations of its generators. In the caseof (6.15) these take the following form.

[K+i ,K

+j ] = i~ εijkK+

k (6.16)

[K−i ,K−j ] = i~ εijkK−k (6.17)

[K+i ,K

−j ] = 0 (6.18)

These generators K+i and K−i both have the same commutation relations as (6.5). This

means that there exists an isomorphism between the Lie algebra of these K+i and K−i and

the Lie algebra of SU(2) and the group elements of these ”rotations” are thus generatedthrough exponentiation. Because these two operators commute their group elements canwork simultaneously and form a new group together. A state |ψ〉 transforms under this groupas follows.

|ψ〉 → |ψ′〉 = eiαiK+i eiβiK

−i |ψ〉 (6.19)

8This obviously equals the definition (2.7) in the classical case.

19

The product in (6.19) between the exponents is a tensor product as they are independent

groups. The representation of the group elements X+(α) = eiαiK+i and X−(β) = eiβiK

−i are

also representations of SU(2). Because of this the representations of the product X+(α) ⊗X−(β) must be the same as the representations of SU(2) ⊗ SU(2). As the representations ofSU(2) are (2l+ 1) dimensional, the product group works on a (2l+ + 1)(2l− + 1) dimensionalspace. The states of this space will be denoted as |l+, l−〉. The Casimir operator inheritedfrom SU(2) takes the form (K±)2 and works on the states as it did in SU(2)9 except for afactor ~2. A constant is not important for the structure of a Lie algebra so this will only beimportant for the physical interpretation.

(K±)2 |l+, l−〉 = l±(l± + 1)~2 |l+, l−〉 (6.20)

With the identity K · L = 0 this can be constrained further. From this identity it can bederived that l+ must equal l− such that in view of (6.19) the states |ψ〉 will be split up in n2

states |s+, s−〉.

The last step is done by another identity.

(−2(K+)2 − 2(K−)2) |l+, l−〉 = (~2 +µα2H−1

2) |l+, l−〉 (6.21)

Or

−2(n− 1)(1

2(n− 1) + 1)~2 |l+, l−〉 = (~2 +

µα2

2E) |l+, l−〉 (6.22)

Here it was used that the constructed states are all eigenstates of H with eigenvalue E. Theequalities above lead after some rearranging to the wanted result.

En = − µα2

2~2n2(6.23)

This equals equation (6.13) if α is the constant corresponding to the Coulomb force. This waythe energy levels of the hydrogen atom have been found as a function of their degeneration.

The next step is to look at the subspaces of this representation that is isomorphic to SU(2)⊗SU(2). The formula for the decomposition of the product space can be obtained from the de-composition given in (6.12). This decomposition was defined as a product of representationsof arbitrary dimensions. Here the representations are both of dimension n.

D(n) ⊗D(n) = D(2n−1) ⊕D(2n−3) ⊕ ...⊕D(3) ⊕D(1) (6.24)

This is a decomposition in irreducible representations of SU(2). The only operator thathas the same Lie algebra as SU(2) is the angular momentum operator L. Therefore thesesubspaces must be angular momentum subspaces. They have the value l = 0, 1, ..., n − 1 forthe angular momentum. This completes the discussion of symmetries for the hydrogen atom.The value for the energy and its n2 degeneration has been found as a result of a 4 dimensionalsymmetry of the Kepler problem. This symmetry is the same as was found in section 4. Fromthis symmetry also follows the decomposition in angular momentum subspaces. So it can beconcluded that all of the important properties have been derived starting with a symmetryof the system itself.

9From now on the notation K± will mean both the group generator and its representation to avoid cum-bersome notation.

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7 Conclusions

The goal of this thesis was to look if one can find the properties of a system by startingfrom its symmetries taking the Kepler problem as an example. It can be concluded thatthe properties of the Kepler problem have been found by this method. In the classical andthe quantum mechanical case the important properties have been derived starting from thesymmetries of the physical system itself. Also the equivalence of the structure of the classicaland quantum mechanical formalism has been discussed. By looking at the symmetries of theKepler problem the corresponding conserved quantities give structure to the set of all Keplerorbits. This structure is conserved in the quantum mechanical formalism. By looking at thesymmetries of the Kepler problem the equivalence of the structures of the classical and thequantum mechanical formalism becomes visible. The structure of the formalisms is essen-tially the same. Because this equivalence wasn’t the main object of this thesis this rathermathematical discussion hasn’t been done as rigorous as the subject deserves. If the readeris interested in an exact treatment of this equivalence he can turn to standard books on Liegroups and Lie algebras.

Another note that has to be made is that in this thesis there has only been looked at the Ke-pler problem. If one wants to analyze the method of symmetries in general a wider approachis needed. Although it could be that the method is only usable to similar physical systems.In the whole discussion the integrability of the Kepler problem and the fact that for negativeenergies the solutions of the Kepler problem are all closed orbits was very important. It istherefore likely that this method is less usable for physical systems that do not have theseproperties. The reason for looking for a more fundamental way of deriving the propertiesof the Kepler problem was to connect them to an intrinsic property of the physical systemitself. This only gives a satisfying result if the symmetries can be understood fully. For therotational symmetry of the angular momentum this is no problem. In this case one could saythat the goal has been achieved but for the 4 dimensional rotational symmetry this is notthe case. Although it has been found that the symmetry is equivalent to the group SO(4) itis not entirely visible what this rotation exactly is. To completely solve this question morework has to be done. Nevertheless it can be said that the properties of the Kepler problemare understood in a more fundamental way by use of the method of symmetries.

21

References

[1] H. Goldstein, Classical Mechanics (2nd edition). Addison Wesley. (1980).

[2] M.J.G. Veltman, B.Q.P.J. de Wit and G. ’t Hooft, Syllabus Lie-groepen in de fysica.University of Utrecht. (2006).

[3] J. Levy Leblond, Conservation Laws for Gauge-Variant Lagrangians. Am. J. of Phys. 39.(1970).

[4] M. Bander and C. Itzykson, Group Theory and the Hydrogen Atom (I). Rev. of Mod.Phys. 38. (1966).

[5] A.J. van Zanten, Representatietheorie I. TH Delft. (1985).

[6] A.J. van Zanten, Representatietheorie II. Th Delft. (1985).

[7] V. Fock, Zur Theorie des Wasserstoffatoms. Zeitschrift fur Physik 98. (1935).

[8] V. Bargmann, Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeitvon V. Fock. Zeitschrift fur Physik 99. (1936).

[9] H. Goldstein, Prehistory of the ”Runge-Lenz” vector. Am. J. Phys. 43. (1975).

[10] W.R. Hamilton, The hodograph, or a new method of expressing in symbolical languagethe Newtonian law of attraction (edited by D. R. Wilkins (2000)), Proceedings of theRoyal Irish Acadamy 3. (1847).

[11] N.W. Ewans, Superintegrability in classical mechanics. Phys. Rev. A 41. (1989).

Illustrations by R.C. Stone.

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8 Appendix A: Representations of SU(2)

The explicit form of the representations of the generators SU(2) are the following. As said

before the representations D(l)i (Ji) are defined by a whole or half valued number l. In the

matrices below the definition ak =√j(j + 1)− k(k − 1) is used.

D(l)i (J1) =

0 12aj 0 ... 0 0 0

12aj 0 1

2aj−1 ... 0 0 00 1

2aj−1 0 ... 0 0 0...

......

......

......

0 0 0 ... 12a−j+2 0 1

2a−j+1

0 0 0 ... 0 12a−j+1 0

(8.1)

D(l)i (J2) =

0 − i2aj 0 ... 0 0 0

− i2aj 0 − i

2aj−1 ... 0 0 00 − i

2aj−1 0 ... 0 0 0...

......

......

......

0 0 0 ... − i2a−j+2 0 − i

2a−j+1

0 0 0 ... 0 − i2a−j+1 0

(8.2)

D(l)i (J3) =

−ij 0 0 ... 0 0 00 −i(j − 1) 0 ... 0 0 00 0 −(j − 2) ... 0 0 0...

......

......

......

0 0 0 ... 0 i(j − 1) 00 0 0 ... 0 0 ij

(8.3)

As a reminder: the representations of the group elements X(α) of SU(2) have then thefollowing form.

D(X(α)) = exp(iαi1

2D

(l)i (Ji)) (8.4)

23