Energy spectrum inverse problem of qdeformed harmonic oscillator and WBK approximation


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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Energy spectrum inverse problem of qdeformed harmonic oscillator and WBK approximation To cite this article: Nguyen Anh Sang et al 06 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content  First, Second Quantization and Q Deformed Harmonic Oscillator Man Van Ngu, Ngo Gia Vinh, Nguyen Tri Lan et al.  Energy spectrum inverse problem of q deformed harmonic oscillator and entanglement of composite bosons Nguyen Anh Sang, Do Thi Thu Thuy, Nguyen Thi Ha Loan et al.  QDeformed Harmonic Oscillator and Morselike Anharmonic Potential Ngo Gia Vinh, Man Van Ngu, Nguyen Tri Lan et al. Recent citations  Energy spectrum inverse problem of q deformed harmonic oscillator and entanglement of composite bosons Nguyen Anh Sang et al This content was downloaded from IP address on 3/0/08 at 0:39
2 Journal of Physics: Conference Series doi:0.088/ /76//009 Energy spectrum inverse problem of qdeformed harmonic oscillator and WBK approximation Nguyen Anh Sang, Do Thi Thu Thuy, Nguyen Thi Ha Loan, Nguyen Tri Lan and Nguyen Ai Viet Hanoi Pedagogical University, Nguyen Van Linh Street, Phuc Yen Town, Vinh Phuc, Vietnam. Institute of Physics,Vietnam Academy of Science and Technology, 0 Dao Tan, Ha Noi, Viet Nam Abstract. Using the connection between qdeformed harmonic oscillator and Morse like anharmonic potential we investigate the energy spectrum inverse problem. Consider some energy levels of energy spectrum of q deformed harmonic oscillator are known, we construct the corresponding Morselike potential then find out the deform parameter q. The application possibility of using the WKB approximation in the energy spectrum inverse problem was discussed for the cases of parabolic potential harmonic oscillator, Morse like potential q deformed harmonic oscillator. so we consider our deformed three levels simple model, where the set parameters of Morse potential and the corresponding set parameters of level deformations are easily and explicitly defined. For practical problems, we propose the deformedthreelevels simple model, where the set parameters of Morse potential and the corresponding setparameters of level deformations are easily and explicitly defined.. Introduction Recently, quantum group and deformed Heisenberg algebras with q deformed harmonic oscillator have been a subject of intensive investigation. This approach is found some applications in various branches of physics and chemistry [ 6]. The method of q deformed quantum mechanics was developed on the base of Heisenberg commutation relation the Heisenberg algebra. The main parameter of this method is the deformation parameter q [0, ]. The Morse potential has an important role in describing the interaction of atoms in diatomic and even polyatomic molecules [7 0] in atomic and molecular physics. Despite its quite simple form, the Morse potential describes very well the vibrations of diatomic molecules. This is because that four particle complex system two heavy atomic nuclei with positive charge and two light electrons with negative charge can be reduced to relative motion between two atomic nuclei in an effective potential which is average Coulomb interaction of nuclei and electron clouds. Morse like potential models just work with a simple one dimensional three parameter effective potential, found many applications in condensed matter, biophysics, nano science and quantum optics. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors and the title of the work, journal citation and DOI. Published under licence by Ltd
3 Journal of Physics: Conference Series doi:0.088/ /76//009 The Morse potential in algebraic approach can be written in terms of the generators of SU. The quantum relation between q deformed harmonic oscillator and Morse potential was considered in [8], where then the anharmonic vibrations in Morse potential have been described as the levels of q deformed harmonic oscillator. The extended SU model qmorse potential also develop to compare with phenomenological Dunham s expansion and experimental data for numbers of diatomic molecules [8, 9]. In this work, considering deformed algebra is mathematical object and atomic effective potential is physical model, we use this relation in inverse way to investigate properties of q deformed harmonic oscillator on the base of the Morse potential. In the our previous works [, ] we have shown that the potential of harmonic oscillator is parabolic with infinity equal step levels, and the potential of qdeformed harmonic oscillator might be described as Morse like anharmonic potential with finite unequal step levels. The relation between the deformation parameter q and the set of parameters of Morse like anharmonic potential was found. In this work, using the connection between q deformed harmonic oscillator and Morse like anharmonic potential we investigate the inverse problem of the energy spectrum. Consider some energy levels of energy spectrum of q deformed harmonic oscillator are known, we construct the corresponding Morse like potential then find out the deform parameter q and associate with it the new parameter δ characterized the level deformation. The possibility of using the WKB approximation in the energy spectrum inverse problem was discussed for the cases of parabolic potential harmonic oscillator and Morse like potential q deformed harmonic oscillator.. Semiclassical Wentzel Kramers Brillouin WKB method.. Bohr Sommerfeld quantization conditions It is well known that the Bohr Sommerfeld quantization conditions are a procedure for selecting out certain discrete set of states of a classical integrable motion as allowed states. These are like the allowed orbits of the Bohr model or standing waves of de Broglie of the atom; the system can only be in one of these states, but not in any states in between. A particle in a one dimensional potential can be described by the Schrödinger equation: [ µ x where µ is the particle mass, is the Planck constant. ] + V x ψ x = Eψ x, The semiclassical Wentzel Kramers Brillouin WKB method is based on the ansatz ψ x = A x exp [is x / ]. In the limit as 0, the action Sx in the exponential satisfies the Hamilton Jacobi equation for the action function we have E = µ ds + V x, 3 dx S x = ds dx = µ [E V x]. 4
4 Journal of Physics: Conference Series doi:0.088/ /76//009 It is then shown that this usually leads to the Bohr Sommerfeld quantum conditions on periodic orbits with n = 0,,, 3,... dx µ [E V x] = π n +, 5 For one dimensional problems, the cyclic integral can be replaced by x x, where x and x are the classical turning points of the motion and E = V x... Application of WKB method to the case of harmonic potential As an example consider the linear harmonic oscillator with where K is the spring constant, and The quantum condition reads which can be solved to give ˆ + E/k E/k V 0 x = Kx, 6 x, = ± E/K. 7 dx µ E Kx / = π n +, 8 E 0n = n + K µ 9 = ω 0 n +, where ω 0 = K/µ. This is one of a small number of cases in which the WKB method gives the exact quantum mechanical energies..3. Application WKB method to the case of Morse potential The semiclassical WKB method applied to one dimensional problems with bound states often reduces to the Sommerfeld Wilson quantization conditions, the cyclic phase space integrals dx µ [E V M x] = π n +. 0 It turns out that this formula gives the exact bound state energies for the Morse oscillator with [ ] V M x = D e kx x0 e kx x 0, where x 0 is the minimum position, D is the depth and k is the width of potential. The requisite integral can be reduced to ˆ x x dx E V M x, 3
5 Journal of Physics: Conference Series doi:0.088/ /76//009 in which x and x are the classical turning points [ x, = x 0 + ] k ln D ± + E. 3 E D The integral can be done "by hand", using the transformation, using the transformation r = exp [k x x 0 ] followed by a contour integration in the complex plane, but can evaluate the integral explicitly, needing only the additional fact that ln = iπ. The result reads π µ k D + E = π n +, 4 which can be rewritten D + E = k n +. 5 µ The above equation can be solved for E to give E = [ D + n + k ]. 6 µ This result is coincided with exact solution see text book of Landau Liftshitz: QM. Denote the highest bound state is = [ µd/ k ] where [...] represents the floor, which for positive numbers is simply the integer part. We have energy spectrum E n = D with n = 0,,, 3,...,, or [ n + ] n n +, 7 max E Mn = E n [ D n ] = ω M + n +, 8 where ω M is the Morse frequency, which equals ω M = D n max = k D µ, 9 The values of D, k, and x 0 of diatomic molecules are taken from experiments. Actually, the Morse experimental parameters are the dissociation parameter D, the fundamental vibrational frequency ω M, the equilibrium internuclear distance x 0. and the reduced mass µ = m m / m + m. The exponential parameter is given by k = ω M µ/d in appropriate units k. The Schrödinger equation for the Morse oscillator is exactly solvable, giving the vibrational eigenvalues ɛ M = ω M n + ω M n +, 0 4D 4
6 Journal of Physics: Conference Series doi:0.088/ /76//009 Unlike the harmonic oscillator, the Morse potential has a finite number of bound vibrational levels with D/ω M. Compare with the Dunham anharmonic parameter relations in the spectroscopy χ e ω M = A = k / Dand ω M = A = k D/µ we have χ e = k Dω M = k µ 4D = k D Introduce the energy differences as and for lowest energy levels E Mn = ω M EMn E Mn, E M = ω M E M = ω M E M3 = ω M According to our Morse Potential for Deformation model MPD model [], we can define a new deformation parameter q M, which is satisfied the equation we have = q M, 4 E M = ω M q M, E M = ω M q M, E M3 = ω M 3q M... 5 From the two first equations we obtain the deformation parameter q M q M = E, 6 M E M and the Morse frequency ω M = E M E M Harmonic approximation for Morse oscillator In this part we find the harmonic approximation for Morse oscillator. From definition of Morse potential easy to see that dv M x dx = V M x = D [ ke kx x 0 ke kx x 0 ], 8 and so d V M x dx = V M x = D [4k e kx x 0 k e kx x 0 ], 9 V M x 0 = D, V M x 0 = 0, V M x 0 = Dk. 30 5
7 Journal of Physics: Conference Series doi:0.088/ /76//009 Using the Taylor series expansion around x = x 0, we found harmonic approximation potential for Morse oscillator V H x V M x 0 + V M x 0 x x 0 + V M x 0 x x 0 = D + Dk x x 0, 3 with the spring constant K / = Dk and corresponding harmonic frequency ω H = K /µ = k D/µ. We have the relation ω i = k i Di /µ, 3 where i = M, H, Inverse problem of deformed harmonic oscillator Suppose that the 3 first energy levels E 0, E, E are known, so we have two energy separations and = E E 0, = E E. 33 Define the deformation parameter q q =, 34 the oscillation frequency ω q ω q =, 35 The highest bound state N q is where [...] represents the integer part. N q = [/ q] = [ ] /, 36 / If <, then 0 < q <, N q = / q, we have the case of deformed harmonic oscillator with finite unequalstep levels, which can be characterized by a Morselike potential V Mq x = D q [e kqx x 0 e kqx x 0 ]. 37 with the energy deep D q Dq = ω q q =, 38 The exponential parameter k q µ k q = ω q = ωq µ q D q = µ. 39 The corresponding parabolic potential V Hq x with the oscillation frequency ω q = = k q Dq /µ is V Hq x = D q + K q x x 0 = D q + µω q x x If the separations are equal =, then q =, N =, we have the case of harmonic oscillator with infinite equalstep levels, which can be characterized by a parabolic potential V H x with the oscillation frequency ω = k D /µ V H x = D + µω x x
8 Journal of Physics: Conference Series doi:0.088/ /76//009 Figure. Morse potential q.0 a.0 δ b a δ b q Figure. The relation between q and δ: a q on δ, and b δ on q. 5. Model example Consider the first three energy levels E 0, E, E are known, we define two parameters = E E 0, and = E E. Introduce the ratio parameter δ as the main parameter of our model δ = E / E = ε / ε. 4 For the case of Morse oscillator 0 < δ <, example three levels Morse oscillator is presented in the figure. For the case of harmonic potential δ =. Taking also two values r 0 and k 0 = / µ/ are the equilibrium distance and imaginary impulse of anharmonic Morse oscillator with q= is harmonic case, we will express all physical values via the energy difference {, δ}or deformation{, q}set parameters. We have the deformation parameter q the relation between q and δ are presented in the Fig. a and b The value is N = = q = δ δ = q, 43 [ ] [ δ = δ q ], 44 where [...]is the integer part, and are presented in the figure 3 as a functions on δ upper steps, and on q lower steps. 7
9 Journal of Physics: Conference Series doi:0.088/ /76// δ Figure 3. The relation value on δ upper steps, and on q lower steps. ω a ω b.0 Δ Δ a δ 4 b q Figure 4. The frequency ω/ : a on δ, and b on q. The frequency ω = δ = q, 45 and its values are presented in the figure 4 as a functions of blue, and on q yellow. The energy deep D q D q δ = δ = [ q q ], 46 are ploted in the figure 5. The imaginary momentumlike width effective mass k q k q = δ =, 47 k 0 q are presented in the figure 6. We found the Morselike potential U M χ = V M [ δ ] = e aχ δ. e aχ δ, 48 ω 0 δ 8
10 Journal of Physics: Conference Series doi:0.088/ /76//009 D q /Δ D q /Δ a δ b q Figure 5. The energy deep of Morse potential D q / : a on δ, and b on q. k/k δ,q Figure 6. The parameter k q of Morse potential on δ upper curve, and on q lower curve. where χ = x/x 0, a = k 0 x 0, and the corresponding harmonic potential U H χ = V H δ = ω 0 δ + δ µx 0 χ. 49 The Morse potential U M and the corresponding harmonic potential U H are presented in the figure 7. The energy spectrum of Morse potential is [ ε n = E n = δ n + n + ], n = 0,,, 3,...,, 50 with first four levels ε 0 = δ 3 ε = δ 9 5 ε = δ ε 3 = δ ,, 5,, 9
11 Journal of Physics: Conference Series doi:0.088/ /76//009 a b Figure 7. a The Morse potential U M and b with the corresponding harmonic potential U H The level difference ε n are ε n = ε n+ ε n, 5 and for some lowest levels ε = δ, ε = δ, ε 3 = δ In the harmonic limit δ,q,, we have ε 0 /,ε 3/,ε 5/,... and ε, ε, ε 3,... In the strong deformation limit δ 0, q /, see the figure 3, we remain only two levels ε 0 / /6 = 0.875, ε 3/ 9/6 =.875, and ε, ε 0. In general, for Morse like potential in WKB approximation the classical turning points are defined by the expression [ χ, n = + ] D Ln ± + E n. 54 k 0 x 0 E n D Given energy spectra E n, we define χ, n, so and Morse like potential. 6. Discussion In this work using the model proposed in [] with the defined connection between qdeformed harmonic oscillator and Morse like anharmonic potential, we investigate the energy spectrum inverse problem. Consider some lowest levels of energy spectrum of qdeformed harmonic oscillator are known, we construct the corresponding Morselike potential with well defined deform parameter q and also new level deformation parameter δ. The possibility of using the WKB approximation in the energy spectrum inverse problem was discussed for both cases of parabolic potential harmonic oscillator and Morselike potential qdeformed harmonic oscillator. Some most important relations between the setparameters {, δ, q} of level deformations and setparameters {D, k, x 0 } of Morse potential are derived explicitly. In the weekdeform limit δ,q ;, we back to the harmonic case with many unique step levels. In the strongdeform limit δ 0, q /;, we go to the twolevels problem, where only the ground n = 0 and first n = levels can be existed and the other n levels are collapsed. 0
12 Journal of Physics: Conference Series doi:0.088/ /76//009 In principal, with a given energy spectra E n we can define the classical turning points χ, n and then construct the effective potential by using the WKB approximation, but it is a inverselike problem and usually difficult to solve. In practical, because for many problems of atom and molecular physics, quantum optics, condensed matter,... only the ground and one or two first excited levels are considered, so we propose to use our deformedthreelevels simple model, where the setparameters of Morse potential and the corresponding setparameters of level deformations are easily and explicitly defined. Our level deformation model might be useful in investigation the problem of entanglement entropy, when the environment or vacuum effects are taken in to account. This will be studied in the our next work. Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development NAFOSTED under grant number References [] Mirzrahi S S, Camargo J P and Dodonov V V 004 J. Phys. A [] Abe S 009 Phys. Rev. E [3] Lavagno A and Swamy P P 00 Physica A [4] Algin A and Senay M 0 Phys. Rev. E [5] Andrade F M and Silva E O 03 Phys. Lett. B [6] Tkachuck V M 03 Phys. Rev. A [7] Dong S H, Lemus R and Frank A 00 Int. J. Quant. Phys [8] Angelova M, Dobrev V and Franck A 004 Eu. Phys. Jour. D 3 7; Angelova M 00 arxiv:condmat/004v [9] Kaplan I G 003 Handbook of Molecular Physics and Quantum Chemistry Wiley [0] Lima E F and Hornos J E M 005 J. Phys. B: At. Mol. Opt. Phys [] Man Van Ngu, Ngo Gia Vinh, Nguyen Tri Lan, Luu Thi Kim Thanh, and Nguyen Ai Viet 05 Journal of Physics: Conference Series [] Ngo Gia Vinh, Man Van Ngu, Do Thi Nga, Nguyen Tri Lan, Luu Thi Kim Thanh and Nguyen Ai Viet 05 Journal of Physics: Conference Series 67 00
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