Emetere, Moses

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The African Review of Physics (2013) 8:0010                                                                                                                     65  Mathematical Modelling of Bloch NMR to Solve the Schrödinger  Time Dependent Equation    Moses E. Emetere*  Department of Physics, Federal University of Technology Minna, Nigeria   Solutions of the Schrödinger are not new in Physics. The Bloch NMR was mathematically modeled to solve th  time dependent Schrödinger equation to investigate various problems in physics and chemistry. The solution was validated on the harmonic oscillator and two energies were consequently solved. The relationship between relaxation ande ergy was found to be negatively parabolic.    1.     Introduction Many solutions have been propounded for Schrödinger’s equation. The application of these solutions is restricted to a few aspects of physics. One of the methods used for solving the Schrödinger equation is the Finite Difference Time Domain (FDTD) method [1]. It was used to solve numerically the Schrödinger equation for ground state and excited state eigenvalues, and eigenfunctions for a single particle in one-, two-, and three-dimensional potential wells. In the past, analytical methods which have been used to solve the Schrödinger equation are: supersymmetry [1], Nikiforov Uvarov method (NU) [3], asymptotic iteration method (AIM) [4,5], hyperviria perturbation [6], exact quantization rule (EQR) [7], perturbative formalism [8], wave function ansatz method [9], and path integral [10]. In all these methods highlighted above, none has been incorporated with the Bloch NMR to solve nuclear spin among other problems in physics and chemistry. The solution of the Schrödinger equation using the mathematical modeling of Bloch NMR has wide range of applications in many fields of physics and chemistry. It can be used for checking and improving models developed to study quantum mechanical systems, improve on the effective analysis of biological systems, analyzing spin transport, nuclear spin dynamics, improvement of the numerical methods and so on.  In Sec. 2, the solution of the Schrödinger is given. In Sec. 3, the solution was validated by applying it to the harmonic solution.      The energy  ______________ *emetere@yahoo.com levels were calculated and features of its simulations were explained. It was observed that the energy and relaxation shows a negative parabolic shape, which explains that T → 0 when the energy levels increases. 2.     Derivation of Solution The Schrödinger equation in its time dependent format is given as  ℏ Ψ(	,) = − ℏ Ψ(	,)	 + Ψ(, )          (1)  From the above expression, − ℏ 	 is known as the kinetic expression of Eqn. (1). The idea of equalizing kinetic energy and magnetic energy was explained in [11] by calculating the theoretical magnetic energy of the spins of classical electrons. It was therefore concluded that the magnetic energy of a moving charge changes as the charge of the electron moves with a changed velocity i.e., kinetic energy. Magnetic field (H) is therefore related to the kinetic energy of the Schrödinger as shown below.  − ℏ 	 = −	. (, )                  (2)  Where, m is the magnetic moment and H(r,t) is the magnetic H- field. The magnetic H-field is defined in a macroscopic equation as   (, ) =  (, ) − (, )           (3)  Where, the   is permeability of free space, (, ) is the magnetization field and (, ) is the The African Review of Physics (2013) 8:0010                                                                                                                     66  magnetic B-field, which is assumed in the above expression as  (, ) =     and   (, ) = !  Where,   is the equilibrium magnetization. From the Bloch NMR equations given as   "#$" = ∆&' − #$(                                  (4)  "#)" = −∆&	 − &! − #)(                 (5)  "#*" = −&' − (#*+#)(,                        (6)  "#$" = -(' );   "#)" = −-(	 );  "#*" = 0  Where, ∆& = & −& 	is the frequency difference between Larmor frequency and the frame of reference,	& = −- is the Rabi frequency, & = −-. is the Larmor frequency,		, ' are the transverse magnetization,	! is the longitudinal magnetization, and   is the equilibrium magnetization Inserting Eqns. (2) and (3) into Eqns. (6) and (1) yields  ℏ Ψ(	,) = −	. &'TΨ(, ) + Ψ(, )   (7)  Both sides must be equal to a constant called E. Therefore, Eqn. (7) yields  ℏ /Ψ(, )/ = 0 = −	. &'TΨ(, )+ Ψ(, )  By separating the variables, we have  	Ψ(, ) = 1()2()  3ℏ( Ψ(	,) = 0                             (8)  (−	. &'T + )X(x) = 0                (9)  The solution of Eqn. (8) is  2 = 6(0) exp 9:3ℏ; = 6(0) exp(&)       (10)  The solution of Eqn. (9)  X(x) = :(+	.<,#)=,>?(	))               (11)  The full wave function must therefore be a linear combination of these fundamental solutions (Eqns. (10) and (11)). Thus, the wave function can be written as  Ψ(, ) = :@(A) BCD(3<)(+	.<,#)=,>?(	))             (12)  The solution (Eqn. (12)) can be written in the following format  Ψ(, ) = :@(A) BCD9EFGℏ;(+	.<,#)=,>?(	))               (13)  Ψ(, ) = :@(A) BCD(3<)(+	.#$H,H>?(	))                   (14)  Ψ(, ) = :@(A) BCD9EFGℏ;(+	.#$H,H>?(	))                   (15)  The solution (Eqns. (12) and (13)) was verified with ℏ Ψ(	,) = Ψ(, )  /Ψ(, )/ = 0ℏ I 06(0) exp 9:3ℏ;(−	. &'T + ())J  Multiplying both sides by ℏ  ℏ /Ψ(, )/ = 0 I 06(0) exp 9:3ℏ;(−	. &'T + ())J ℏ /Ψ(, )/ = 0Ψ(, )  The above gives us the following equation.  0Ψ(, ) = Ψ(, )  3.     Application of Solution to the Harmonic Oscillator The application of Eqn. (12) to harmonic oscillator, which is characterized by a quadratic potential,  = K	 , yields the following Schrödinger equation  − ℏ2 MΨ()M + N2 Ψ() = 0OΨ()  The above equation was solved using   The African Review of Physics (2013) 8:0010                                                                                                                     67  Ψ() = 06(0)(−	. &'T + K	 )  Since, the probability of a particle being found anywhere on the x-axis is 1, the above equation was therefore normalized to know the value of 6(0).   6(0) = 	. &'T  The energies obtained are  0 = − ℏ (1 − N) = − ℏ (1 − Q(	) )             (16)  0 = 	−	. &'T + () = −	. &'T + N2        (17) 4.     Results and Discussion  Fig. 1 describes a different idea of a particle in thebox. The energy levels of harmonic oscillator for potential (V) are listed in Table 1 (shown below)  Table 1: Harmonic Oscillator’s energy levels   Potential (V) Energy  (Eqn. (16)) (J) Energy  (Eqn. (17)) (J) 1 − ℏ4 1−	. &'T 2 0 2−	. &'T 3 ℏ4 3−	. &'T 4 ℏ2 4−	. &'T 5 3ℏ4  5−	. &'T  Fig. 2 shows the same effect as in Fig. 1. This means that by application the solution (Eqn. (17)) in operation is almost equal to the solutions in Eq. (16). The effect of relaxation on both the Energy levels and potentials were shown in Fig. 3. The energy and relaxation shows a negative parabolic shape, which explains that T → 0 when the energy levels increase. The positive parabolic relation between energy levels and potentials is the usual relation as proven by former researchers. A solution was derived under the influence of two relaxations, T	&	T as can be seen in Eqns. (14) and (15), This has the potential of applying the solutions to magnetic resonance inference (MRI); analysis of dual spins (hole [12,13] and electron spins [14,16]) as present in semiconductors, spin dependent hyperfine interaction (HFI), and anisotropic part of the Zeeman interaction (AZI) [16] amongst others. Also, figure 3 describes the relationship between potential and relaxation negatively linear.    Fig.1: The graph of Energy against (-2/k) potential. A graphical solution of Eqn. (16).   Fig.2: The graph of energy against potential. A graphical solution of Eqn. (17).     Fig.3: A general solution of Eqn. (17) when T1<100. 1 1.5 2 2.5 3 3.5 4x 10400.511.522.533.5x 10-282V/kEEnergy versus potential with the quantum well100 200 300 400 500 600 700 800 900 1000-5.3-5.2-5.1-5-4.9-4.8-4.7x 106potential VEnergy EE against V02040608010005101520-6-4-20246x 105relaxation T1second solution of the harmonic oscillationpotential VEnergy EThe African Review of Physics (2013) 8:0010                                                                                                                     68  5.     Conclusion The introduction of the Bloch NMR equation into Schrödinger equation has shown success in predicting well established features of the harmonic oscillator. The concept of this paper is defining a clear research pathway towards investigating into spin dynamics, biological structure, and chemical bonding structure amongst other. References [1]  I. W. Sudiarta and D. J. W. Geldart, J. Phys. A: Math. Theor. 40, 1885 (2007). [2]  D. A. Morales, Chem. Phys. Lett. 394, 68 (2004). [3]  C. Tezcan and R. Sever, Int. J. Theor. Phys. 48, 337 (2009). [4]  Hakan Ciftci, Richard L. Hall and Nasser Saad, Phys. Rev. A 72, 022101 (2005). [5]  O. Bayrak and I. Boztosun, J. Phys. A: Math. Gen. 39, 6955 (2006). [6]  J. B. Killingbeck, A. Grosjean and G. Jolicard, J. Chem. Phys. 116, 447 (2002). [7]  Z. Q. Ma and B. W. Xu, Int. J. Mod. Phys. E 14, 599 (2005). [8]  S. M. Ikhdair and R. Sever, J. Mol. Struc.-Theochem. 809, 103 (2007). [9]  S. H. Dong, Int. J. Theor. Phys. 39, 1119 (2000). [10]  N. K. Pak and I. Sokmen, Phys. Rev. A 30, 1629 (1984). [11] Carel van der Togt, Galilean Electrodynamics 17(6), 110 (2006). [12]  A. Dargys, Acta Physica Polonica A 1(107), 46 (2004). [13]  C. E. Swenbwerg and N. E. Geacintov, in Organic Molecular Photophysics, Vol.1, edited by J. B. Birks (Wiley, NY,1973). [14]  G. Finkelstein, H. Shtrikman and I. Bar-Joseph, Phys. Rev. Lett. 74, 976 (1995). [15]  C. Adelmann, X. Lou, J. Strand, C. J. Palmstrom and P. A. Crowell, Phys. Rev. B 71, 121301 (2005). [16]  U. E. Steiner and T. Ulrich, Chem. Rev. 89, 51 (1989).      Received: 12 November, 2012 Accepted: 1 March, 2013 

Mathematical Modelling of Bloch NMR to Solve the Schrödinger

Time Dependent Equation

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Mathematical Modelling of Bloch NMR to Solve the Schrödinger Time Dependent Equation

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