Path Integrals with Μ1(X) and Μ2(X) the ﬁrst and Second Moments of the Transition Probabilities
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1 Introduction to path integrals with ¹1(x) and ¹2(x) the ¯rst and second moments of the transition probabilities. v. January 22, 2006 Another approach is to use path integrals. We will use Phys 719 - M. Hilke the example of a simple brownian motion (the random walk) to illustrate the concept of the path integral (or Wiener integral) in this context. CONTENT ² Classical stochastic dynamics DISCRETE RANDOM WALK ² Brownian motion (random walk) ² Quantum dynamics The discrete random walk describes a particle (or per- son) moving along ¯xed segments for ¯xed time intervals ² Free particle (of unit 1). The choice of direction for each new segment ² Particle in a potential is random. We will assume that only perpendicular di- rection are possible (4 possibilities in 2D and 6 in 3D). ² Driven harmonic oscillator The important quantity is the probability to reach x at 0 0 ² Semiclassical approximation time t, when the random walker started at x at time t : ² Statistical description (imaginary time) P (x; t; x0; t0): (3) ² Quantum dissipative systems For this probability the following normalization condi- tions apply: INTRODUCTION X 0 0 0 0 P (x; t; x ; t) = ±x;x0 and for (t > t ) P (x; t; x ; t ) = 1: Path integrals are used in a variety of ¯elds, including x stochastic dynamics, polymer physics, protein folding, (4) ¯eld theories, quantum mechanics, quantum ¯eld theo- This leads to ries, quantum gravity and string theory. The basic idea 1 X is to sum up all contributing paths. Here we will overview P (x; t + 1; x0; t0) = P (»; t; x0; t0); (5) the technique be starting on classical dynamics, in par- 2D h»i ticular, the random walk problem before we discuss the quantum case by looking at a particle in a potential, and where h»i represent the nearest neighbors of x. This is ¯nish with an example of a quantum dissipative system, solved by the Fourier transform: where this technique bares all its fruits. Z ¼ D 0 0 d k ikx ~ P (x; t; x ; t ) = D e P (k; t); (6) CLASSICAL STOCHASTIC DYNAMICS ¡¼ (2¼) The classical equation of motion for a particle in a which leads to ﬂuctuating environment is simply given by D 1 X 0 P~(k; t+1) = cos(k )P~(k; t) with P~(k; t0) = e¡ikx : dv D ¹ m = F (t) ¡ v=B (1) ¹=1 dt (7) where m is the mass, v the velocity and B describes the Hence, friction of the particle. This equation is the Langevin Ã !t¡t0 equation if F (t) is a ﬂuctuating force. These equa- Z ¼ D D d k 0 1 X tions are often di±cult to solve and one possibility is P (x; t; x0; t0) = eik(x¡x ) cos(k ) ; (2¼)D D ¹ to look at the probability density ½(x; t) instead. This ¡¼ ¹=1 approach leads to expressing ½ in terms of a master- (8) equation (Fokker-Planck) which takes on the following where we used 7 iteratively. Now, instead of using the form: probability we introduce the probability density p = P=aD, where a ia the step size, which will allow us to take the limit a ! 0, for the continuous case with the @½ @ @2 = ¡ (¹ ½) + (¹ ½) (2) variable change (k ! ak, x ! x=a). Hence, @t @x 1 @x2 2 2 Ã !(t¡t0)2D=a2 Z 1 D D d k 0 1 X p(x; t; x0; 0) = eik(x¡x ) cos(ak ) ; (9) (2¼)D D ¹ ¡1 ¹=1 | {z } 'e¡tk2 1 PD 2 2 where we used D ¹=1 cos(ak¹) ' (1 ¡ a k =2D + ¢ ¢ ¢ ). It is now possible to describe the evolution from an Hence, initial point (xi; 0) to a ¯nal point (xf ; t) by inserting an intermediate point (x1; t1) and integrating over it, i.e., Z 1 D 0 d k ik(x¡x0)¡tk2 p(x; t; x ; 0) = D e ¡1 (2¼) 0 2 1 ¡ (x¡x ) = e 4t ; (10) 4¼tD=2 which is the standard di®usion equation. The normaliza- Z D tion of the probability density requires p(xf ; t; xi; 0) = d x1p(xf ; t; x1; t1)p(x1; t1; xi; 0) (0 < t1 < tf ): Z (13) dDxp(x; t; x0; t0) = 1 (11) We can perform a similar decomposition for n interme- diate points, with xf = xn and x0 = xi. This then leads and to the summation over n paths, which is the main idea of the path integral. Thus, using (13) and (10) we lim p(x; t; x0; t0) = ±D(x ¡ x0) (12) obtain t!0 Z n¡1 2 Y D 1 Pn¡1 (xj+1¡xj ) d xj 1 ¡ p(x ; t; x ; 0) = e 4 j=0 tj+1¡tj : (14) f i 4¼(t ¡ t )D=2 D=2 j=1 j+1 j 4¼t1 Since Z nX¡1 (x ¡ x )2 t j+1 j = dsx_ 2; (15) t ¡ t j=0 j+1 j 0 we de¯ne the path integral as Z xf R ¡ 1 t dsx_ 2 p(xf ; t; xi; 0) = Dx(s)e 4 0 : (16) xi Here D denotes the multiple integrals de¯ned in (14) and represents the sum over all paths (Wiener integral). We will see in the next sections that a very similar expression exists in quantum mechanics. In classical systems this path integral is useful when there exist many ways to get form one point to the other (like in stochastic processes) and where it is then necessary to sum over all possible paths. Quantum Mechanics Feynman Path Integral ⇒ ⇒ with Written in eigenfunction basis: ⇒ Splitting the time evolution in 2: ⇒ Free particle: ⇒ using and ⇒ In terms of the classical action: (classical free path) ⇒ ⇒ Particle in a potential: { ⇒ using 0 and ⇒ c The Feynman path integral Driven harmonic oscillator: Classical path = Classical path+quantum fluctuations fluctuations ⇓ with ; Hence, Normalization: but and ⇒ Semiclassical approximation : 0 ⇒ ⇒ with Imaginary time (statistical properties): The equilibrium density operator: where is the quantum partition function But Shows that for hence with Applying this to the harmonic oscillator: Dissipative Systems: { Effective action: (the density matrix) (initially independent) (tracing out the environment) (tracing out the environment) ⇒ Path Integrals and Their Application to Dissipative Quantum Systems Gert-Ludwig Ingold Institut f¨ur Physik, Universit¨at Augsburg, D-86135 Augsburg to be published in “Coherent Evolution in Noisy Environments”, Lecture Notes in Physics, http://link.springer.de/series/lnpp/ c Springer Verlag, Berlin-Heidelberg-New York Path Integrals and Their Application to Dissipative Quantum Systems Gert-Ludwig Ingold Institut f¨ur Physik, Universit¨at Augsburg, D-86135 Augsburg, Germany 1 Introduction The coupling of a system to its environment is a recurrent subject in this collec- tion of lecture notes. The consequences of such a coupling are threefold. First of all, energy may irreversibly be transferred from the system to the environment thereby giving rise to the phenomenon of dissipation. In addition, the ﬂuctuating force exerted by the environment on the system causes ﬂuctuations of the system degree of freedom which manifest itself for example as Brownian motion. While these two eﬀects occur both for classical as well as quantum systems, there exists a third phenomenon which is speciﬁc to the quantum world. As a consequence of the entanglement between system and environmental degrees of freedom a coherent superposition of quantum states may be destroyed in a process referred to as decoherence. This eﬀect is of major concern if one wants to implement a quantum computer. Therefore, decoherence is discussed in detail in Chap. 5. Quantum computation, however, is by no means the only topic where the coupling to an environment is relevant. In fact, virtually no real system can be considered as completely isolated from its surroundings. Therefore, the phenom- ena listed in the previous paragraph play a role in many areas of physics and chemistry and a series of methods has been developed to address this situation. Some approaches like the master equations discussed in Chap. 2 are particularly well suited if the coupling to the environment is weak, a situation desired in quantum computing. On the other hand, in many solid state systems, the envi- ronmental coupling can be so strong that weak coupling theories are no longer valid. This is the regime where the path integral approach has proven to be very useful. It would be beyond the scope of this chapter even to attempt to give a com- plete overview of the use of path integrals in the description of dissipative quan- tum systems. In particular for a two-level system coupled to harmonic oscillator degrees of freedom, the so-called spin-boson model, quite a number of approx- imations have been developed which are useful in their respective parameter regimes. This chapter rather attempts to give an introduction to path integrals for readers unfamiliar with but interested in this method and its application to dissipative quantum systems. In this spirit, Sect. 2 gives an introduction to path integrals. Some aspects discussed in this section are not necessarily closely related to the problem of dissipative systems. They rather serve to illustrate the path integral approach and to convey to the reader the beauty and power of this approach. In Sect. 3 2 Gert-Ludwig Ingold we elaborate on the general idea of the coupling of a system to an environment. The path integral formalism is employed to eliminate the environmental degrees of freedom and thus to obtain an eﬀective description of the system degree of freedom. The results provide the basis for a discussion of the damped harmonic oscillator in Sect.