Relativistic Schrödinger Wave Equation for Hydrogen Atom Using Factorization Method

Open Journal of Microphysics, 2013, 3, 1-7 http://dx.doi.org/10.4236/ojm.2013.31001 Published Online February 2013 (http://www.scirp.org/journal/ojm)

Mohammad Reza Pahlavani, Hossein Rahbar, Mohsen Ghezelbash Department of Physics, Faculty of sciences, University of Mazandaran, Babolsar, Iran Email: [email protected], [email protected], [email protected], mailto:[email protected]

Received December 15, 2012; revised January 17, 2013; accepted January 25, 2013

ABSTRACT In this investigation a simple method developed by introducing spin to Schrödinger equation to study the relativistic hydrogen atom. By separating Schrödinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrödinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.

Keywords: Relativistic Schrödinger Wave Equation; Factorization Method; Ladder Operators; Supersymmetry; Spin

1. Introduction lead us to have ladder operators which are represent the generators of respective algebra for relativistic particle. Inserting spin to Schrödinger equation as a relativistic cor- rection, in the base of Pauli exclusion principle with two 2. Relativistic Schrödinger Wave Equation spinors is a context of perturbation theories [1]. Several other relativistic wave equations dealing with various The general form of Schrödinger equation consist of an- aspects of spin have been put forth to address large vari- gular momentum and spin can be define as [22], ety of problems. Klein-Gordon equation for spin-(0) par- 22 2 24 2 ticles [2,3], wave equations for describing relativistic crmcEVrΔΨt  0  Ψ, (1) 1 dynamics of a system of two interacting spin-  parti- where Δ is given by, 2 t cles [4-6], Breit equation for two electrons [7] also called 2211  2 r  t 22r LS   (2) two body Dirac equation, generalized Bruit equation for r rr 2 two fermions [8], Duffin-Kemmer-Petiau (DKP) theory [9-11], of scalar and vector field for describing interact- where L is the orbital angular momentum operator and tion of relativistic spin-(0) and spin-(1) bosons [12-20] simply introduce as, are examples of this topic. Some authors include Poin- 2 2211  care invariant theory of classical spinning particles [21], L  sin  22 (3) sin  sin   quantum mechanical embedding of spinning particles  and spin dependent gauge transformation between clas- and S is the operator associated to the spin. The parame- sical and quantum mechanics. All these theories fall un- ter  read as, der the class of perturbation theories and no account for 1 inserting spin into the dynamics of motion. The paper or- 24 22 2mc0  E ganized as follow: We introduce the relativistic Schrö-   (4) dinger equation in Section 2. Then, in Section 3, we use c mathematical aspect and obtain the exact solution for this Substituting Equation (2) in Equation (1) leads us to ob- wave equation. This approach improved using the so tain following expression for Schrödinger equation, called supersymmetric quantum mechanics in framework 11   2  of shape invariance in Section 4. Also we study given rmcEV224   r 222 0  problem using the factorization method. These results rcrrr2 

1 2  1d2 d 1 1  21 2 2 11 S  2   isin  dd242 222 2    rrsin sin r  2 jj1  22 R 0  r  (5) In order to obtain the exact solution for the equation The radial and angular parts are separated by apply- (10), we need to consider the radial wave function R  ing  rRrY  ,  in Equation (5). This substitu- as, tion leads us to derive an equation which separated in  totwo parts. One part related to coordinate r and other part RF e 2 (11) depends on coordinate  , , so both parts had to equal a constant, say Γ. Thus Equation (5) gives us a radial dif- Here F   is a polynomial of finite order in  af- ferential equation, ter substituting of this definition in Equation (10), result the following second order differential equation for 1   Γ  1 2  rm224c EVF   , 222 20  rrcrr 2 r 2   Rr 0 FF1    (6) 1 (12)   2 and an angular differential equation, 2 jj1  2 F  0 1    2   11YY,,  2  isin   22 sin sin  Where  is,  (7) 21 S 2  (13)  2  YY,  Γ  ,0    Now we have to modify this equation with the associ- In the following section we will attempt to obtain so- ated Laguerre differential equation. For the real parame- lutions of these equations for coulomb potential for spin- ter   1and   1 this differential equation in the in- 1 terval   0, is defined as follows [23],  electrons as relativistic simple hydrogen atom.   2 ,, LLnm,,  1 nm (14) 3. The Relativistic Hydrogen Atom mmm1 , nLnm,   0. In order to solve the radial part of the Relativistic 222 Schrödinger wave equation, we define new parameters where indices n and m are non-negative integers with as, 0  mn. So it is required to define function F   22 as, 22mc00 E mc E , (8) 12cc Ffg    () (15) 2 By substituting this definition in Equation (12) we and  12. Now we substitute Equation (8) in Equa- tion (6), so the radial part of Schrödinger equation is re- have, written as, g  ff22       2   1 22Γ 21 1 g  22r    222   rrrr 24 r r2 r 2   ggj  1 j1 2 Rr 0  (16) gg 2  Ke2 f  0 Where   , K is an electrical constant and  is a c real constant which is identified by Γ2 with eigenva- By modifying this equation with the associated La- lues jj1 2 . Now, by introducing new variable, guerre differential equation (14), in the first step we con-   Kr and substituting it in Equation (9) one obtain, clude the function f   is corresponding to the asso-

, Table 1. Radial wave function and energy spectrum. Here n ciated Laguerre function Lnm,   . The Rodrigues rep- 1 resentation for associated Laguerre differential equation and m are the quantum numbers and C  . we is given by,  ρ2edρ ρ nm  ,anm, , d nassume αβ 12, . Lnm,  m e (17)   d 2    e n m Rnm,   Enm,

In which a ,  is the normalization coefficient 2 12 nm,  2 2 10 2eC   mc 1  0  and is also obtained by, 3

1 12  m 1  2 2 2  mc 12 m  11  2eC 0  anm, ,1 (18) nm11 n  2 12  2 2 20 22C   1e mc 1   0  In addition to the Equation (17) for function f   , 7 2 12 1  this modification leads us to obtain the function g    2 2 21 2   mc 1  221eC 0  as, 5 12 11 2  2 2 22 22 2eC   mc 1  gC e (19) 0   3

2 12 Here C is the normalization coefficient and the para- 3 2 2 30 C 8244e2  mc 1  0  meter  is evaluated by, 6 11

2 12 1   11m  2 2 31 2 2   mc 1   n (20) 2eC  2 4 1e 0  22 2 9 2 12  2 2 According to parameters ,,12 and  , one can 32 22C 1  e mc 1   0  7 derive the energy levels Enm, in Equation (20) for bound 2 12 states as, 1   2 2 2   33 mc0 1  1 2eC   5 2  12 2 4 32   2 2 2 40 C 693e mc0 1   3 15 2  Emcnm,01  2 2 12 1 11  2  m 2 2 32  2  n  41 C418183e   mc0 1   3 13 22 2  2 12 2 2  2 2 (21) 42 C 263e mc 1   0  3 11 Finally the corresponding wave functions R  for 12 1 2 2  2 2 these bound states, according to the Equations (11), (15) 43 C2 23e mc 1   0  and (19) can be written as, 3 9 2 12 1 4   2 2   44 Ce mc 1  22, 0  RCnm,,  e Lnm, (22) 6 7 In order to represent an exact view of obtained results, the radial wave function Rnm,   and values of energy spectrum Enm, are showed in Table 1 for the different quantum numbers n and m. Also, in order to show the effect of spin on the energy spectrum, the obtained energy spectrum of radial part from solving relativistic Schrödinger equation are illus- trated in Figure 1 and Figure 2 as function of quantum numbers n and m. We may also derive the eigen function of the angular part of the relativistic Schrödinger equation similar to the solution of the radial part. The angular Equation (7) can be further separated by substituting, Y , ΘΦ . Figure 1. Energy spectrum for n  1, ,5 .

In order to obtain the wave function Θx , we modify this equation with the associated Jacobi differential equation. Here for the real parameters ,  1, this  , equation corresponding Pnm, in the interval x  1,1 is introduced as [24],

2, , 12x  Pxnm,,   xPnm x mm   x nn 1  2 (29) 1 x , Pxnm,  0 where, the indices n and m are non-negative integers with 0  mn and for m = 0 Equation (29) converts to the differential equation corresponding to the Jacobi Po- Figure 2. Energy spectrum for n  6, ,10 . lynomials. After the modification of Equations (28) and (29), in We take Φ as follow, first step one can obtain wx  as, ddxx  x 1 im   e (23) wx Ce.2211x2x2 (30) 2π We conclude that the function vx in Equation (27) So for azimuthal part we have, , is corresponding to the associated Jacobi function Pxnm,   1ddΘ as solution of the Equation (29) which has the following sin  sin d d  Rodriguez representation, (24) 2 2 S  anm, , Px,  22LS ΓΘ 2 0 nm,  mm sin    11xx22  (31) where  equal to square of quantum number m. The nm d nn L·S term in Equation (24) point to the spin-orbit interact- 11xx tion energy. This term simply consider as a perturbation dx to the final solution, so we ignore it. Here we define a Here anm, ,   is the normalization coefficient S 2 which for nm is given by, new constant parameter,   2  Γ, so the Equation  anm, ,  (24) is rewritten by following expression, 11m nm   C ,     2n nm111 n n Θ cot Θ  2 0 (25)  sin  (32) By introducing a new variable x  cos , one can re- In which C ,  is an arbitrary real constant inde- write Equation (25) as,   pendent of n and m. Therefore we have following rela-  2    tion for Θ x, 12xxxx    2  x0 (26) 1 x ddxxx   2211xx22, Now we consider wave function as, Θnm,,x  CPe. nm x (33) x vxwx   (27) Finally, according to the Equations (23) and (29) the angular wave function is given by, Thus the Equation (26) will become, ddxxx   C 22 wx YP,ee i,m2211xxcos 12xvx2212 x xvx  nm,, nm    2π wx (28) (34) wx wx    12xx2   vx 0Parameter C is the normalization coefficient and easily  2  wx wx 1 x evaluated using normalization condition.

4. Factorization Method to Wave Equations trum, the relations among the energy spectra of the various Hamiltonians, derivation of algebraic solutions and In recent years supersymmetry and shape invariance in etc. In previous section, we determine the radial and an- quantum mechanics have undergone a spectacular deve- gular wave functions of relativistic Hydrogen atom. In lopment. The concepts of shape invariance are developed this section we apply the factorization method to radial in several branches of physics, such as atomi, nuclear and and angular parts of Relativistic Schrödinger wave equa- mathematical physics as well as quantum optics. Super- tion. In the first step, we consider the radial part of wave symmetry in quantum mechanics is based upon the fac- equation obtained in previous section. As mentioned in torization method in the framework of shape invariance. refs [43,44], one can factorize the associated Laguerre Factorization method goes back to Darboux, but was differential equation as the following shape invariance developed by Schrödinger in order to apply it to quantum equations with respect to the parameters n and m, mechanics [25,26]. There is a discussion of the factoriza- ,, tion method in the review article of Infield and Hull [27], Anm,,,rA nm rL nm r B nm ,, L nm  r where they have been shown a large variety of the se- ,, A rA  rL r B L() r (35) cond-order differential equation with different boundary nm,, nm n 1, m nm ,1 n, m conditions set in six different types of factorizations. If a where quantum mechanics problem admit context of super- Bnmn    (36) symmetry, one able to factorize the Hamiltonian of quan- nm, tum states in terms of a multiplication of the first-order and its associated differential operators are, differential operators as the shape invariance equation. In d m Arr   rn this approach, the Hamiltonians decomposed once in nm, d2r successive multiplication of lowering and raising opera- d m tors, in such a way that the corresponding quantum states Ar   r n (37) of successive levels are their Eigen states of them. These nm, d2r Hamiltonian are called partner and supersymmetric of We note that the shape invariance Equations (35) can each other. also be written as the lowering and raising relations, In fact, three separate subject, i.e. the factorization me-  ,, thod, the supersymmetry in the quantum mechanics and Anm,1,,,rrL n m  r B nm L nm r (38) the shape invariance, nowadays converged at a point. ArLr ,,  BL  r The idea of supersymmetry in the context of quantum nm,, nm nm ,1, n m mechanics was first study by Nicolai and Witten and la- Therefore, we obtain the raising and lowering opera- ter by Cooper and Freedman et al. [28-30]. Recently, Gen- tors for the radial part of the relativistic Hydrogen atom. denshtein put forward the concept of shape invariance in Next we apply factorization method to the angular part of the context of the supersymmetric quantum mechanics the Relativistic Schrödinger wave equation. The shape [31]. As yet, according to the factorization method, many invariance equations of the associated Jacobi differential studies on the one-dimensional shape invariance poten- equation respect to the parameters n and m given by tial in the framework of the supersymmetric quantum [45,46], mechanics have been carried out [32-38]. One of the most ArA  rP,,  r C P  r well-known one-dimension quantum mechanical systems nm,,, nm nm nm ,, nm (39) ,, is the quantum harmonic oscillator [39]. There are other Anm,,rA nm  rP n 1,,1, m  r C nm P n m() r solvable systems with, say, a Morse potential, Scarf po- where tential, Eckart potential, and many others [40-42]. These solvable potentials have established a tight connection 4nm n n nm Cnm,  (40) with the pioneering work of Infield and Hull on factori- 2n 2 zation and algebraic solution of bound state problems. It Therefore raising and lowering operators can be evalu- should be noted that most of the solvable potentials are ated as, shape invariance. On this basis, the one dimension partner Hamiltonian is connected by supersymmetry trans-  2 d nm Axnm, 1   nx  formations. The spectra of two partner Hamiltonians are d2xn identical, expect for the ground state. Supersymmetry nm played important role in analyzing of the quantum me-  2 d  Axnxnm, 1    chanical systems, since it can consider remarkable prop- d2xn erties including degeneracy structure of the energy spec- (41)

Also in the case of the shape invariance respect to m Jacobi differential equation, give us the exact angular we have, wave functions. The resulting energy levels of Hydrogen ,, atom in this theory are exactly match the results obtained AmxA m  xP nm,,,  x D nm P nm  x (42) using relativistic Dirac equation. ,, AmxA m  xP nm,1  x D nm , P nm ,1  x where REFERENCES

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