Lindblad Equation for Harmonic Oscillator: Uncertainty Relation Depending on Temperature

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1 Applied Mathematics, 07, 8, ISSN Online: ISSN Print: Lindblad Equation for Harmonic Oscillator: Uncertainty Relation Depending on Temperature Boris V Bondarev Moscow Aviation Institute, Moscow, Russia How to cite this paper: Bondarev, BV (07 Lindblad Equation for Harmonic Oscillator: Uncertainty Relation Depending on Temperature Applied Mathematics, 8, Received: December 7, 06 Accepted: October 30, 07 Published: November, 07 Copyright 07 by author and Scientific Research Publishing Inc This work is licensed under the Creative Commons Attribution International License (CC BY 40 Open Access Abstract Specific nonequilibrium states of the quantum harmonic oscillator described by the Lindblad equation have been hereby suggested This equation makes it possible to determine time-varying effects produced by statistical operator or statistical matri Thus, respective representation-varied equilibrium statistical matries have been found Specific mean value equations have been found and their equilibrium solutions have been obtained Keywords Statistical Operator, Density Matri, Lindblad Equation, Harmonic Oscillator Lindblad Equation Statistical operator ˆ or density matri is basically applied as the quantum mechanics tool, any information of the nonequilibrium process proceeding within the tested system may be gained from [] [] [3] [4] [5] When the process concerned proceeds within the system which fails interacting with its environment, statistical operator ˆ will satisfy Liouville-von Neumann equation as follows: iħˆ = Hˆ ˆ ( With provision for the fact that the system interacts with any environment, a new equation shall be produced [4]-[8] Lindblad is the first one who offered the equation describing interaction of the system with a thermostat [] This work is devoted to Markovian equation, which hereby describes nonequilibrium quantum harmonic oscillator performance DOI: 0436/am078 Nov, Applied Mathematics

2 We will write the kinetic equation for a quantum harmonic oscillator as follows: (,, (,, iħˆ = Hˆ ˆ + i A aˆ ˆ aˆ + aˆ ˆˆ a + i B aˆ ˆ aˆ + aˆ ˆˆ a ħ ħ, ( ( aˆ aˆ Hˆ = ħ ω + +, (3 A and B are constants Operator â is formulated as follows: ( κ aˆ = ipˆ m + ˆ ħ ω, (4 ω = κ m Equation ( is very precise to describe time varying state of the thermostat-interacted quantum harmonic oscillator and its equilibrium state Energy Representation Now, we will define the wave functions describing specific energy state ( The very functions satisfy the equation as follows: E ϕ ϕ Hˆ ϕ =, ( n n n n En ( n = ħ ω +, ( n = 0,,, As referred to energy representation, the matri elements of statistical operator ˆ will be formulated by the equation as follows: ˆϕ Wave functions satisfy the following equations: * mn = ϕn n d (3 aˆ ϕ = nϕ, aˆ ϕ = n+ ϕ (4 + n n n n+ With provision for the above formulas the following matri-formed equation ( is derived: ( n n + A ( n+ ( n + ( n+ n = iω nn nn n+, n + nn ( + + B nn n, n n n + nn (5 Now, we will write the equation for diagonal elements of density matri nn = wn, n w is the probability referred to oscillator state ϕ n The equation produced has the form as follows: w n = A n+ wn+ nwn+ Bnwn n+ wn (6 This kinetic equation describes particular harmonic oscillator state transitions In this case, there may be gained coefficients A and B as follows: ( ep( β ω, ( ep( β ω A= P ħ B= P ħ, (7 DOI: 0436/am Applied Mathematics

3 P is probability of transition per unit time; ( kt β = B is reciprocal temperature Equation (6 has specific oscillator state equilibrium distribution, which satisfies the following equation: n A n+ w nw + B w n + w = 0 n+ n n n (8 This equation is solved by the method as follows: under the following condition wn 3 Mean Value of Coordinate ( q n = q (9 q= BA= ep( βħ ω (0 Mean value b assigned by operator ˆb is defined as b = Tr b ˆ ˆ (3 For gaining mean value a the respective equation may be derived from Formula ( Using the equality of: + + aa ˆ ˆ aˆ aˆ =, (3 we will get the equation as follows: a = iω A+ B a (33 Now, we can find the derivatives from mean values and p By applying Formula (4 we will get: ( ω ( κ ip m+ κ = i A + B ip m+ Then, we will try to equate both the real and imaginary parts of this equation: = ( A B + p m, p (34 = κ ( A B p If we eliminate p from this set of equations, we can obtain the mean coordinate equation ω + A B + + A B = 0 (35 The above Equation (35 provides the following solution: = ( cosω + sinω ep ( t C t C t A Bt, (36 C and C are arbitrary constants 4 Mean Oscillator Energy Now, we will find the time derivative from aa + By applying the above equality (3 we will produce the following derivative from Equation (: aa + + A Baa + = B (4 DOI: 0436/am Applied Mathematics

4 We can define harmonic oscillator time-varying energy effects inserting the following formula in Equation (4: + aa= H ħ ω Thus, the following differential equation is derived: The solution of the equation is: H + A B H =? A+ B (4 ħω = ep ( + ħω ( A+ B ( A B H t C A B t, (43 C is an arbitrary constant The Equation (4 has specific stationary solution: H = ħ ω A+ B A B (44 Since constants A and B are related (7, the stationary solution obeys the formula as follows: If T = 0, then H ( ω βħω ( βħω e ( e ( cth( H = ħω + = ħ ω βħ ω (45 = ħ If T increases to infinity, then H = kt B 5 Kinetic Equation Epressed in Terms of Coordinate and Momentum Operators Let us epress the Equation ( in terms of operators ˆ and ˆp For this purpose, we will firstly write the Equation ( as follows: iħa( ( aˆ ˆ aˆ aˆ ˆ ˆ ˆ ˆ ˆ iħ ˆ = Hˆ ˆ ˆHˆ aa ˆ ˆˆ aa ˆ ˆ ˆ ˆaa ˆ ˆ + iħb a aa Since the energy operator is equal to: Hˆ pˆ m κ ˆ (5 = +, (5 we will insert it in Equation (5 along with Formula (4 to obtain the following one: iħ ˆ= pˆ ( m + κˆ ˆ ˆ pˆ ( m + κˆ ( ω ( ˆ ˆ ˆˆˆ + ˆˆ κ( ˆ ˆ ˆˆˆ ˆˆ i A+ B p pp p m+ + ˆ ˆˆ ˆ ˆˆ ˆ ( A B( p p + iħ 6 Coordinate Representation (53 In coordinate representation the density matri looks like this: = ( t,, The coordinate and momentum operators are: ˆ = pˆ = iħ, Using the above values we can write Equation (53 by the formula as follows: DOI: 0436/am Applied Mathematics

5 ( m( iκ ( ħ( = iħ t ( A B ( ħω ħ m κ( ( A B( (6 Physical interpretation of density matri implies that the following epression is the probability density Let us introduce new variables In this case (, ( t,, wt = (6 = +, = (63 = +, = Referring to density matri ( t,, will get the equation as follows: t = iħ m iκ ħ and using the above new variables we ( A B ( ħω ħ ( m κ ε( , (64 In this case ( A B ( A B th ( ε = ħω + = ħω βħ ω (65 ( t,,0 = wt (, (66 We will find the solution of Equation (64 as follows: Reciprocal transformation t,, = π f t, k, ep ik dk (67 = ( Taking into account (66 we will obtain: f t, k, f t,, ep ik d (68 f t,0,0 = f t,,0 d = wt, d= (69 Thus, in view of function (68 the following equation is formed: f= ħkm f + κ ħ f t k ( ħω ħ k ( m + κ + ε( + A+ B k k f (60 This equation has an equilibrium solution which satisfies both equations as follows: ħkm f + κ ħ f = 0, (6 κ + ε ( + f m k k k ħ k + f f = 0 (6 We will write the performance equation of the above Formula (6: m d ħ k = dkκ DOI: 0436/am Applied Mathematics

6 This equation has the solution as follows: ħ k m + κ = const This formula implies that the general solution of Equation (6 takes the form as follows: f = f ( E, E = k m + κ ħ We put this function into Equation (6 to get the following formula: df de+ f ε = 0 Taking into account condition (69 this equation has the following solution: = ep ( f E E ε Thus, the equilibrium solution of Equation (60 takes the form as follows: { } (, = ep ( + κ ( ε f k ħ k m (63 We will find the equilibrium density matri by formula (67: ik { } ( = ( π ħ + κ ( ε, ep k m e dk Integration brings us to the formula: (, = α πep α σ ( 4α, (64 α = σth βħω, σ = mω ħ Using Formulas (63 we will get the equation as follows: (, = α πep α( + 4 σ ( ( 4α (65 Using Formula (6 we will get equilibrium probability density [9] α ep( α w 7 Momentum Representation = π (66 In momentum representation the coordinate and momentum operators are: ˆ = iħ pˆ = p In this representation the density matri looks like this: = ( t, p, p enables to write Equation (53 as follows: p, ( ħ ( ħκ ( ( A B ( ħω ( p p m κħ ( p p ( A B( p p p p = i m p p + i t p p Physical interpretation of density matri ( t, p, p This (7 implies that the epres- DOI: 0436/am Applied Mathematics

7 sion (, ( t, p, p w t p = (7 is the probability density to detect the state when an oscillator have impulse p 8 Wigner Function In order to better understand the physical meaning of various kinetic state summands we will derive the equation for Wigner function w= wtp (,,, which is a quantum analog of classical distribution function and can be defined with the use of density matri ( t,, by the relation: wtp (,, = ( π ( t, + ħq, ħq ep( ipq dq (8 If the density matri depends on and, than wtp,, = π ħ t, =, ep ip ħ d (8 Reciprocal transformation Since there is Formula (69 = ( ħ t,, wt,, pep ip dp (83 t,,0 d =, Wigner function satisfies the normalization requirement wtp (,, dd p= (84 we will get the equation for Wigner function tw= pm w+ κ pw (85 + ( A+ B ( ħω ( m w+ κ pw+ ε( + + p p w ħ ħ From Equation (64 for density matri ( t,, The equation obtained is much different from its quantum analog of Fokker- Planck equation Summands containing derivatives w and w p can be interpreted as those describing phase space diffusion And still, it is necessary to add that it is rather hard to find physical meaning of the formula in parentheses that follows coefficient ε The equilibrium solution of Equation (85 should at the same time be a solution for the following equations: pm w+ κ w= 0, (86 κ p ε( p p ħ + ħ = (87 m w w p w 0 General solution of the Equation (86 is the function: w= w E, (88 E p m = + κ We will insert this function in Equation (87 and get the following equation: DOI: 0436/am Applied Mathematics

8 Ed w de + + µ E dw de+ µ w= 0, (89 th µ = ħω βħ ω (80 The solution of this equation is the function: e E w E = C µ (8 Wigner function can be obtained by Formula (8 inserting in it equilibrium function (64 We have: = α α σ ( α ( w, p πħ π ep 4 ep ip ħ d (8 Integration brings us to the equilibrium function { } wp, = µω π ep µ p m + κ (83 This function can be represented as: (, α π ep( α γ π ep( γ wp = p, (84 ( m γ = µ (85 Let us find the mean value p pwp, dd p Calculation gives the following formula: = (86 βħω βħω p = ħ 4 e + e (87 This formula leads to the following result If T = 0, the uncertainty is equal to p = ħ 4 If T, than p ( ktω = B 9 The Lindblad Equation Is a First Order Approimation In work [7] it was proved that the Lindblad equation can be derived the quantum equation for a small system that interacts with the equilibrium system from the equation equations of Liouville-von Neumannas The Lindblad equation can be written as iħ ˆ = Hˆ ˆ + λi Dˆ ( ˆ ħ +, (9 here λ is the order parameter Statistical operator ϱ write as well ˆ = ˆ + λ ˆ + (9 0 Let us substitute the operator (9 in Equation (9 We have iħ ˆ 0 + λ ˆ ˆ ˆ ˆ ˆ + = H, 0 + λ + + λi D( ˆ ˆ ħ 0 + λ + (93 DOI: 0436/am Applied Mathematics

9 If the order parameter λ = 0, we get the unperturbed statistical operator ˆ : 0 iħˆ ˆ ˆ 0 = H 0 (94 The first value of λ, gives Equilibrium values obey the equations: iħˆ ˆ ˆ ˆ = H + i D( ˆ ħ 0, (95 Hˆ ˆ 0 = 0, (96 ˆ ˆ ˆ ˆ H + iħd 0 = 0 (97 Equation (97 is equivalent to the equation D ˆ ˆ = 0 (98 So, the equilibrium values satisfy the following equations 0 Hˆ ˆ 0 = 0, Dˆ ( ˆ 0 = 0 (99 Eamples of such equations are the Equations (6, (6 and (86, (87 0 Conclusion We considered the equation proposed by Lindblad for the statistical operator describing nonequilibrium state of quantum harmonic oscillator From this equation, first we obtained the density matri equation in energy representation and the equation for the diagonal elements of this matri We formulated the epressions defining physical meaning of Lindblad equation coefficients Then we derived the equation for the mean value of a coordinate and found its general solution We demonstrated that the mean coordinate value eponentially decreases in time We obtained the equation for the mean oscillator energy and its general solution We found the equilibrium mean energy value This value is a monotonic decreasing function of temperature We formulated Lindblad equation using coordinate and momentum operators We obtained the density matri equation in the coordinate representation From this equation, we derived the formula for equilibrium density matri We wrote the density matri equation in the momentum representation We obtained Wigner function equation and found the respective equilibrium state function We found the uncertainty relation for various temperatures by applying Wigner equilibrium function References [] von Neumann, J (964 Mathematical Basis of Quantum Mechanics Nauka, Moscow [] Blokhintzev, DI (96 Fundamental Quantum Mechanics Higher School, Moscow [3] Landau, LD and Lifshits, EM (963 Quantum Mechanics Nauka, Moscow [4] Blum, K (98 Density Matri Theory and Application Plenum, New York and DOI: 0436/am Applied Mathematics

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