Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom

Transcription

1 Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom Miguel Lorente 1 Departamento de Física, Universidad de Oviedo, Oviedo, Spain The Kravchuk and Meixner polynomials of discrete variable are introduced for the discrete models of the harmonic oscillator and hydrogen atom. Starting from Rodrigues formula we construct raising and lowering operators, commutation and anticommutation relations. The physical properties of discrete models are figured out through the equivalence with the continuous models obtained by limit process. PACS: 0.0.+b, Bz, Fd Key words: orthogonal polynomials; difference equation; raising and lowering operators; quantum oscillator; hydrogen atom. 1 Introduction The method of finite difference is becoming more powerful in physics for different reasons: difference equations are more suitable to computational physics and numerical calculations; lattice gauge theories explore the physical properties of the discrete models before the limit is taken; some modern theories have proposed physical models with discrete space and time [1]. In recent papers [], [3] we have presented the mathematical properties of hypergeometric functions of continuous and discrete variable. We have worked out general formulas for the differential/difference equation, recurrence relations, raising and lowering operators, commutation and anticommutation relations. The starting point is the general properties of classical orthogonal polynomials of continuous and discrete variable [4] of hypergeometric type, in 1 mlp@pinon.ccu.uniovi.es Preprint submitted to Elsevier Preprint 16 January 004

2 particular the Rodrigues formula from which the raising and lowering operators are derived. Similar results were obtained with more sofisticated method using the factorization of the hamiltonian [6] [7]. In those papers the term oscillator is used for the hamiltonian that has equally spaced eigenvalues. In this paper we study the physical properties of two simple examples, the harmonic oscillator and the hydrogen atom on the lattice. We make the Ansatz that the discrete model has the same elements hamiltonian, eigenvalues, eigenvectors, expectation values, dispersion relations as the continuous one, provided the difference equation, the raising and lowering operator the commutation and anticommutation relations become in the limit the equivalent elements in the continuous case. To this scheme we can add the evolution of the fields with discrete time, using some difference equation that replace Heisenberg equation. [8] Some applications of discrete models have been presented elsewhere. Bijker et al. [9] have apply the SO3 algebra of the Wigner functions to theonedimensional anharmonic Morse oscillator. Bank and Ismail have apply Laguerre functions to the attractiv Coulomb potential [10] The quantum harmonic oscillator of discrete variable We start from the orthogonal polynomials of a discrete variable, the Kravchuk polynomials K p n x and the corresponding normalized Kravchuk functions [1] K n p x =d 1 n ρx k p n x, 1 where d n = N! n!n n! pqn is a normalization constant, ρx = N!px q N x x!n x! pqn is the weight function, with p>0, q > 0, p+ q =1, x =0, 1,...N +1. The Kravchuk functions satisfy the orthonormality condition N x=0 K n p xkp n x =δ nn, and the following difference and recurrence equations []: pqn xx +1K n p x +1 + pqn x +1xK n p x 1 + [xp q Np+ n] K n p x =0, 3

3 pqn nx +1K n+1x p + pqn n +1nK n 1x+[nq p p+np x] K n p x =0, 4 From the properties of the Kravchuk polynomials we can construct raising and lowering operators for the Kravchuk functions [] L + x, nk n p x =pqx + n NKp n x + pqn x +1xK n p x 1 = pqn nn +1K n+1x, p 5 L x, nk n p x =pqx + n NKp n x + pqn xx +1K n p x +1= pqn n +1nK n 1x. p 6 The raising operator satisfies K n p x = pq n N n! N!n! where K p N!p 0 x = x q N x x!n x! n 1 k=0 L + x, n 1 kk p 0 x, is the solution of the difference equation L x, 0K p 0 x =0. It can be proved that the raising and lowering operators L + x, n andl x, n are mutually adjoint with respect to the scalar product. If we define the difference equation 3 as the operator equation Hx, nk p n x = 0, we can factorize this equation as follows L + x, n 1L x, n =pqn n +1n + pqx + n 1 NHx, n, 7 L x, n +1L + x, n =pqn nn +1+pqx + n +1 NHx, n. 8 Now we make connection between the Kravchuk function and the Wigner functions that appear in the generalized spherical functions [4] 1 m m d j mm β =Kp n x, 9 where j = N/,m= j n, m = j x, p =sin β/,q=cos β/. Then formulas 3 to 8 can be written down in terms of the Wigner functions, namely, 3

4 1 sin β j + m j m +1d j m,m 1 β + 1 j sin β m j + m +1d j m,m +1 β+m m cos βd j m,m β =0, 3a 1 sin β j + mj m +1d j m 1,m β + 1 j sin β mj + m +1d j m+1,m β m m cos βd j m,m β =0, L + m,m d j m,m β=sin β m + m d j m,m β + 1 sin β j m j + m +1d j m,m +1 β = 1 sin β j + mj m +1d j m 1,m β, L m,m d j m,m β=sin β m + m d j m,m β + 1 sin β j + m j m +1d j m,m 1 β = 1 sin β j mj + m +1d j m+1,m β. 4a 5a 6a Notice that 3a and 4a are equivalent if we interchange m m and take in account the general property of Wigner functions d j m,m β = 1m m d j m,mβ. 10 The same property is satisfied between 5a and 6a. Expresions 5a and 6a can be obtained directly from the properties of Wigner functions. In fact, it is known [4, formula ] that d m cos β dβ dj m,m β+m d j m,m sin β β = j mj + m +1d j m+1,m β, 11 d m cos β dβ dj m,m β+m d j m,m j sin β β = + mj m +1d j m 1,m β, 1 4

5 The last equation 11, after interchanging m m and using 9, can be transformed into d dβ dj m,m β m m cos β d j m,m j sin β β = + m j m +1d j m,m 1 β. 1a Combining 11 and 1a we obtain 6a and by similar method we obtain 5a In order to give a physical interpretation of the difference equation and raising and lowering operators for the Kravchuk functions we take the limit when N goes to infinity and the discrete variable x becomes continuous s. First of all, we take the limit of Kravchuk functions. We write K p n x = { n!n n! N!pq n }1 Npq n / { }1 N!p x q N x / n n! x!n x! Npq n / n!k p n x = n { }1 1 / { }1 / 1 e s Hn s =ψ n n s, 13 n! πnpq where the last braket becomes the weight for the Hermite functions and the functions ψ n s are the solution of the continuous harmonic oscillator up to the constant Npq 1 / 4. In order to get the continuous limit of 4 we multiply it by / Npq and substitute x = Np + Npq s ; after simplification we get 1 n N + 1 n 1 N n +1K p n+1x n K n 1x p p 1n s + K p Npq n x =0. In the limit N this equation becomes the familiar recurrence relation for the normalized Hermite functions n +1ψ n+1 s+ nψ n 1 s =sψ n s 14 Before we take the limit of 5 we redifine the raising operator, substracting from it one half equation 3, namely, 5

6 L + x, nk n p x = 1 { = [x Np+np q]k n p x 1 + } pqn xx +1K n p pqn x +1xK n p x 1 pqn nn +1K n+1x p 15 We divide this expression by h Npq Npq where h Npq =1.Aftersimplification we get x 1 L + x, nk p Npq n x = 1 [ ] np q s + K p Npq n x 1 h 1 1 = 1 n N = N p q Nq s 1+ Np s + 1 K n x +1 Np p Nq s + 1 q 1+ Nq Np s n +1K p n+1x { 1 s 1 { 1 = s d h 0 ds K n x 1 } h [ψ ns + h ψ n s h] } ψ n s = n +1ψ n s, 16 Similarly from 6 we get 1 L x, n K p x Npq n N h 0 { 1 s + d } ψ n s = nψ n s. 17 ds Therefore the raising and lowering operators for the Kravchuk functions become, in the limit, creation and annihilation operators for the normalized Hermite functions. We still have an other connection between the raising and lowering operators of Wigner functions with the generators of the SO3 algebra. From 5a and 6a we define A + d j mm β 1 j + mj m +1 L + d j Npq mm β = d j m 1,m β, 18 j 6

7 A d j mm β 1 j mj + m +1 L d j Npq mm β = d j m+1,m β. 19 j Multiplying both expressions by the spherical harmonics Y jm, adding for m and using the property of Wigner functions Y jm = d j m,m Y jm, 0 m we obtain j + mj m +1 A + Y jm = j Y j,m 1 = 1 j J Y jm 1 A Y jm = j mj + m +1 j Y j,m+1 = 1 j J + Y jm where J +,J are the generators of SO3 algebra. For the commutation relations of these operators we have AA + A + AY jm = m j Y jm = 1 j J zy jm = 1 n Y jm, 3 j we substitute 0 in this expression and then take the limit: [ AA + ] d j mm β = 1 n j d j [ mm β ] a, a + ψ n s. j For the anticommutation relation we have from 7 and 8: AA + + A + A Y jm = 1 j jj +1 m Y jm = 1 j J J z Yjm. 4 Again substituting 0 and taking the limit AA + + A + A d j mm β { } = n +1 n d j mm β aa + + a + a ψ n s =n +1ψ n s j j This correspondence suggests that the operator algebra for the quantum harmonic oscillator on the lattice is expanded by the generators of the SO3 groups. The commutation relations [J +,J ]=J z play the rol of the Heisenberg algebra, and the anticommutation relation multiply by ω/ play the rol 7

8 of the Hamiltonian. In order to complete the picture we define the position and momemtum operators on the lattice as follows: X : i 1 / A + A + d j mm Mω β =i Mω 1 / m m cos β d j j sin β mm β 1 / A A + M ω d j mm β = M ω P : j + m j m +1 j 1 / j d j m,m 1 β m j + m +1 d j m,m j +1 β with dispersion with respect to the state K n p x X n = X n = P n = P n = M ω AA + + A + A Mω = n +1 n n Mω j AA + + A + A n = M ω from which the uncertainty relation follows: X n P n = n +1 n j n +1 n j The eigenvalues of the Hamilton operator on the lattice are connected with the index m = j n of the eigenvectors d j mm β. These eigenvalues are equally separated by ω but finite m = j,...+ j. The eigenvalues of the position operator on the lattice are connected with the index m = j x of d j mm β. These eigenvalues are equally separated by but finite Mω m = j,...+ j. Therefore the Planck constant plays a role with respect to the discrete space coordinate similar to the discrete energy eigenvalues. 3 Wave equation for the hidrogen atom with discrete variables Our model is based on the properties of generalized Laguerre polynomials as continuous limit of the Meixner polynomials of discrete variable. We start from the generalized Laguerre functions ψns α =d 1 n ρ 1 s L α ns 5 8

9 with α> 1, d n =Γn + α +1/n!, ρ 1s =s α+1 e s, that satisfy the orthonormality condition 0 ψ α ns ψ α n s s 1 ds = ρ nn 6 from the differencial equation and Rodrigues formula for the Laguerre polynomials [4] we deduce the following properties for the Laguerre functions 5 i differential equation [ λ ψn α s+ s 1 ] 4 α 1 s ii Recurrence relations ψns α =0, λ = n + 1 α n + α +1n +1ψn+1 α s n + α nψn 1 α s+n + α +1 s ψα n s =0. 8 iii Raising operator L + s, n ψns= α 1 n + α +1 s ψα ns s d ds ψα ns = n +1n + α +1ψn+1s α 9 iv Lowering operator L s, n ψn α s= 1 n + α +1 s ψα n s+s d ds ψα n s = = nn + α ψn 1 α s 30 from 9and 30 we get the factorization of 7 { d L s, n +1L + s, n =n +1n + α +1 s ds + λ s 1 } 4 α 1 s 31 { d L + s, n 1 L s, n =nn + α s ds + λ s 1 } 4 α 1 s 3 Notice that 7 corresponds to some self adjoint operator of Sturm-Lioville type. Also 9 and 30 are mutually adjoint operators with respect to the scalar product 6. 9

10 Similarly, in the discrete case, we defined the normalized Meixner functions M γ,µ n where m n γ,µ x are the Meixner polynomials, d n = x d 1 n ρ 1 x m n γ,µ x 33 n!γn + γ µ n 1 µ γ Γγ, ρ 1x = µx Γx + γ +1 Γx +1Γγ, and γ,µ are some constants 0 <µ<1, the orthonormality condition γ > 0. The functions 33 satisfy x=0 and the following properties: i Difference equation M n γ,µ x M γ,µ n x 1 µx + γ = ρ nn 34 µx + γx +1x + γ M n x +1 x + γ +1 + µx + γxm n x 1 [µx + γ+x n1 µ] M n x = 035 ii Recurrence relation µn + γn +1M n+1 x µn + γ 1nM n 1 x +µx + µn + µγ + n x M n x = 0 36 iii Raising operator L + x, n M n x= µx + γ + n M n x+ µx + γx M n x 1 = µn + γn +1M n+1 x 37 iii Lowering operator L x, n M n x= µx + γ + n M n x µx + γx +1x + γ + M n x +1 x + γ +1 = µn + γ 1n M n 1 x 38 A redefinition of 37 and 38 can be obtained substracting one half the difference equation 35 from both: 10

11 L + x, nm n x = 1 nµ + µγ + n1 µ+µ 1x M nx + 1 µx + γx +1x + γ µx + γxm n x 1 M n x +1 x + γ +1 37a L x, nm n x = 1 nµ + µγ + n1 µ+µ 1x M nx + 1 µx + γx +1x + γ M n x +1 µx + γxm n x 1 x + γ +1 38a It can be proved that the difference equation 35corresponds to some selfadjoint operator of Sturm-Lioville type. Also the raising and lowering operators 37 and 38 are mutually adjoint with respect to the scalar product 34. The anticommutation relations for the raising and lowering operators 37a and 38a are 1 { L + x, n 1L x, n+l x, n +1L + x, n } = 1 µγ +µ +1n +µ 1x 4 µ x + γ x +1x + γ +1x + E + x + γ 1 x 4 x + γ + x + γx +1+ x + γ x x + γ 1 x 1 E + 1 µx + γx +1x + γ µ +1 M n x +1 x + γ +1 µx + γxm n x 1 39 The commutation relations read L + x, n 1 L x, n L x, n +1L + x, n = µn + γ 40 In order to make connection between the Meixner functions of discrete variable 33 and Laguerre functions of continuous variable 5 we substitute γ = α +1,µ=1 h, x = s and then take the limit h 0. h 11

12 The explicite expression of 33 in terms of α, h,ands reads M n γ,µ x = µn+1 n! Γn + γ = 1 hn+1 n! exp Γn + α +1 Γx + γ +1 µ x Γx +1 m n γ,µ x n! { } s 1 n ln1 h M h α+1 x α+1 n α+1 + O x n! where we have used the asymptotic expansion Γx + a 1 Γx + b = xa b 1+O, x x s n In the limit we obtain 5, namely Also M γ,µ n x h 0 x hx s n! Γn + α +1 e s s α+1 L α n s =ψα n s 36 h a h a h [ ] h 0 In order to make application to the hydrogen atom we take the reduced radial equation [5] d [ u ν dρ + ρ 1 ] ll +1 uρ = 0 4 ρ 41 where ρ 8M E r, ν = Ze µ E If we compair 41 with 7 we must have ρ s, ll+1 = α 1/4, hence, α =l +1, and ν = n + α+1 = n + l +1;sincel is integer, α and ν should be also integer. The energy eigenvalues are given by E νl = 1 M Ze 1 ν, ν =1,

13 For fixed ν we still have degeneracy for l =0, 1,...ν 1 The corresponding eignvectors are given by 5, which after the substituion α =l +1, and n = ν l 1, become ψ νl ρ = { ν l 1! ν + l! } 1 ρ l+1 e ρ L l+1 ν l 1 ρ 43 From the connection between the Meixner and Laguerre functions given above, we can make the ansatz of a discrete model for the hydrogen atom where the reduced radial equation is substituted by the difference equation 35 with γ = α +1=l +, n = ν l 1, x =0, 1,... The raising and lowering opertors 9, 30, respect to the radial quantum number n, are substituted by the raising and lowering operators 37 and 38 with respect to n or ν = n + l +1. The anticommutation relations , which is proportional to hamiltonian of the hydrogen atom for the radial part, is substituted by the anticommutation realtions 39. The commutation relations for the raising and lowering operators defining the Lie algebra of the SU1,1 group, are substituted by equation 40. The expectation value of the discrete variable x with respect to the Meixner functions M n γ,µ x is x nγ = M γ,µ n x xm n γ,µ x, that can be calculated with the help of the recurrence relation 36. Finally the term ll + 1 in the Hamiltonian for the continuous and discrete case can be interpreted as the eigenvalues of the Casimir operator for the SO3 group, L = 1 L + L + L L + + L 3 where the operators L +, L y L 3 are acting on discrete space see formulas 5 6. Acknowledgement This work has been partially supported by D.G.I.C.Y.T. under contract PB The autor wants to express his gratitude to Profs. A. Ronveaux and Y. Smirnov for valuable conversations. 13

14 References [1] M. Lorente, Quantum Mechanics on discrete space en M. Ferrero, A van der Merwe ed. New Developments on Fundamental Problems in Quantum Physics. Kluwer Academic, Dordrecht 1997, p [] M. Lorente, Creation and annihilation operators for orthogonal polynomials of continuous and discrete variable, Electron.Trans.Num.Anal.E.T.N.A. 9, [3] M. Lorente, Raising and lowering operators, factorization and differential/ difference operators of hypergeometric type. J. Phys. A: Math. Gen [4] A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal polynomials of a discrete variable, Springer, Berlin [5] A. Galindo, P. Pascual, Mecánica Cuántica I. Eudema, Madrid 1989, p [6] N.M. Atakishiyev, E. I. Jafarov, S.M. Nagiyev, K.B. Wolf, Meixner oscillators Rev. Mex. Fis. 44, [7] N.M. Atakishiyev, S.K. Suslov, Difference Analog of the harmonic oscillator Theor. Math. Phys. 85, [8] M. Lorente, On some integrable one-dimensional quantum mechanical systems Phys. Lett, B. 3, [9] A. Frank, R. Lemus, R. Bijker, F. Prez Bernal, J.M. Arias, A general algebraic model for molecular vibrational spectroscopy Annals of Phys. 5, [10] E. Bank, M.E.H. Ismail, The Attractive Coulomb Potential polynomials, Constr. Approx. 1,