10 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 Β© Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Potential Dependent Frictional Schrodinger Equation Ebtisam A. Mohameda*, Mubarak Dirarb, Ibrahim A. I. Hammadc, A. Elfakid, Ibrahim Alfakie aJazan University, Department of Physics, Jazan, Saudi Arabia aKordofan University, Department of Physics, El-Obeid, Sudan b,d,eSudan University of Science and Technology, Department of Physics, Khartoum, Sudan cAlzaiem Alazhari University, Department of Physics, Omdurman, Sudan aEmail: ebtisam99@yahoo.com bEmail: mubarakdirar@gmail.com cEmail: kshiky1986@gmail.com Abstract By treating particles as harmonic oscillator is obtained the friction energy related to the momentum. The energy and the corresponding Newtonian operator is found. This result in a new Schrodinger equation accounting for the effect of friction. This new equation shows that the energy and mass are quantized, if one treats particles as strings. The radioactive decay law and collision probability is also derived. Key words: friction; string; harmonic oscillator; radioactive decay law; collision probability. 1. Introduction Quantum mechanics include two independent formulations. The first formulation, called matrix mechanics, was developed by Heisenberg (1925) to describe atomic structure starting from the observed spectral lines of atoms. Heisenberg founded his theory on the notion that the only allowed values of photon are due to the transition of electrons between energy levels of atoms as discrete quanta. Expressing dynamical quantities such as energy, position, momentum and angular momentum in terms of matrices, he obtained an eigenvalue that describes the dynamics of microscopic systems; the diagonalization of the Hamiltonian matrix yields the energy spectrum and the state vectors of the system. Matrix mechanics was very successful in accounting for the discrete quanta of light emitted and absorbed by atoms.The second formulation, called wave mechanics, was due to SchrΓΆdinger (1926). It is a generalization of the de Broglie’s postulate. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 39, No 1, pp 10-18 11 De Broglie assumes that particles like electrons behave as waves. This method, describes the dynamics of microscopic matter by means of a wave equation, called the Schrodinger equation. Schrodinger equation describes successfully the behavior of atoms. Despite the remarkable successes of quantum equations, but they suffer from noticeable setbacks. For example, the quantum equation cannot differentiate between the behavior of two particles subjected to the same potential, but one moves in free space and the other moves inside matter. This is in direct conflict whit experimental observations [1].This is since the particle move in a media is affected by fraction. Friction is the force resisting the relative motion of solid surfaces, fluid layers, as an example of friction; we can consider a body moving rapidly against a stationary background. Its kinetic energy is dissipated, generating heat and entropy in the environment, the amount of dissipation is proportional to the velocity [2]. The fraction is observed in micro-mechanical systems at low temperatures, in superfluid theory, and even in quantum cosmology [3]. In particular, friction is the term widely used in descriptions of ion collisions. Frictional forces depend in this case on position and their range is comparable with the nuclear radius [4]. Thus, this work is concerned with deriving Schrodinger equation for quantum system suffering from fraction. This is done in section 2. Applications for harmonic oscillator and radioactive decay low are in sections (3) and (4) respectively [5, 6, 7]. 2. Schrodinger equation for frictional medium According to Plank and de Broglie hypothesis the quantum quanta are treated as wave packets. Pure waves is a wave packet consisting of single wave having specific wave length .while a localized particle is a wave packet having a very large of interfering waves having different wave lengths . This means that any quantum system is a single or aggregate of oscillators. Moreover, according to string theory matter building blocks are treated as vibrating string. Motivated by all there hypothesis, the energy dissipated by fraction can be derived consider now a fractional force Ff in terms of mass m, relaxation time 𝜏𝜏 and velocity Ο… to be Ef = mv 𝜏𝜏 (1) Considering matter building blocks as oscillators Ο… = Ο…Β°eiwt (2) Thus, the displacement is given by: π‘₯π‘₯ = �𝑣𝑣𝑣𝑣𝑣𝑣 = 𝜐𝜐𝜊𝜊 �𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖𝑣𝑣𝑣𝑣 = 𝜐𝜐𝜊𝜊 𝑖𝑖𝑖𝑖 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑣𝑣 𝑖𝑖𝑖𝑖 (3) The total dissipative energy 𝐸𝐸𝑓𝑓 is given by: Ef = �𝐹𝐹𝑓𝑓 .𝑣𝑣π‘₯π‘₯ = m iw𝜏𝜏 οΏ½πœπœπ‘£π‘£πœπœ = π‘šπ‘šπ‘£π‘£2 2π‘–π‘–π‘–π‘–πœπœ = π‘–π‘–π‘šπ‘šπ‘£π‘£2 2𝑖𝑖2πœπœπ‘–π‘– = βˆ’π‘–π‘–π‘šπ‘šπ‘£π‘£2 2π‘–π‘–πœπœ = βˆ’ 𝑖𝑖 π‘–π‘–πœπœ οΏ½ 1 2 π‘šπ‘šπ‘£π‘£2οΏ½ = βˆ’π‘–π‘– π‘–π‘–πœπœ οΏ½ 𝑃𝑃2 2π‘šπ‘š οΏ½ (4) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 39, No 1, pp 10-18 12 But according to Newtonian mechanics the total energy can be expressed in terms of the kinetic and potential energy V in the form 𝐸𝐸 = 𝐾𝐾 + 𝑉𝑉 = 𝑃𝑃 2 2𝑀𝑀 + 𝑉𝑉 (5) Thus according to Eq. (5) and Eq. (4) 𝐸𝐸𝑓𝑓 is given by 𝐸𝐸𝑓𝑓 = βˆ’π‘–π‘– π‘–π‘–πœπœ (E βˆ’ V) (6) But using plank hypothesis the energy E is given by: 𝐸𝐸 = β„πœ”πœ” (7) In view of Eqs. (6) and (7) the frictional energy is given by Ef = βˆ’π‘–π‘–β„ β„π‘–π‘–πœπœ (E βˆ’ V) = 𝑖𝑖ℏ 𝜏𝜏𝐸𝐸 (V βˆ’ E) Ef = 𝑖𝑖ℏ 𝜏𝜏 οΏ½V 𝐸𝐸 βˆ’ 1οΏ½ (8) Thus the Hamiltonian classical relation for a particle in a fractional medium is given by 𝐸𝐸 = 𝐻𝐻 = 𝑃𝑃2 2π‘šπ‘š + 𝑉𝑉 + 𝑖𝑖ℏ 𝜏𝜏 �𝑉𝑉 𝐸𝐸 βˆ’ 1οΏ½ = 𝑃𝑃2 2π‘šπ‘š + 𝑉𝑉 + 𝑖𝑖ℏ 𝜏𝜏 οΏ½π‘‰π‘‰βˆ’πΈπΈ 𝐸𝐸 οΏ½ (9) Therefore 𝐸𝐸2 = �𝑃𝑃 2 2π‘šπ‘š + 𝑉𝑉�𝐸𝐸 + 𝑖𝑖ℏ 𝜏𝜏 (𝑉𝑉 βˆ’ 𝐸𝐸) (10) To find the Schrodinger equation corresponding to this relation multiplies both sides of Eq. (10) by Ξ¨ to get: 𝐸𝐸2Ξ¨ = �𝑃𝑃 2 2π‘šπ‘š + 𝑉𝑉�𝐸𝐸Ψ + 𝑖𝑖ℏ 𝜏𝜏 (𝑉𝑉 βˆ’ 𝐸𝐸)Ξ¨ (11) Considering the wave function Ξ¨ = 𝐴𝐴𝑒𝑒 𝑖𝑖 ℏ .(π‘π‘π‘π‘βˆ’πΈπΈπ‘–π‘–) (12) Hence πœ•πœ•Ξ¨ πœ•πœ•π‘£π‘£ = βˆ’ 𝑖𝑖 ℏ EΞ¨ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 39, No 1, pp 10-18 13 𝐸𝐸Ψ = 𝑖𝑖ℏ πœ•πœ•Ξ¨ πœ•πœ•π‘£π‘£ πœ•πœ•2Ξ¨ πœ•πœ•π‘–π‘–2 = βˆ’ 𝑖𝑖2 ℏ2 𝐸𝐸2Ξ¨ (13) βˆ’β„2 πœ•πœ• 2Ξ¨ πœ•πœ•π‘–π‘–2 = 𝐸𝐸2Ξ¨ (14) Similarly differentiating the wave function respect to x yields πœ•πœ•Ξ¨ πœ•πœ•π‘₯π‘₯ = 𝑖𝑖 ℏ PΞ¨ 𝑖𝑖ℏ πœ•πœ•Ξ¨ πœ•πœ•π‘£π‘£ = PΞ¨ πœ•πœ•2Ξ¨ πœ•πœ•π‘₯π‘₯2 = 𝑖𝑖2 ℏ2 𝑃𝑃2Ξ¨ βˆ’β„2 πœ•πœ• 2Ξ¨ πœ•πœ•π‘π‘2 = βˆ’β„2βˆ‡2Ξ¨ = 𝑃𝑃2Ξ¨ (15) Thus inserting Eqs. (13), (14) and (15) into Eq. (11) yields βˆ’β„2 πœ•πœ•2Ξ¨ πœ•πœ•π‘₯π‘₯2 = οΏ½βˆ’ ℏ2 2π‘šπ‘š βˆ‡2 + 𝑉𝑉� 𝑖𝑖ℏ πœ•πœ•Ξ¨ πœ•πœ•π‘£π‘£ + 𝑖𝑖ℏ 𝜏𝜏 οΏ½βˆ’π‘–π‘–β„ πœ•πœ•Ξ¨ πœ•πœ•π‘£π‘£ + 𝑉𝑉Ψ� βˆ’β„2 πœ•πœ• 2Ξ¨ πœ•πœ•π‘π‘2 = 𝑖𝑖ℏ οΏ½βˆ’ ℏ2 2π‘šπ‘š βˆ‡2 + 𝑉𝑉� πœ•πœ•Ξ¨ πœ•πœ•π‘–π‘– + ℏ2 𝜏𝜏 πœ•πœ•Ξ¨ πœ•πœ•π‘–π‘– + 𝑖𝑖ℏ 𝜏𝜏 𝑉𝑉Ψ (16) 3. Harmonic oscillator solution To see how fraction force consider the solution of Eq. (12) in the form Ξ¨ = π‘’π‘’βˆ’π‘–π‘– 𝐸𝐸 ℏ𝑖𝑖𝑒𝑒(𝑣𝑣) = 𝑓𝑓(𝑣𝑣)𝑒𝑒(𝑣𝑣) = 𝑓𝑓𝑒𝑒 πœ•πœ•Ξ¨ πœ•πœ•π‘£π‘£ = βˆ’π‘–π‘– 𝐸𝐸 ℏ 𝑓𝑓𝑒𝑒 πœ•πœ•2Ξ¨ πœ•πœ•π‘π‘2 = 𝑖𝑖2𝐸𝐸2 ℏ2 𝑓𝑓𝑒𝑒 = βˆ’πΈπΈ2 ℏ2 𝑓𝑓𝑒𝑒 (17) A direct substitution in Eq. (16) gives 𝐸𝐸2𝑓𝑓𝑒𝑒 = 𝑖𝑖ℏ οΏ½βˆ’ ℏ2 2π‘šπ‘š βˆ‡2𝑒𝑒 + 𝑉𝑉𝑒𝑒� 𝑓𝑓 οΏ½βˆ’π‘–π‘–πΈπΈ ℏ οΏ½ βˆ’ 𝑖𝑖 𝐸𝐸ℏ 2 β„πœπœ 𝑓𝑓𝑒𝑒 + 𝑖𝑖 ℏ 𝜏𝜏 𝑉𝑉𝑓𝑓𝑒𝑒 (18) Dividing both sides of Eq. (18) by f yields American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 39, No 1, pp 10-18 14 𝐸𝐸2𝑒𝑒 = +𝐸𝐸 οΏ½βˆ’ ℏ2 2π‘šπ‘š βˆ‡2𝑒𝑒 + 𝑉𝑉𝑒𝑒� βˆ’ 𝑖𝑖 𝐸𝐸ℏ 𝜏𝜏 𝑒𝑒 + 𝑖𝑖 ℏ 𝜏𝜏 𝑉𝑉𝑒𝑒 (19) Dividing both sides of Eq. (19) by +E yields �𝐸𝐸 + 𝑖𝑖ℏ 𝜏𝜏 οΏ½ 𝑒𝑒 = βˆ’ ℏ2 2π‘šπ‘š βˆ‡2𝑒𝑒 + 𝑉𝑉 οΏ½1 + 𝑖𝑖ℏ 𝜏𝜏𝐸𝐸 �𝑒𝑒 βˆ’ ℏ2 2π‘šπ‘š βˆ‡2𝑒𝑒 + 𝑐𝑐1𝑉𝑉𝑒𝑒 = 𝐸𝐸1𝑒𝑒 (20) Where 𝑐𝑐1 = 1 + 𝑖𝑖ℏ 𝜏𝜏𝐸𝐸 𝐸𝐸1 = 𝐸𝐸 + 𝑖𝑖ℏ 𝜏𝜏 (21) For harmonic oscillator one finds 𝑉𝑉 = 1 2 π‘˜π‘˜π‘₯π‘₯2 (22) Thus substituting this expression in Eq. (20) gives βˆ’ ℏ2 2π‘šπ‘š βˆ‡2u + 𝑐𝑐1 1 2 kπ‘₯π‘₯2 = 𝐸𝐸1u (23) Let now π‘˜π‘˜πœŠπœŠ = 𝑐𝑐1π‘˜π‘˜ (24) Therefore equation (23) became βˆ’ ℏ2 2π‘šπ‘š βˆ‡2u + 1 2 π‘˜π‘˜πœŠπœŠπ‘₯π‘₯2 = 𝐸𝐸1u (25) Thus substituting Eq. (21) into Eq. (25) gives 𝐸𝐸1 = 𝐸𝐸 + 𝑖𝑖ℏ 𝜏𝜏 = �𝑛𝑛 + 1 2 οΏ½ β„πœ”πœ” (26) 𝐸𝐸 = �𝑛𝑛 + 1 2 οΏ½ β„πœ”πœ” βˆ’ 𝑖𝑖ℏ 𝜏𝜏 (27) The frequency is given according to Eq. (24) and Eq. (21) to be π‘˜π‘˜πœŠπœŠ = π‘šπ‘šπœ”πœ”2 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 39, No 1, pp 10-18 15 𝑐𝑐1π‘˜π‘˜ = οΏ½1 + 𝑖𝑖ℏ 𝜏𝜏𝐸𝐸 οΏ½ π‘˜π‘˜ = π‘šπ‘šπœ”πœ”2 �𝐸𝐸 + 𝑖𝑖ℏ 𝜏𝜏 οΏ½ π‘˜π‘˜ = π‘šπ‘šπœ”πœ”2𝐸𝐸 (28) Thus 𝐸𝐸 = οΏ½π‘šπ‘šπœ”πœ”2 π‘˜π‘˜ βˆ’ 1οΏ½ βˆ’1 𝑖𝑖ℏ 𝜏𝜏 (29) From (3-12) and (3.13) 0 = βˆ’ π‘šπ‘šπœ”πœ”2 π‘˜π‘˜ + �𝑛𝑛 + 1 2 οΏ½ β„πœ”πœ” π‘šπ‘š = οΏ½1 + 𝑖𝑖 𝜏𝜏(𝑛𝑛+12) οΏ½ π‘˜π‘˜ πœ”πœ”2 (30) Thus, from Eq. (30) one finds the mass is quantized 4. radioactive decay low and collision probability Consider now Eq. (16) for constant potential π‘‰π‘‰πœŠπœŠ Using the separation of variables let the wave function Ξ¨ be in the form Ξ¨(π‘Ÿπ‘Ÿ, 𝑣𝑣) = 𝑓𝑓(𝑣𝑣)𝑒𝑒(π‘Ÿπ‘Ÿ) = 𝑓𝑓𝑒𝑒 (31) A direct substitution of equation (31) in equation (16) gives βˆ’β„2𝑒𝑒 πœ•πœ•2𝑓𝑓 πœ•πœ•π‘£π‘£2 = οΏ½βˆ’ ℏ2 2π‘šπ‘š βˆ‡2 + π‘‰π‘‰πœŠπœŠοΏ½π‘’π‘’ �𝑖𝑖ℏ πœ•πœ•π‘“π‘“ πœ•πœ•π‘£π‘£ οΏ½ + 𝑖𝑖ℏ 𝜏𝜏 π‘‰π‘‰πœŠπœŠπ‘’π‘’π‘“π‘“ + ℏ2 𝜏𝜏 𝑒𝑒 πœ•πœ•π‘“π‘“ πœ•πœ•π‘£π‘£ Thus οΏ½βˆ’β„2 πœ•πœ• 2𝑓𝑓 πœ•πœ•π‘–π‘–2 βˆ’ 𝑖𝑖ℏ 𝜏𝜏 π‘‰π‘‰πœŠπœŠπ‘“π‘“ βˆ’ ℏ2 𝜏𝜏 πœ•πœ•π‘“π‘“ πœ•πœ•π‘–π‘– οΏ½ 𝑒𝑒 = οΏ½βˆ’ ℏ2 2π‘šπ‘š βˆ‡2 + π‘‰π‘‰πœŠπœŠοΏ½ 𝑒𝑒 �𝑖𝑖ℏ πœ•πœ•π‘“π‘“ πœ•πœ•π‘–π‘– οΏ½ (32) Divide both sides of Eq. (32) by fu to get �𝑖𝑖ℏ πœ•πœ•π‘“π‘“ πœ•πœ•π‘–π‘– οΏ½ βˆ’1 οΏ½βˆ’β„2 πœ•πœ• 2𝑓𝑓 πœ•πœ•π‘–π‘–2 βˆ’ π‘–π‘–π‘‰π‘‰πœŠπœŠβ„ 𝜏𝜏 𝑓𝑓 βˆ’ ℏ2 𝜏𝜏 πœ•πœ•π‘“π‘“ πœ•πœ•π‘–π‘– οΏ½ = 1 𝑒𝑒 οΏ½βˆ’ ℏ2 2π‘šπ‘š βˆ‡2 + π‘‰π‘‰πœŠπœŠοΏ½ 𝑒𝑒 = 𝐸𝐸𝜊𝜊 (33) Taking the time part of Eq. (33) only gives βˆ’β„2 πœ•πœ• 2𝑓𝑓 πœ•πœ•π‘–π‘–2 βˆ’ π‘–π‘–π‘‰π‘‰πœŠπœŠβ„ 𝜏𝜏 𝑓𝑓 βˆ’ ℏ2 𝜏𝜏 πœ•πœ•π‘“π‘“ πœ•πœ•π‘–π‘– = π‘–π‘–β„πΈπΈπœŠπœŠ πœ•πœ•π‘“π‘“ πœ•πœ•π‘–π‘– (34) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 39, No 1, pp 10-18 16 Consider the case when the potential vanishes π‘‰π‘‰πœŠπœŠ = 0 (35) Hence βˆ’β„2 πœ•πœ• 2𝑓𝑓 πœ•πœ•π‘–π‘–2 βˆ’ ℏ2 𝑖𝑖 πœ•πœ•f πœ•πœ•π‘–π‘– = iβ„πΈπΈπœŠπœŠ πœ•πœ•f πœ•πœ•π‘–π‘– (36) Consider now a solution f = Aeβˆ’ i ℏEt πœ•πœ•f πœ•πœ•π‘£π‘£ = βˆ’π‘–π‘– ℏ 𝐸𝐸𝑓𝑓 πœ•πœ•2𝑓𝑓 πœ•πœ•π‘–π‘–2 = + 𝑖𝑖2 ℏ2 𝐸𝐸2𝑓𝑓 = βˆ’πΈπΈ2 ℏ2 𝑓𝑓 (37) Inserting Eq. (37) in Eq. (36) yields 𝐸𝐸2𝑓𝑓 + iℏ 𝜏𝜏 𝐸𝐸𝑓𝑓 = π‘–π‘–β„πΈπΈπœŠπœŠ οΏ½βˆ’ 𝑖𝑖 ℏ 𝐸𝐸𝑓𝑓� (38) Dividing both sides of Eq. (38) by f gives 𝐸𝐸2 + 𝑖𝑖ℏ 𝜏𝜏 𝐸𝐸 = 𝐸𝐸𝜊𝜊𝐸𝐸 (39) Rearranging both sides of Eq. (39) gives 𝐸𝐸2 = �𝐸𝐸𝜊𝜊 βˆ’ 𝑖𝑖ℏ 𝜏𝜏 οΏ½ 𝐸𝐸 (40) Dividing both sides of Eq. (4) by E gives 𝐸𝐸 = �𝐸𝐸𝜊𝜊 βˆ’ 𝑖𝑖ℏ 𝜏𝜏 οΏ½ (41) Inserting Eq. (41) in Eq. (47) gives 𝑓𝑓 = 𝐴𝐴𝑒𝑒 βˆ’π‘–π‘– ℏ οΏ½πΈπΈπœŠπœŠβˆ’ 𝑖𝑖ℏ 𝜏𝜏 �𝑖𝑖 = 𝐴𝐴𝑒𝑒 .βˆ’ 𝑑𝑑 πœπœπ‘’π‘’βˆ’ 𝑖𝑖 β„πΈπΈπœŠπœŠπ‘–π‘– Hence 𝑓𝑓 = 𝐴𝐴𝑒𝑒 βˆ’π‘‘π‘‘ 𝜏𝜏 π‘’π‘’βˆ’ 𝑖𝑖 β„πΈπΈπœŠπœŠπ‘–π‘– (42) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 39, No 1, pp 10-18 17 Since the probability and number of particles are given by 𝑛𝑛 = |𝑓𝑓|2 = 𝑓𝑓𝑓𝑓\ = 𝐴𝐴2𝑒𝑒 βˆ’2𝑑𝑑 𝜏𝜏 (43) Eq. (43) is the ordinary radioactive decay low with πœ†πœ† = 2 𝜏𝜏 , π‘›π‘›πœŠπœŠ = 𝐴𝐴2 (44) i.e. 𝑛𝑛 = π‘›π‘›πœŠπœŠπ‘’π‘’βˆ’πœ†πœ†π‘–π‘– (45) This expression also gives collision probability p with 𝑝𝑝 = 𝑛𝑛 π‘π‘πœŠπœŠ = 𝐴𝐴2 𝜏𝜏𝜊𝜊 = 𝜏𝜏 2⁄ (46) To get 𝑝𝑝 = π‘π‘πœŠπœŠ 𝑒𝑒 βˆ’π‘‘π‘‘ 𝜏𝜏𝜊𝜊 (47) Eq. (47) is the ordinary collision probability relation. 5. Discussion New Schrodinger equation for frictional medium was derived by using de Broglie hypothesis about the wave nature of atomic particles, besides assuming that particles are vibrating strings. The two hypotheses require that for frictional medium the classical energy is that of a harmonic oscillator [see Eqs. (2), (3), (4) and (5)] Using canonical quantization method by replacing the momentum and energy with their corresponding operators, [see Eqs. (12), (13) and (14)], a new frictional Schrodinger equation was derived [see Eq. (15) ]. It is very interesting to note that using separation of variables the time dependent part was used to derive radioactive decay law as shown by Eq. (43) and the collision probability as shown by Eq. (47). Solving for harmonic oscillator the expression for energy is quantized with additional frictional term [see Eq. (27)]. The solution of Eq. (27) shows that the mass is quantized. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 39, No 1, pp 10-18 18 6. Conclusion The Schrodinger equation for frictional medium shows that the energy and mass are quantized. It is also used to derive radioactive decay low and collision probability. References [1]. David S. Saxson, Elementary Quantum Mechanics, Dovered. P.cm, ISBN-10: 048648596X, (2012). [2]. Richard P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw–Hill, ISBN- 10: 0070206503, (1965). [3]. B. Bhushan, In Tribology and Mechanics of Magnetic Storage Devices, 2nd ed., Springer- verlag, New York, ISBN-13:978-1-4612-7517-6, (1990). [4]. RoumenTsekov, Quantum friction, Chin. Phys. Lett. 29. 120504 [arXiv 1203.2421], DOI: 10.1088/0256-307X/29/12/120504,( 2012). [5]. T. Srokowski, Position Dependent Friction In Quantum Mechanics , Institute of Nuclear Physics, Radzikoirskiego 152, PL-31-342 , unpublished, KrakΓ³w, Poland, (1972). [6]. Nouredine Zettili, Quantum Mechanics Concepts and Applications, 2nd ed, Jacksonville State University, Jacksonville, USA, ISBN: 978-0-470-02678-6, (2009). [7]. G. Aruldhas, Quantum Mechanics, 2nd ed, PHI leaming private limited. New Delhi, ISBN 10: 8120336356, (2009).