Adv. Studies Theor. Phys., Vol. 6, 212, no. 18, 887-892 The Octupole Field Effect on the H Atom Spectrum in Noncommutative Space Ahmed Al-Jamel, Hatem Widyan and Eqab M. Rabei Physics Department, Al Al-Bayt University, Mafraq 25113, Jordan aaljamel@gmail.com Abstract An expression for the octupole field potential in noncommutative space is obtained and its consequences on the hydrogen atom ground state energy is studied using the Schrodinger perturbation theory. It is found that there is no first-order correction for all states, while there is a nonvanishing second order shift on the ground state energy, from which we extracted the octupole selection rules Δm = ±3, ±1 and Δl =1, 3. PACS:31.15.-P, 3.65.Fd, 11.1.Nx Keywords: noncommutative, H-atom, octupole field 1 Introduction Within the frame of noncommutative space, the determination of the atomic response properties to an external electric field is of a great deal of interest these days. Linear Stark effect has been studied in [5] in which they showed that there is no Stark shift linear in the electric field induced by the noncommutativity of the coordinates and in particular the ground state remains unchanged. Like the ordinary Stark effect of the hydrogen atom, in order to obtain correction for ground state energy one has to go to a second order in perturbation theory. This was carried out in [7], in which they showed that there is a quadratic Stark shift in the electric field for the ground state energy induced by the noncommutativity of the coordinates. In [6], an expression for the noncommutative quadrupole potential has been obtained, the second-order noncommutative quadrupole effect on the ground-state energy of the hydrogen atom has been computed, and as a result, a sum rule was obtained. This computation was done using both the second-order perturbation theory and the exact computation using the Dalgarno-Lewis method. Here we extend the work done in [7] and [6] by studying the effect of octupole field on the H-atom spectrum in noncommutative space. This is of great
888 Ahmed Al-Jamel et.al. interest when studying the electric 2 l -pole moment sum rules for the hydrogen atom, for instance, in the calculation of interatomic forces. In section 2, the octupole field in noncommutative space is constructed, from which we study its influence on the H-atom energy levels and selections rules are extracted. In section 3, summary and conclusions are presented. 2 The octupole field in noncommutative space Within the framework of noncommutative (NC)quantum mechanics, the coordinate and momentum operators obey the commutation relations: [x i, x j ]=iθ ij, [x i, p j ]=i hδ ij, [p i, p j ] = (1) where θ ij is a real antisymmetric matrix, and each of its entries has dimensions of (length) 2. For computational purposes, it is more convenient to change to the new coordinates x i and p i using the Bopp shift x i = x i + 1 2 h θ ijp j, p i = p i, (2) such as the new coordinates x i, p i satisfy the usual canonical commutation relations. When the electron in the H-atom is subjected to a perturbation by an octupole electrostatic potential eλp 3 (cos ϑ)r 3, where P 3 (cos ϑ) = 1(5 2 cos3 ϑ 3 cos ϑ) is the Legendre polynomial of the third degree (l = 3), λ is the strength of the field and e is the electric charge, then the total Hamiltonian of the system is H = p2 2m e2 r eλp 3(cos ϑ)r 3. (3) We construct the perturbation term in NC space as follows Ĥ 1 = eλ ˆP 3 (cos ϑ)ˆrˆrˆr = eλẑ [5ẑẑ 3ˆrˆr] (4) 2 Defining the vector θ whose components are written in terms of the NC parameters θ ij as θ i = ɛ ijk θ jk, and using the transformation relations (2), one can show that the perturbation that takes into account the noncommutativity of coordinates is Ĥ 1 = eλp 3 (cos ϑ)r 3 5eλz2 2 h ( θ p) z 3eλz 4 h ( L θ) eλz2 4 h ( θ p) z (5z 2 3r 2 )+O(θ 2 ) (5)
Octupole field effect on the H atom spectrum 889 where L is the orbital angular momentum. The first term eλp 3 (cos θ)r 3 is the original perturbation whereas the rest of terms correspond to the noncommutative octupole potential energy H NC H NC = eλ 2 h [5z2 ( θ p) z + 3z 2 ( L θ)+ 1 2 ( θ p) z (5z 2 3r 2 )]. (6) Using z 2 ( θ p) z = θ x z 2 p y θ y z 2 p x and elaborating each operator as follows: z 2 p y = z 2 (zp y + p z y)+ z 2 (zp y p z y) (7) = z 2 (zp y + p z y) z 2 L x, and similarly, z 2 p x = z 2 (zp x + p z x)+ z 2 L y, (8) and using the fact p j = μ [x i h j,h ] where H = p2 noncommutative octupole field in Eq.(6) as e2 2μ r, one can rewrite the H NC = eλ 2 h θ x[ 3m i h (z2 yh zh zy 1 4 x2 yh + 1 4 xh xy 1 2 yh y 2 + 1 2 H y 3 ) 3 2 zl x 3 4 xl z]+ eλ 2 h θ y[ 3m i h ( z2 xh + zh zx + 1 4 y2 xh 1 4 yh yx + 1 2 xh x 2 1 2 H x 3 ) 3 2 zl y 3 4 yl z] 3eλ 4 h θ zzl z. (9) We need next to consider the effect of this noncommutative part, H NC, on the energy levels of the hydrogen atom using the perturbation theory. Thus in the needed matrix elements, we have radial integrals and angular integrals. The latters contain primarily the product of five spherical harmonics Y m 1 l 1 (ϑ, φ)y m 2 l 2 (ϑ, φ)y m 3 l 3 (ϑ, φ)y m 4 l 4 (ϑ, φ)y m 5 l 5 (ϑ, φ) (1) which can be reduced into the product of three spherical harmonics using the formula [2] Y m 1 l 1 (ϑ, φ)y m 2 l 2 (ϑ, φ) = l m ( l1 l 2 l m 1 m 2 m (2l 1 + 1)(2l 2 +1) 4π(2l +1) ) ( l1 l 2 l ) ( 1) m Y m l (ϑ, φ) (11) The integral of the product of three spherical harmonics of the form 2π π Y m 1 l 1 (ϑ, φ)y m 2 = (2l 1 + 1)(2l 2 +1) 4π(2l 3 +1) l 2 (ϑ, φ)y m 3(ϑ, φ) sin(ϑ)dϑdφ ( l1 l 2 l 3 l 3 )( l1 l 2 l 3 m 1 m 2 m 3 ) (12)
89 Ahmed Al-Jamel et.al. can be evaluated using the Wigner 3j symbol, which has the selection rules m 1 = m 2 + m 3 and l 2 l 3 <l 1 <l 2 + l 3. Using the above formulas and z = r 4π Y 3 1 (ϑ, φ), x = r 2π 1 [Y 3 1 (ϑ, φ) Y1 1(ϑ, φ)], y = ir 2π 1 [Y 3 1 (ϑ, φ) + Y1 1 (ϑ, φ)], and the orthonormality relation of spherical harmonics, one can evaluate the needed angular integrals. The first-order perturbation correction to the energy levels due to H NC is given by =< nlm H NC nlm >. (13) ΔE 1(NC) n The angular integrals can be computed using Eq.(12). We have found that all the terms in Eq.(12) gives integrals of the form 2π π dωyl m (ϑ, φ)y m l (ϑ, φ) Y ±1 Y ±1 Yl 3 (ϑ, φ) 1 (ϑ, φ) (ϑ, φ), (14) and 2π { } π Y dωyl m (ϑ, φ)y1 m±1 l±2 (ϑ, φ) (ϑ, φ) Yl m±1, (15) (ϑ, φ) which are always vanish according to the rules of evaluating Eq (15). Hence, ΔE 1(NC) n = (16) for all energy levels. Therefore, the noncommutativity of space does not change the energy levels of all states in the first-order perturbation. As usual, one then needs to go to the second-order in perturbation theory in order to find the correction to the ground-state energy. The second-order correction for the ground state energy is ΔE 2(NC) 1 = n 1,l,m n, l, m H (NC) 1,, 2 En. (17) E 1 As done in the case of the first order we can reduce the angular integrals for each term in H NC into a form given by Eq.(12). We have found that the term in θ z gives zero contribution, while the terms in θ x and θ y give finite values for some values of l and m. Hence we deduce the selection rules Δm = ±3, ±1 and Δl =1, 3. Also, we have noticed that there is no cancellations from the all terms within each of θ x and θ y. If we take a special case where θ x = θ y θ z, the terms in θ x do not cancel the terms in θ y. InSome authors [3, 4] assume that θ x = θ y = then in this case there will be no contribution to the second order. Therefore, to get nonzero contribution we assume that θ x and θ y, which gives ΔE 2(NC) 1 = 9μ2 e 2 λ 2 A n θ x + B n θ y 2 4 h 4 En. (18) E 1 n 1
Octupole field effect on the H atom spectrum 891 where A n and B n are mainly resulted from integrations over the radial parts corresponding to the principal quantum number n A n = <nlm z2 yh zh zy 1 4 x2 yh + 1 4 xh xy 1 2 yh y 2 + 1 2 H y 3 1 > B n = <nlm z 2 xh + zh zx + 1 4 y2 xh 1 4 yh xy + 1 2 xh x 2 1 2 H x 3 1 >. It should be mentioned that an integral over the continuous states should be added to the above sum to include the scattering states, which makes our calculations and selection rules approximate. It is customarily to compare our noncommutative results with the corresponding commutative results. If we assume θ x = θ y = θ, then one can calculate the first few terms of the summations in the above result. Using Mathematica V7.., and the atomic units (μ = e = h = 1), the result is ΔE 2(NC) 1 = 9λ 2 θ 2( 1.4533 +.3528 +.2547 +.1734 +.1139 +.759 +.517 +.365 +.274 +... ) Note that the contributions is decreasing as n is increasing. Therefore, there is a positive contribution to the octupole effect from the noncommutativity of coordinates, which was not appeared in the first order in perturbation theory. The commutative second-order correction to the octupole field is [1] where a = h2 μe 2. Therefore, for θ x = θ y θ, then ΔE 2(C) 1 = λ 2 (5)(7!)(a7 ) 3 (2) 7 (19) Δ = ΔE2(NC) 1 ΔE 2(C) 1 = 216θ2 (5)(7!)e 2 a 9 n 1 A n + B n 2. (2) En E1 Thus, Δ θ 2, from which one can infer an upper limit on the noncommutative parameter θ once the precision experimental measurements of the energy shift is carried out and the sum is computed somehow, or approximated by taking the first few terms only. It should be mentioned that it is not straightforward to use the Dalgarno-Lewis exact method as in [7, 6] due to the complicated form of H NC. Finally, it is interesting to compare our main results of the selection rules and the θ dependent of the energy corrections with that for dipole and quadrupole fields. The comparison is presented in Table1 for the case θ x = θ y = θ.
892 Ahmed Al-Jamel et.al. Table 1: Comparsion of selection rules and θ-dependent of ΔE 2(NC) 1 [7, 6]. Field selection rules θ dependent Dipole (l =1) Δl =1,Δm = ±1 E θ 2 Quadrupole (l =2) Δl =2,Δm = ±1 θ 2 Octupole (l =3) Δl =1, 3, Δm = ±3, ±1 θ 2 3 Conclusions In this paper we have computed the noncommutative octupole field. We have shown that there is no first-order contribution for all energy levels of the H- atom due to the noncommutativity of the coordinates. However, we have obtained a second-order correction to the ground state, which is quadratically dependent on the strength of the field eλ and on the noncommutative parameter as θ 2. As a result of these calculations, selection rules were deduced and found to be Δm = ±3, ±1 and Δl =1, 3. We believe that it is difficult to find the exact result using Dalgarno-Lewis method due to the long and complex form of the noncommutative potential. References [1] Charles Schwartz, Calculations in Shrodinger pertbation theory, Annals of Physics: 2, 156-169 (1959). [2] J. J. Sakurai, Modern Qunatum mechanics, Addison Wesley (1994). [3] K. Li, J. Wang, C. Chen, Representation of noncommutative phase space, Mod. Phys. Lett. A 2, 2165 (25). [4] LI Kang and Chamoun Nidal, Hydrogen Atom Spectrum in Noncommutative Phase Space, Chin. Phys.Lett. 23, 1122 (26). [5] M. Chaichian, M. M. Sheikh-Jabbari and A. Turenu, Hydrogen atom spectrum and the lamb shift in noncommutative QED, Phys. Rev.Lett. 86 2716-9 (21). [6] N. Chair, A. Al-Jamel, M. Sarhan, M. Abu Sini, and E. R. Rabie, The noncommutative quadrupole field effect for the H-atom, Phys. A: Math. Theor. 44 9536 (6pp)(211). [7] N. Chair and M. A. Dalabeeh, The noncommutative quadratic Stark effect for the H-atom, J. Phys. A: Math. Gen.38, 1553-155 (25). Received: April, 212