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Scattering of Waves in the Phase Space, Quantum Mechanics, and Irreversibility EJTP 12, No. 32 (2015) 43–60 Electronic Journal of Theoretical Physics Scattering of Waves in the Phase Space, Quantum Mechanics, and Irreversibility E. M. Beniaminov∗† Russian State University for the Humanities, Miusskaya sq. 6, Moscow, GSP-3, 125993, Russia Received 29 November 2014, Accepted 19 December 2014, Published 10 January 2015 Abstract: We give an example of a mathematical model describing quantum mechanical processes interacting with medium. As a model, we consider the process of heat scattering of a wave function defined on the phase space. We consider the case when the heat diffusion takes place only with respect to momenta. We state and study the corresponding modified Kramers equation for this process. We consider the consequent approximations to this equation in powers of the quantity inverse to the medium resistance per unit of mass of the particle in the process. The approximations are constructed similarly to statistical physics, where from the usual Kramers equation for the evolution of probability density of the Brownian motion of a particle in the phase space, one deduces an approximate description of this process by the Fokker–Planck equation for the density of probability distribution in the configuration space. We prove that the zero (invertible) approximation to our model with respect to the large parameter of the medium resistance, yields the usual quantum mechanical description by the Schr¨odinger equation with the standard Hamilton operator. We deduce the next approximation to the model with respect to the negative power of the medium resistance coefficient. As a result we obtain the modified Schr¨odinger equation taking into account dissipation of the process in the initial model, and explaining the decoherence of the wave function. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Quantum Mechanics; Kramers Equation; Waves in the Phase Space; Diffusion Scattering; Asymptotic Solutions; Decoherence PACS (2010): 03.65.-w; 03.65.Nk; 66.30.Ma; 03.65.Yz; 02.60.-x 1. Introduction In this paper we continue the study of the generalized Kramers equation introduced in the paper [1]. ∗ Email: [email protected] † Tel:+79162231515 44 Electronic Journal of Theoretical Physics 12, No. 32 (2015) 43–60 In [1], the generalized Kramers equation arose as a mathematical model of the scat- tering process of waves in the phase space under the action of medium in the heat equi- librium. This process is irreversible. In [1] it has been shown that, for certain parameters of the model, the process described by the generalized Kramers equation, is the compo- sition of a rapid transitional process and a slow process. The slow process, as proved in loc. cit., is approximately described by the Schr¨odinger equation used for description of quantum processes. The obtained approximation of the slow process is reversible. Thus, we have given an example of an irreversible process (heat scattering of waves in the phase space), for which it has been shown that the standard reversible quantum mechanical description of the motion of a particle arises as the asymptotic description of this process in the zeroth approximation. The purpose of the present paper is to construct an approximate equation describing the slow part of the process given by the generalized Kramers equation, with the precision showing dissipative effects in this process. As a result, we derive the modified Schr¨odinger equation taking into account dissipation of the process. The paper consists of four Sections and the Appendix. In the next Section we present the setting of the problem, the generalized Kramers equation for the heat scattering process of waves on the phase space, and the properties of the operators involved in the generalized Kramers equation. In Section 3 we recall our earlier results describing the process given by the generalized Kramers equation. We also give a method for the approximate description of this process in negative powers of the medium resistance coefficient, and state the main result of the paper, Theorem 4. In this Theorem we write down an equation describing the slow component of the studied process, taking into account dissipation. This equation is the modified Schr¨odinger equation, in which the process is not invertible. We show that consequences of the modified Schr¨odinger equation are the effects of decoherence and spontaneous jumps between the levels. In Section 4 we state some further possible directions of research and possibilities of comparison of the presented model with experiment. Proof of the main Theorem 4 of the paper is given in the Appendix. 2. Mathematical Setting of the Problem, and the Main Prop- erties of the Operators in the Equation Thus, we consider the following mathematical model of a process. A state of the process at each moment of time t ∈ R is given by a complex valued function ϕ[x, p, t] on the phase space (x, p) ∈ R2n,wheren is the dimension of the configuration space. Coordinates in the configuration space are given by a tuple of numbers x ∈ Rn, and momenta by a tuple of numbers p ∈ Rn. (Below everywhere we will write the arguments of a function in the square brackets, in order to distinguish an argument of a function from multiplication by a variable.) The generalized Kramers equation for a function ϕ[x, p, t] is defined as the following Electronic Journal of Theoretical Physics 12, No. 32 (2015) 43–60 45 equation: ∂ϕ = Aϕ + γBϕ, (1) ∂t n n 2 ∂V ∂ϕ pj ∂ϕ i pj where Aϕ = − − V − ϕ (2) ∂x ∂p m ∂x m j=1 j j j j=1 2 n ∂ ∂ ∂ϕ and Bϕ = pj + i ϕ + kBTm ; ∂p ∂x ∂p j=1 j j j m isthemassoftheparticle;V [x] is the potential function of external forces acting on the particle; i is the imaginary unit; is the Planck constant; γ = β/m is the medium resistance coefficient β per unit of mass of the particle; kB is the Boltzmann constant; T is the temperature of the medium. If we pass in equation (1) to the following dimensionless variables: √ p kBTm V [x] p = √ ,x = x, V [x]= , (3) kBTm kBT then in the new variables equation (1) takes the following form: ∂ϕ = Aϕ + γBϕ, (4) ∂t n n 2 kBT ∂V ∂ϕ ∂ϕ (pj) where Aϕ = − p − i V − ϕ (5) ∂x ∂p j ∂x j=1 j j j j=1 2 n ∂ ∂ ∂ϕ and Bϕ = p + i ϕ + . ∂p j ∂x ∂p j=1 j j j Note that the operator A is skew Hermitian, and the operator B is neither skew Hermitian neither self-adjoint. The operator B determines the scattering process of the wave function with respect to momenta and hence non-invertibility of the process. In this paper we consider the case when γ is a large quantity, i. e. the impact of the operator B on the general evolution process of the wave function is large. Properties of the operator B are presented in the following Theorem. Theorem 1. The operator B given by expression (5) has a full set of eigenfunc- tions (in the class of functions ϕ[x, p] tending to zero at infinity) with the eigenvalues 0, −1, −2,.... Respectively, the operator B is presented in the following form: ∞ B = − kPk, (6) k=0 where Pk are the projection operators onto the eigenspaces of the operator B with eigen- values −k. The projection operators Pk satisfy the relations PkPk = Pk,PkPk =0 for k = k ,PkB = B Pk = −kPk (7) 46 Electronic Journal of Theoretical Physics 12, No. 32 (2015) 43–60 ∞ and E = Pk, (8) k=0 where E is the identity operator. Proof of this Theorem will be given simultaneously with the proof of the following Theorem describing the form of the projection operators Pk. Hk p def H p ...H p Denote by k1...kn [ ] = k1 [ 1] kn [ n] the product of Hermite polynomials [7] of the corresponding variables, where k = k1 + ... + kn is the sum of degrees of the Hermite polynomials in the product. By definition, the Hermite polynomial is given by the expression p2 ∂ kj p2 H p def − − . kj [ j] =exp exp (9) 2 ∂pj 2 Let Sk1,...,kn and Ik1,...,kn be the operators given by the expressions def k ∂ ψk ,...,k = Sk ,...,k [ϕ] = H p + i ϕ[x ,p ]dp , (10) 1 n 1 n k1...kn ∂x Rn ϕ I ψ def k1,...,kn = k1,...,kn [ k1,...,kn ] = (11) 2 1 1 1 k − (p −s ) is(x−x) ... ψ x H p − s e 2 e ds dx . 3n/2 k1,...,kn [ ] k1...kn [ ] (2π) k1! kn! R2n Theorem 2. The projection operators Pk have the form k Pk = Ik1,...,kn Sk1,...,kn , (12) k1,...,kn=0 k1+...+kn=k S I and the operators k1,...,kn , k1,...,kn satisfy the relations S I ψ δ ...δ ψ , k1,...,kn k1,...,kn k1,...,kn = k1,k1 kn,kn k1,...,kn (13) δ k k k k where kiki equals 0 if i = i, and equals 1 if i = i. In particular, formulas (10), (11) and Theorem 2 imply that def ψ[x ]=S0¯[ϕ] = ϕ[x ,p]dp , (14) Rn 2 def 1 − (p −s ) is(x−x) ϕ0[x ,p]=I0¯[ψ] = ψ[x ]e 2 e ds dx , (15) (2π)3n/2 R2n 2 1 − (p −s ) is(x−x) P0ϕ = ϕ[x ,p ]dp e 2 e ds dx , (16) (2π)3n/2 R3n n 1 ∂ P1ϕ = p + i ϕ[x ,p ]dp × π 3n/2 j ∂x j=1 (2 ) j R3n 2 − (p −s ) is(x−x) (pj − sj)e 2 e ds dx .
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