Supersymmetry for the Hydrogen Atom

Transcription

1 Supesymmety fo the Hydogen Atom Victo Östesjö Faculty of Technology and Science FYGC05, Examensabete fysik kandidatnivå Macus Beg Jügen Fuchs Mas 015

2 Faculty of Technology and Science Victo Östesjö Supesymmety fo the Hydogen Atom Physics Degee Poject of 15 ECTS, Bachelo Level Date/Tem: Supeviso: Examine: VT015 Macus Beg Jügen Fuchs Kalstads univesitet Kalstad Tfn Fax

3 Abstact In this thesis it will be shown that the hydogen atom has a SU SU symmety geneated by the quantum mechanical angula momentum and Runge-Lenz vecto opeatos. Additionally, the hydogenic atom will be studied with supesymmetic methods to identify a supesymmety that elates diffeent such systems. This thesis is intended to pesent the mateial in a manne accessible to people without backgound in Lie goups and supesymmety, as well as fill in some calculations between steps that ae not spelt out in the litteatue.

4 Contents 1 Intoduction 4 The Hydogen Atom 6.1 Angula Momentum Runge-Lenz Vecto Supesymmety Supesymmetic Famewok Hydogen Atom Conclusions 0 5 Appendix A Symmety Goups Appendix B 3 3

5 1 Intoduction In this thesis, we look at symmety goups that can be geneated with constants of motion fo the hydogen atom. Fist we look at the moe well known case of foming a SU using the components of the quantum mechanical angula momentum as geneatos, as pe efeence [1], which is the main efeence fo section. Afte that we will use the Runge-Lenz vecto, known to be a conseved quantity fo some otating systems, paticulay the 1 -potential, to fom a lage symmety goup. The Runge-Lenz vecto is also commonly known as the Laplace- Runge-Lenz vecto, though none of the thee wee fist to fom it. This lage goup is the SU SU Lie goup, which can be shown using step opeatos analogously as fo angula momentum. The point of this is that Lie goups ae known stuctues, to which known methods apply. The way we fom the SU SU makes it look like the goup of infinitesimal otations in fou dimensions, though the units ae, of couse, not the same. Both of these symmeties coespond to the degeneacy of the enegy levels of the hydogen atom. They ae elated via the conseved quantities. This is not new knowledge; fo example, is was discussed by Pauli aleady back in 196 [8]. Futhemoe, we will use the notion of supesymmety to look at some additional symmety of hydogenic atoms. Fo this, we will equie some moe geneal infomation on how to apply supesymmety to quantum mechanical systems. The specific appoach we will take to applying supesymmety in section 3 comes fom efeence [3], but the basic famewok existed ealie, see e.g. efeence [9]. It has since been efeed to in a numbe of othe woks, such as efeence [5]. Futhe eading can also be done in efeence [6] Supesymmety oiginally comes fom theoetical high-enegy physics. Thee, Q is an opeato that opeates on a boson, poducing a femion. This gives an exta paticle fo each paticle in the standad model and gives us the minimal supesymmetic standad model. This model does not contain any othe additional heavie paticles. As compaed to the standad model, the supesymmetic vesion alleviates the so-called hieachy poblem. Moe on this can be found e.g. in efeence [4]. The pupose of consideing these diffeent types of symmety togethe is that we ae able to daw paallels between thei stuctue and popeties. None of the consideations in this thesis ae tuly new. The thesis contains no mateial that cannot be found elsewhee, but gathes the elevant infomation in one place and pesents some calculations that have been skipped in the oiginal 4

6 liteatue. Compae fo example efeence [7], which discusses much the same mateial, as well as othe things, without going though the SU SU symmety extensively. In the entie thesis, ou model fo hydogenic atoms will be the most idealized vesion. As such we will always have one electon and neglect such things as coections fo spin, elativistic effects o finite size of the nucleus. Some backgound on the geneatos of SU SU can be found in Appendix A. 5

7 The Hydogen Atom Known conseved quantities of the hydogen atom include the angula momentum L and the Runge-Lenz vecto A. As such, both commute with the Hamiltonian H. We will not pove this, but it can be found elsewhee, as in efeence [1], section 9. We will geneally be woking with these conseved quantities as quantum mechanical opeatos, which ae used on state vectos. The state vectos ae usually descibed using wave functions in position o momentum space. In position space x is a multiplicative opeato and p is a diffeential opeato. In momentum space it is the othe way aound. Two quantum mechanical opeatos do not necessaily commute, but as long as two opeatos ae constucted fom x and p thei commutato can be detemined fom the canonical commutation elations fo x and p. In the following subsections it will be shown that the angula momentum opeatos geneates a SU and that the angula momentum opeatos and the Runge-Lenz vecto opeatos togethe geneate a SU SU..1 Angula Momentum We wish to show that L foms a goup with geneatos, such that [L i, L j ] = i ɛ ijk L k..1 k=1 In quantum mechanics, we define angula momentum as L = 1 x p p x. whee x and p ae the opeatos fo position and momentum, espectively. We need to define L in this way athe than the way we do classically, because x and p do not commute. The angula momentum is a conseved quantity, and accodingly satisfies [L, H] = 0. H fo this system is taken as As pe the ules fo the coss poduct, we have H = p m k..3 L 1 = x p 3 x 3 p.4 6

8 L = x 3 p 1 x 1 p 3.5 L 3 = x 1 p x p 1..6 These ae the same fomulas as in the classical case, except that we ae woking with opeatos. Fom now on, we will only give the calculation of a single sample equation fo a component, followed by the moe geneal summation fomula when elevant. We need to check the commutatos fo the components of L, but fist we detemine those fo L i with x i and p i. Recall that the canonical commutation elations ae and [x i, p j ] = i δ ij.7 fo i, j = 1,, 3. [x i, x j ] = [p i, p j ] = 0.8 The following commutatos ae easy, as L i does not contain x i o p i. [L i, x i ] = [L i, p i ] = 0..9 The othe commutatos we need to actually calculate. We will be using the elations and [A, BC] = B[A, C] + [A, B]C.10 [AB, C] = A[B, C] + [A, C]B..11 Using equations.7,.8,.10 and.11, we find that 7

9 [L 1, x ] = [x p 3 x 3 p, x ] = [x p 3, x ] [x 3 p, x ] = x [p 3, x ] + [x, x ]p 3 x 3 [p, x ] [x 3, x ]p.1 = i x 3 0 = i x 3. Calculating all the commutatos of L 1, L and L 3 with all the components of x and p analogously, we can see that we have and [L i, x j ] = i ɛ ijk x k.13 k=1 [L i, p j ] = i ɛ ijk p k..14 k=1 Having these commutatos, we can continue to calculate [L i, L j ]. Tivially, we have Fo the commutatos of diffeent components, we get [L i, L i ] = [L 1, L ] = [L 1, x 3 p 1 x 1 p 3 ] = [L 1, x 3 p 1 ] [L 1, x 1 p 3 ] = x 3 [L 1, p 1 ] + [L 1, x 3 ]p 1 x 1 [L 1, p 3 ] [L 1, x 1 ]p 3 = 0 i x p 1 + i x 1 p 0 = i x 1 p x p 1 = i L Similaity, we find that [L 3, L 1 ] = i L.17 and 8

10 [L, L 3 ] = i L Equations togethe with.15 give us.1. This is what we needed fo the components of L to be the geneatos of SU. As such, we can also define the step opeatos L ± : The step opeatos satisfy the commutation elations L ± = L 1 ± il..19 [L +, L ] = L 3.0 and [L 3, L ± ] = ± L ±..1. Runge-Lenz Vecto The Runge-Lenz Vecto is anothe quantity that is conseved fo the system we ae consideing. Classically it is given by A = p L mk x. whee = x, m is the mass of the electon and k is the constant in the electical foce F as pe F = kx 3..3 In quantum mechanics, the Runge-Lenz vecto is defined as A = 1 p L L p mk x.4 whee p, L, x, ae now quantum mechanical opeatos. This means that the Runge-Lenz vecto does not have the same fom in quantum mechanics as it has classically. The eason fo this is that the components of p and L do not commute fo diffeent indices of the vectos, in the same manne as x and p. A is a conseved quantity, that is [A, H] = 0. Explicitly, we get A in components as 9

11 A 1 = 1 p L 3 p 3 L L p 3 L 3 p mk x 1 = 1 p L 3 p 3 L L p 3 + L 3 p mk x 1 = 1 p L 3 p 3 L p 3 L i p 1 + p L 3 + i p 1 mk x 1 = 1 p L 3 p 3 L p 3 L + i p 1 + p L 3 + i p 1 mk x 1 = p L 3 p 3 L + i p 1 mk x 1.5 and analogously fo A and A 3. Note that we have used equation.14. We can summaize the components of A as A i = j,k=1 ɛ ijk p j L k + i p i mk x i..6 We need to calculate [A i, A j ] and [L i, A j ]. Howeve, to do so we need to find a numbe of othe commutatos fist. We need the following elations, which ae equations..3 in efeence []. F x [p i, F x] = i,.7 x i [x i, F x] = 0..8 Equations.7 and.8 ae valid fo functions F x with convegent Taylo seies about zeo. Fo example, we can easily check that it holds fo F x = x.9 and F x = x..30 This gives us 10

12 [ p 1, x ] 1 = i x 1 x 1 x 1 + x + x 3 x + x 3 = i x 1 + x + x = i x + x 3 3, while [ p 1, x ] = i x 1 x 1 x = i x 1 + x + x 3 = i x 1x 3. x x 1 + x + x.3 By the same methods, we can detemine all 9 commutatos of the type [p i, x i ]. The othes ae omitted to save space. Using the ealie commutatos, we can find [L 1, x 1 ] = [x p 3 x 3 p, x 1 ] = [x p 3, x 1 ] [x 3p, x 1 ] = x [p 3, x 1 ] + [x, x 1 ]p 3 x 3 [p, x 1 ] [x 3, x 1 ]p = x i x 1x x 3 3 i x 1x 0 3 = In the same manne, we get all the othe commutatos. The esults can be summaized as fo i, j = 1,, 3. [L i, x j ] = k=1 i ɛ ijk x k At this point, we can handle [L i, A j ]. We get.34 11

13 [L 1, A 1 ] = [L 1, p L 3 p 3 L + i p 1 mk x 1 ] = [L 1, p L 3 ] [L 1, p 3 L ] + i [L 1, p 1 ] mk[l 1, x 1 ] = p [L 1, L 3 ] + [L 1, p ]L 3 p 3 [L 1, L ] [L 1, p 3 ]L = i p L + i p 3 L 3 i p 3 L 3 + i p L = 0. By analogous calculations, we get [L i, A j ] = fo i, j = 1,, 3. i ɛ ijk A k.36 k=1 Finally, we also calculate [A i, A j ]. Tivially, [A i, A i ] = Fo the othe commutatos, we will need [p i, A j ] and [A i, x j ]. We have [p 1, A 1 ] = [p 1, p L 3 ] [p 1, p 3 L ] + i [p 1, p 1 ] mk[p 1, x 1 ] = p [p 1, L 3 ] + [p 1, p ]L 3 p 3 [p 1, L ] [p 1, p 3 ]L + 0 mk[p 1, x 1 ] = i p p + 0 i p 3 p mk[p 1, x 1 ] = i p + p 3 3 mk[p 1, x 1 ].38 and [A 1, x ] = [p L 3 p 3 L + i p 1 mk x 1, x ] = p [L 3, x ] + [p, x ]L 3 p 3 [L, x ] [p 3, x ]L + i [p 1, x ] mk[x 1, x ] = i p x 1 + i x 1 + x 3 L 3 0 i x x 3 3 L + i x 1x space. Analogous calculations fo the emaining components ae omitted to save Now we have all the commutatos we need to calculate [A i, A j ]. As the cal- 1

14 culation is a bit long, it is given in Appendix B as equations calculations fo the othe two commutato follow analogously. To give all thee commutatos at once, we can wite [A i, A j ] = i m The Hɛ ijk L k..40 We see that we need to define a nomalized K to get the pope commutatos fo a SU SU. We can define it as K = k=1 A mh.41 In ode fo this expession to make sense, we must conside H to be eplaced by an eigenvalue. This implies in paticula that we must estict ouselves to woking with the bound states, as these ae the ones with negative enegies. We may then teat H as a constant, as it commutes with all othe opeatos involved. We then get as well as [L i, K j ] = i ɛ ijk K k.4 k=1 [K i, K j ] = i ɛ ijk L k..43 k=1 To efomulate this as the commutation elations of a SU SU, we define the ladde opeatos L ± as L ± = 1 L ± K.44 Clealy, these ae also conseved quantities. By diect calculation fom equations.1,.41,.4 and.43, we have the following commutatos: [L + i, L+ i ] = i ɛ ijk L + k,.45 k=1 [L i, L i ] = i ɛ ijk L k,.46 k=1 [L + i, L i ] = Thus, we do indeed have two sepaate SU symmeties. 13

15 3 Supesymmety 3.1 Supesymmetic Famewok Thee is also anothe symmety that can be identified in hydogenic atoms. It is called a supesymmety, and will be discussed in this section. Following efeence [3] p , we conside two one-dimensional systems with potentials V ± = U 8 U 4 with U = Ux as some given function that chaacteizes the systems. 3.1 We summaize the two Hamiltonians fo these systems by a single Hamiltonian with a -matix stuctue, accoding to H = H 0 0 H + = p m V 0 0 V The cucial obsevation is now that this combined Hamiltonian can be ewitten in tems of opeatos Q and Q, defined as and whee and With these opeatos, we have and Q = p i U Q = p + i U σ + = σ = H + H σ σ = 1 QQ 3.7 = 1 QQ. 3.8 It follows in paticula that the Hamiltonian can be expessed as 14

16 H = 1 QQ + QQ. 3.9 As a shothand fo equations 3.7 and 3.8, we will use H + = 1 QQ 3.10 and H = 1 QQ The specta of H + and H ae elated to each othe by the opeato Q. We can see this in the following manne: Assuming ψ n to be an eigenstate of H +, with we get H + ψ n = E n ψ n, 3.1 H Qψ n = 1 QQQψ n = Q 1 QQψ n = QE n ψ n = E n Qψ n This means that H + and H have the same eigenvalues, except that the lowest enegy state will be missing fo H. Indeed, it is easily veified tbat ψ 0 = e 1 Ux satisfies Qψ 0 = Togethe with equations 3.10 and 3.11 we get that ψ 0 is an eigenstate of H + with eigenvalue 0. Howeve, Qψ 0 = 0 is not a non-tivial eigenstate of H. On the othe hand, povided that the function U behaves sufficiently well fo x ±, the state Qψ n is nomalizable and is thus a valid non-tivial eigenstate of H. Thus, H + has one eigenvalue that is not an eigenvalue fo H, while all the othes ae the same. Conceptually the so obtained supesymmety is athe diffeent than the Lie goup symmeties in section, but thee ae some computational paallels. Fo example, the opeatos σ ± in equations 3.5 and 3.6 ae simila to the step opeatos L ± fom equation

17 3. Hydogen Atom We will now analyse the spectum of hydogenic atoms with the help of this famewok. We will be following efeence [3], p We conside the adial pat of the Schödinge equation, i.e. 1 d dy 1 y ll χ y nl y = E n χ nl y 3.16 as pe efeence [3], equation 9. Hee, 1 ll+1 is the Coulumb attaction and y y the angula momentum baie. Also, we ae using the abbeviation is y = Z me Z is the atomic numbe of the hydogenic atom. This y is a dimensionless quantity, which causes the eigenvalues we will obtain to become dimensionless as well. Despite this, we will efe to such eigenvalues as those they othewise coespond to, such as in equation 3.18 below. The enegy eigenvalues ae E n = 1 n It tuns out that we can expess the Hamiltonian of this system in the supesymmetic fom fom equation 3.1: the coesponding function Uy is Uy = y l + 1 lny, 3.19 l + 1 which we can see by using U to epoduce the potential as V + as given in equation 3. up to an ielevant additive constant. Indeed, using equation 3.1, we get V + y = U U 8 4 = 1 1 l + 1 y = 1 1 l y + 1 l + l y = 1 1 l y l + 1 y ll + 1 y. l + 1 y 3.0 Having U, we also calculate 16

18 V y = U + U 8 4 = 1 1 l + 1 y = 1 + l + 1 y + l + 1 y 1 l + 1 y + l + l l + 1 y 1 l + 3l + = 1 l y + 1 = 1 1 l y + 1 y l + 1l + y. 3.1 It is now cucial to note that V is of the same fom as V +, and diffes fom it only by the eplacement and l l We can also calculate Q and Q. As pe equations 3.3 and 3.4, they become Q = Q = p 1 l σ l + 1 y p + 1 l + 1 σ. 3.4 l + 1 y Finding the eigenvalues of the coesponding Hamiltonians is a little involved, but can be achieved by using the vaiable x = lny fo H +, which gives us a so-called Mose potential. Shifting the lowest enegy eigenvalue to zeo, we get a new potential Ṽ + x = 1 n ex e x n The Hamiltonian H + has, as pe efeence [3], the eigenvalues. 3.5 E n,l = 1 n + ln l 1, 3.6 with 0 l n 1. Fo this potential, we have Using this, we find that Ũx = 1 n ex + n x

19 Ṽ x = 1 n ex 1 1 e x n n. 3.8 Accoding to the discussion in section 3.1, Ṽ has the same eigenvalues as Ṽ+, except that it is lacking the eigenvalue zeo. l = n 1. This is the eigenvalue fo which Refomulating this in tems of the vaiable y, equation 3.16 becomes H + yχ nl y = E n χ nl y 3.9 At this point, it is a good idea to check that inseting equation 3.19 into equation 3.14 does give us the state we expect. ψ 0 = e 1 y l+1 l+1 lny = e y l+1 e l+1 lny = y l+1 e y l Inseting l = 0, which is coect fo n = 1, and equation 3.17, we get ψ 0 = Z me e Z me Apat fom the constant, which is a matte of nomalization, this is the state we wee expecting. Continuing, we get H yχ nl y = 1 d dy = 1 n χ nly n y + 1 ll + 1 χ y nl y 3.3 By dividing the entie equation by 1 1 n and defining z = 1 1 y 3.33 n we get H χ nl z = 1 d dz 1 z + 1 = = 1 n 1 1 χ nlz n 1 n 1 χ nlz. ll + 1 χ z nl z

20 Since y contains Z, we can view the change y z as taking Z Z1 1/n. Now the only diffeence between equations 3.3 and 3.34 is that the fist applies to n while the second applies to n 1. As such, ou supesymmetic opeato Q implements the simultaneous change of n n and Z Z1 1/n Thee examples of this tansition, with the zeo eigenvalue missing fo H ae shown in figue 1. The figue makes it clea that these supesymmetical tansitions indeed elate hydogenic atoms of diffeent chage. He + H Li + He + Be + Li + Figue 1: Supesymmetic spectum of simultaneous tansition n n 1 and Z Z1 1/n fo a n = Z = b n = Z = 3 c n = Z = 4 The usefulness of the supesymmetical opeatos Q and Q we have constucted may be a bit difficult to evaluate. We can intepet the esult as having found a symmety between states fo hydogenic ions of diffeent atomic numbe Z, such that we have equal enegy eigenvalue fo n and Z and fo n 1 and Z1 1/n, with both ions having the same l. This symmety is pesent in this simple model of a hydogenic atom. Howeve, when compaing this theoetic pediction with measuements, one must be awae of the fact that a moe accuate teatment of a system like this should involve othe tems, such as elativistic petubations. In addition to this, the tansition epesented by Q cannot usually take place, as it involves a simultaneous change of state of the electon and the chage of the nucleus. Still, it is inteesting that we can find this additional symmety fo the model of the hydogenic atom. 19

21 4 Conclusions In this thesis, we wee to look ove the standad symmeties fo the hydogen atom and some futhe supesymmety that can be applied when studying the most simple model of a hydogenic atom. To do so, we had to go though seveal diffeent steps. We have calculated the commutatos fo L and A fo the hydogen atom, seeing that we can use them to fom the geneatos of a SU SU symmety. To do so, we had to intoduce the opeato K of equation.41 and detemine the commutatos [L i, L j ], [L i, K j ] and [K i, K j ] fo i, j = 1,, 3. We did this in smalle steps, foming othe commutatos, which we then used to keep the calculations elatively bief. We have also applied supesymmety to the hydogenic atom, finding a simila symmety fo simultaneous tansitions of n n 1 and Z Z1 1/n. This equied us to fist eview the supesymmetic famewok. This involved using Hamiltonians H ± with potentials V ± fomed fom the deivatives of a chaacteistic function U of the system. We then applied this to the simplest model of a hydogenic atom, finding V ±, Q ± and H ±. The esulting supesymmetic elations wee intepeted as a symmety between diffeent hydogenic ions. Both of the symmeties we have consideed ae boken if we apply coections to the Hamiltonians involved, but the esults ae still impotant as long as the coections ae sufficiently small. It is woth noting that the standad quantum mechanical symmety fom section shaes some popeties with the supesymmety fom section 3. Fo example, they both involve step opeatos. We conclude that they both ae symmeties that apply to the hydogenic atom, even if they ae of athe diffeent types. 0

22 5 Appendix A This section is devoted to showing how the goup SO3 is defined with the skewsymmetic epsilon tenso, as well as its commutation elations. The skew-symmetic epsilon tenso ɛ ijk is defined as taking the value of 1 fo even pemutations of i, j, k and 1 fo odd pemutations. It is othewise equal to zeo. This makes it useful fo some summation fomulas, such as the coss poduct. Fo the coss-pocuct C = A B, the components C i can be expessed though the components of A and B as C i = ɛ ijk A j B k Symmety Goups j,k=1 Descibing a goup using geneatos is often useful. In ou case, the actual Lie goups have an infinite numbe of elements, but have a small, finite, numbe of geneatos. The geneatos fom a Lie algeba. Often, we can descibe an element E of a Lie goup with geneatos A 1 though A n as n E = expi b k A k. 5. We will only be using the goup SU and the SU SU in this thesis, and fo these two goups equation 5. holds. As pe efeence [1] SU is a Lie goup with geneatos A 1, A and A 3 such that [A i, A j ] = ɛ ijk A k. 5.3 k=1 j,k=1 In quantum mechanics, it is conventional to escale the opeatos such that thee is also a facto of i. The SU SU is simila, consisting of two diffeent SU. It has thee moe geneatos B 1, B and B 3, such that [B i, B j ] = ɛ ijk A k, 5.4 j,k=1 [A i, B j ] = ɛ ijk B k. 5.5 j,k=1 1

23 These ae not manifestly the commutatos fo two diffeent SU symmeties, but instead fo the goup SO4 of otations in fou dimensions, that has 6 geneatos. This SO4 is howeve the same as SU SU, which can be made explicit by efomulating equations by foming step opeatos. This is done as pe equation.44.

24 6 Appendix B Equations has been moved to this appendix to save space in section.. In the calculation, most of the esults fo commutatos fom section ae used, as well as the ules fo woking with commutatos. Hee, only the calculation of [A 1, A ] is given, but the commutatos with othe components of the Runge-Lenz vecto can be calculated analogously. Fist, we use the definition of A 1 and the ules fo addition and multiplication with constant in a commutato. [A 1, A ] = [ ] p L 3 p 3 L + i p 1 mk x 1, A = p [L 3, A ] + [p, A ]L3 p 3 [L, A ] [p 3, A ]L +i [p 1, A ] + mk [ 6.1 ] A, x 1. Then, we inset peviously known commutatos fom section. [A 1, A ] = i p A 1 + i p 1 + p 3 + mki x 1 +x 3 3 L 3 i p p 3 mki x x 3 3 L + i i p 1 p mki x 1x x +mki 1 x 3 L x +x 3 x L 3 3 p 1 + i x 1x Now, we ty to put multiplicative constants outside the expessions containing opeatos, and goup simila expessions togethe. [A 1, A ] = i p A 1 p 1 + p 3L 3 p p 3 L x + 1 +x 3 L x 1x 3 L 3 + x 1x 3 L x +x 3 x L 3 3 p 1 +i p 1 p + mk x 1x 3 + x 1x At this point, we inset the definition of A 1, and continue simplifying the expession. [A 1, A ] = i p p L 3 p 3 L + p 3 L + i p 1 mk x 1 p 1 + p L 3 x +mk 1 +x +x 3 L 3 3 x 3 x p At this point, seveal tems cancel each othe out. We keep simplifying. [A 1, A ] = i p x L 3 i p 1 p + mk p 1 x p 1 + L i p 1 p = i p L 3 + mk 1L mkp x 1 p 1 x 1. 3

25 Finally, we identify L 3 and finish simplifying the expession. [A 1, A ] = i p L 3 + mk 1L 3 + mk 1L 3 = i m L 3. p m k 6.6 Refeences [1] G. t Hooft, B.Q.P.J. de Wit and M.J.G. Veltman, Lie Goups In Physics, Utecht Univesity, 5/06/07 [] J. J. Sakuai and J. Napolitano, Moden Quantum Mechanics, Peason, nd Edition 011 [3] R. W. Haymake and A. R. P. Rau, Supesymmety in quantum mechanics, Am.J.Phys , p [4] S. P. Matin, A Supesymmety Pime, axiv:hep-ph/ , 1997 [5] F. Coope, A. Khae and U. Sukhatme, Supesymmety and quantum mechanics, Physics Repots , p [6] R. Tangeman, J. Tjon, Exact supesymmety in the non-elativistic hydogen atom, Physical Review A , p [7] A. Valence, T. Mogan, H. Begeon, Eigensolution of the Coulumb Hamiltonian via Supesymmety, Am.J.Phys , p [8] W. Pauli, Übe das Wassestoffspektum vom Standpunkt de neuen Quantenmechanik [On the hydogen spectum fom the standpoint of the new quantum mechanics], Z. Phys, , p [9] E. Witten, Dynamical beaking of supesymmety, Nucl. Phys , p