Electron Spin Precession for the Time Fractional Pauli Equation. Hosein Nasrolahpour

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1 respacetime Journal December 0 Vol. Issue 3 pp Electron Spin recession for the Time Fractional auli Equation Hosein Nasrolahpour Dept. of hysics Faculty of Basic Sci. Univ. of Maandaran.O. Box Babolsar Iran Hadaf Institute of Higher Education.O. Box Sari Maandaran Iran Abstract In this work we aim to extend the application of the fractional calculus in the realm of quantum mechanics. We present a time fractional auli equation containing Caputo fractional derivative. By use of the new equation we study the electron spin precession problem in a homogeneous constant magnetic field. Keywords: Fractional quantum mechanics; Time fractional auli equation; Electron spin precession - Introduction Fractional derivatives provide a relevant tool for understanding the dynamics of the complex systems. Recently fractional calculus has found interest and applications in the domain of quantum mechanics and field theory (see [5] and refs. in). Fractional quantum mechanics (FQM) is in fact generaliation of standard quantum mechanics to discuss quantum phenomena in fractal and complex systems. Up to now several different approaches on FQM have been investigated [6-9]. In FQM space fractional Schrödinger equation has been obtained by askin [9]. He constructed the fractional quantum mechanics using the évy path integral and considered some aspects of the space fractional quantum system. Space fractional Schrödinger equation similar to the standard one satisfies the Markovian evolution law. However one can introduce the time fractional Schrödinger equation to describe non- Markovian evolution in quantum mechanics [0]. In Refs. [0 ] some properties of time fractional Schrödinger equation have been studied. It is showed that Hamiltonian in this case is non-hermitian and non-local in time. Furthermore it is found [0] that for free particle case the wave function evolves in such a way that the value of total probability increases to the value (where is the order of derivative in the time fractional Schrödinger equation) in the limit t.the probability structure of time fractional Schrödinger equation in the case of low-level fractionality [0 ] also has been discussed in []. It has been proved that for the basis state the probability density increases initially and then fluctuates around a constant value in later times. In the standard quantum mechanics the well-known auli equation (the non-relativistic evolution equation of spin particles) is given by [] e [ ( A ) e B. B ] i m c t () address: hnasrolahpour@gmail.com

2 Where A is the three-component magnetic vector potential and is the electric scalar potential and Ψ = Ψ Ψ is the two component spinor wave function and e B is the so-called Bohr magneton. mc As we mentioned above quantum phenomena in disordered systems such as random fractal structure can be discussed by the fractional quantum mechanics. In disordered systems spatial and temporal fuctuations often obey Gaussian statistics due to the central limit theorem. However it has been found in recent years that anomalous broad évy type distributions of the fuctuations can occur in many complex systems [4].With this motivation we aim to present a fractional auli equation containing Caputo fractional derivative and study the electron spin precession within this framework. In the following fractional calculus [-4] is briefly reviewed in Sec.. The time fractional auli equation is presented in Sec. 3. The time-dependent spin function and expectation values for the spin observables are obtained in terms of the Mittag-effler functions. The probability for spin-up and spin-down also are calculated in this section. In Sec. 4 we will present our conclusions. - Fractional calculus.. The Caputo fractional derivative operator Fractional calculus (FC) has a long history when the derivative of order has been described by eibni in a letter to 'Hospital in 695 [3]. FC is the calculus of derivatives and integrals with arbitrary order which unify and generalie the notions of integer order differentiation and n-fold integration which have found many applications in recent studies to model a variety processes from classical to quantum physics [4-]. In contrast with the definition of the integer order derivative the definition of the fractional one is not unique. In fact there exists several definitions including Grünwald - etnikov Riemann - iouville Weyl Ries and Caputo for fractional order derivative. Fractional differential equations defined in terms of Caputo derivatives require standard boundary (initial) conditions. For this reason in this paper we only use the Caputo fractional derivative. The left (forward) Caputo fractional derivative of a function f (t) is defined by t c n ( n ) 0Dt f ( t ) ( t ) f ( ) d 0 t 0 ( n ) 0 () Where n is an integer number and is the order of the derivative such that n-< <n and ( n f ) ( ) denotes the n-th derivative of the function f ( ). aplace transform to Caputo's fractional derivative gives n c m ( m ) 0 t m 0 { D f ( t )} s F ( s ) s f (0) (3) Where F ( s ) is the aplace transform of f ( t )... The Mittag-effler function The Mittag-effler function is a generaliation of the exponential function which plays an important role in the fractional calculus. The Mittag-effler function is such a one-parameter function defined by the series expansion as

3 E k ( ) ( k ) α C 0 C k 0 (4) And its general two-parameter representations is defined as E k ( ) ( k ) α β C 0 C k 0 (5) Where C is the set of complex numbers and ( ) denotes the gamma function. aplace transform for Mittag-effler function is very useful in solving fractional differential equations. The aplace transformations for several Mittag-effler functions are summaried below s { E ( t )} s { t E ( t )} s t E t { ( )} s s (6a) (6b) (6c) Where s. 3- Time fractional auli equation As we mentioned in section one can introduce the time fractional Schrödinger equation to describe non- Markovian evolution in quantum mechanics. In this section we generalie the time fractional Schrödinger equation [ 3] and obtain the following time fractional auli equation e [ ( A ) e B. B ] i 0 m c t (7) where M c T is a scaled lanck constant. The parameters lanck time respectively which are defined as M and T are lanck mass and T G c 5 M c G (8) where G and c are the gravitational constant and the speed of light in vacuum respectively. We now use the Eq. (7) to discuss the electron spin precession problem in a homogeneous constant magnetic field. For this purpose we first assume that the electron is fixed at a certain location and its spin is the only degree of freedom. Also let the magnetic field consist of a constant field B in the Z direction B B k 0 Non-Markovian systems appear in many branches of physics such as quantum optics solid state physics quantum information processing and quantum chemistry (see for example the Ref. []).

4 Therefore that part of the time fractional auli equation Eq. (7) which contains the spin yields ( i ) c D t 0 (9) eb Where are the so-called armor frequency and the auli matrices for a spin particles are mc as below x 0 0 y 0 i i Since the Hamiltonian of our system is a x matrix the spin function in arbitrary time t must be written as a column matrix of two components and can be derived as below i e cos( ) E ( i ( t ) ) a( t ) ( t ) b( t ) i e sin( ) E ( i ( t ) ) (0) Where and are arbitrary phase constants. Now we able to calculate the expectation values for the observables s x s y s. Then we will have s e E i t e E i t 4 i ( ) ( ) ( ) sin( )[ ( ( ( ) i )) x t x ( ( ( ) )) ] i s e E i t e E i t 4 i ( ) ( ) ( ) sin( )[ ( ( ( ) i )) y t y ( ( ( ) )) ] s E i t E i t t ( ) cos( )[ ( ( ) ) ( ( ) )] () () (3) Here we can see explicitly that as Eq. (3) gives s t cos( )[ E( i ( t )) E( i ( t ))] cos( ) (4) So we have s t s t 0 as expected from the standard quantum mechanics. Furthermore for the special case of we have s e erfc i t erfc i t cos( ) t ( )[ ( )] t (5) Where erfc ( ) denoted the complementary error function [3] which is defined by

5 t erfc ( ) e dt (6) Also by use of Eq. (0) we can calculate the probability for spin-up and spin-down at t 0. So we have a( t ) cos ( )[ E ( i ( t ) ) E ( i ( t ) )] b( t ) sin ( )[ E ( i ( t ) ) E ( i ( t ) )]. We can see explicitly that as above equations gives. But for the arbitrary case of ( 0 )we have t t cos (( ) ) sin (( ) ) tot tot Where is obtained in terms of the sine and cosine Mittag-effler functions defined as ( ) ( ) ( ) ( ) cos ( ) sin ( ) ( ) (( ) ) n n n n n 0 n n 0 n (7) (8) (9) (0) 4- Discussion and Conclusion Fractional quantum mechanics is a generaliation of traditional quantum mechanics to discuss quantum phenomena in disordered systems such as random fractal structure. In disordered systems spatial and temporal fuctuations often obey Gaussian statistics due to the central limit theorem. However it has been found in recent years that anomalous broad évy type distributions of the fuctuations can occur in many complex systems [4].With this motivation a time fractional auli equation is introduced. In this paper we have studied the spin precession in the framework of fractional dynamics. For this purpose: first we have presented the time fractional auli equation. Utiliing this equation we have derived the time-dependent spin function for our system. Then using the Eq. (0) we have obtained the expectation values for the observables s x s y s. The following result can be easily deduced: By use of the standard auli equation we have s (cos( t )sin( )sin( t )sin( )cos( )) Obviously the above relation can be deduced from Eqs. (-3) as and we will have a conserved spin component in the field direction while the spin precesses around the axis with twice the armor frequency (we know that this is due to the gyromagnetic factor of the spin). But as we can see from the Eq. (3) we have: s t s t 0 0 () These results show that spin component in the field direction s is not conserved for the arbitrary case of.therefore spin precession phenomena is complicated in the framework of fractional dynamics(also as a result we can consider modifications to the gyromagnetic factor of the spin in this framework). Furthermore we have calculated the time dependent probability for spin-up and spin-down at t 0 for the arbitrary case of ( 0 ). As a consequence of these results we can predict a transition from one spin state to the other spin state in our system when time flows. In this paper we only ()

6 consider the case of 0 it is also interesting to consider the case of for the Eq. (7) and derive new results. We hope to do this in future work. Recently fractional calculus has found interest and applications in the context of relativistic quantum mechanics and field theory [5-3]. We hope to study some applications of fractional calculus in these interesting areas of physics as well. Acknowledgement The author is grateful to Richard Herrmann for many useful discussions and explanations. References [] J. J. Sakurai Modern Quantum Mechanics (Addison-Wesley New York 994). W. Greiner Quantum Mechanics: An Introduction 4th ed. (Springer Berlin 00). [] I. odlubny Fractional Differential Equations (Academic ress New York 999). [3] K. S. Miller B. Ross An introduction to the fractional calculus and fractional differential equations ( John Wiley & Sons Inc. 993). K. B. Oldham J. Spanier the Fractional Calculus (Academic ress NewYork 974). [4] R. Hilfer Applications of Fractional Calculus in hysics ( World Scientific Singapore 000). G. M. Zaslavsky Hamiltonian Chaos and Fractional Dynamics (Oxford Univ. ress Oxford 005). R. Metler and J. Klafter hys. Rep. 339(000) 77. R. Metler and J. Klafter J. hys. A 37(004)6 08. [5] H. Nasrolahpour Applications of fractional derivative in particle physics (MSc Thesis) ublished By University of Maandaran 008. [6]. Nottale Fractal space-time and micro-physics ( World Scientific Singapore993).. Nottale rog. hys. (005). M. N. Celerier. Nottale J. hys. A Math. Gen. 37 (004) 93. [7] N. askin hys.ett. A 68 (000) 98. [8] N. askin hys. Rev. E 6 (000) 335. [9] N. askin hys. Rev. E 66 (00) [0] M. Naber J. Math. hys. 45(004) [] A. Tofighi Acta hys. ol. A. Vol. 6 (009)4-8. [] M. Bhatti Int. J. Contemp. Math. Sci. (007) 943. [3] S. I. Muslih Om. Agrawal Dumitru Baleanu Int. J. Theor. hys. 49 (00) [4] R. Herrmann hysica A. 389 (00) [5] R. Herrmann J.hys.G: Nucl. art. hys 34 (007) 607. [6] R. Herrmann hysica A. 389 (00) [7] E. Goldfain Comm. Non. Sci. Num. Siml. 3 (008) [8] E. Goldfain Chaos Solitons & Fractals (004) [9] V. E. Tarasov hysics etters A 37 (006) V. E. Tarasov Theor. and Math. hys. 58() (009) [0] V. E. Tarasov G.M. Zaslavsky hysica A 368(006) 399. [] A. Tofighi H.Nasrolahpour hysica A 374(007) 4. [] H.-. Breuer and F. etruccione The Theory of Open Quantum Systems (Oxford Univ. ress Oxford 007). C.W. ai. Maletinsky A. Badolato and A. Imamoglu hys. Rev. ett. 96(006)67403 D. Aharonov A. Kitaev and J. reskill hys. Rev. ett.96 (006) Á. Rivas S. F. Huelga M. B. lenio hys. Rev. ett. 05(00) J. Shao J. Chem. hys. 0(004) 5053.

7 [3] M. A. Abramowit I. A. Stegun (Eds.) Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables (Dover New York 97). [4]. Maass F. Scheffler hysica A 34 (00) [5]. Zavada J.Appl.Math. (00) [6] A. Raspini hys. Scr.64 (00)0. [7] E. Goldfain Chaos Solitons & Fractals 8 (006) [8] E. Goldfain Comm. Non. Sci. Num. Siml. 3(008) [9] E. Goldfain Comm. in Nonlin. Dynamics and Numer. Simulation 4 (009) [30] R. Herrmann hys. ett. A 37(008) [3] S. I. Muslih Om. Agrawal D. Baleanu J. hys. A: Math. Theor. 43 (00) [3] R. A. El-Nabulsi Chaos Solitons & Fractals 4 (009) R. A. El-Nabulsi Chaos Solitons & Fractals 4 (009) R. A. El-Nabulsi Int. J. Mod. Geom. Meth. Mod. hys. 5(008)863.