Ambiguities on the Hamiltonian formulation of the free falling particle with quadratic dissipation

Ambiguities on the Hamiltonian formulation of the free falling particle with quadratic dissipation

G. López, P. López, X. E. López, and R. I. Castro

Facultad de Ciencias de la UNAM
Apartado Postal 70-348, Coyoacán 04511 México D.F.
Abstract

For a free falling particle moving in a media which has quadratic velocity force effect on the particle, two equivalent constants of motion, with units of energy, two Lagrangians, and two Hamiltonians are deduced. These quantities describe the dynamics of the same classical system. However, their quantization and the associated statistical mechanics (for an ensemble of particles) describe two completely different quantum and statistical systems. This is shown at first order in the dissipative parameter.

PACS: 03.20.+i, 03.30.+p, 03.65.-w

1 Introduction

It is well known that the Lagrangian (therefore the Hamiltonian) formulation for some systems of more than one dimension may not exists (Douglas 1941). Fortunately for our study of the nature up to now, most of our physical systems have avoided this problem, and the whole quantum and statistical mechanics of non-dissipative systems can be given in terms of a Lagrangian or Hamiltonian formulation. Now, for dissipative systems there have been two main approaches. The first one consists of keeping the same Hamiltonian formalism for the whole system where the interacting background is included, as a result, one brings about a master equation with the dissipation and diffusion parameters appearing as part of the solution (Caldeira and Legget 1983, Unruh and Zurek 1989, and Hu et al 1992). This approach has its own merit, but it will not be followed in this paper. We will follow the second approach which consists in to obtain a phenomenological velocity depending Hamiltonian, representing a classical dissipative system, and to proceed to make the usual quantization (or statistical mechanics) with this Hamiltonian.

Within this last approach, one can, additionally, study the mathematical consistence of the the Hamiltonian formalism in quantum and statical mechanics. It is also known that even for one-dimensional systems, where the existence of their Lagrangian is guaranteed (Darboux 1894), the Lagrangian and Hamiltonian formulations are not free from problems (Havas 1973, Okubo 1980, Dodonov et al 1981, Marmon et al 1985, Glauber et al 1984, López 1998, and López 1999). One of the main problems is the implication on the quantization of the associated classical system when different Hamiltonians describe the same classical system (López 2002). This ambiguity has already been studied for the harmonic oscillator with dissipation and some general system (López 1996). In this paper, we want to show explicitly this ambiguity by studying the free falling particle within a medium which has the effect on the particle of producing the dissipation. This dissipation depends quadratically on the velocity of the particle. Firstly, two constants of motion are deduced for this system. Secondly, with these constants of motion two Lagrangian and two Hamiltonian are obtained using a known procedure (López 1996 and López and Hernández 1989). Finally, using the Hamiltonian expression at first order in the dissipation parameter, the resulting eigenvalues of their associated quantum Hamiltonian and their associated statistical mechanics properties (for an ensemble of particles) are shown.

2 Constants of Motion

The motion of the particle of mass falling under a constant gravitational force, , where is the constant acceleration due to gravity, which is within a dissipative medium which has the effect on the particle of producing a force proportional to the square of the velocity of the particle, for , can be described by the following autonomous dynamical system

 dxdt=v ,dvdt=−g+αmv2 , (1)

where the variable represents the vertical position of the particle, and represents its velocity. A constant of motion for this system is a function such that , i.e. it satisfies the following equation (López 1999)

 v∂K∂x+(−g+αmv2)∂K∂v=0 . (2)

The general solution of this equation is given by (John 1974)

 Kα(x,v)=G(C(x,v)) , (3)

where is an arbitrary function of the characteristic curve . This characteristic curve can be given in two different ways as

 C1=−mg2αln(1−αmgv2)+gx , (4)

or

 C2=(1−αmgv2)e−2αx/m . (5)

Considering that one must obtain the usual constant of motion (Energy) expression for equal to zero, the functionality of in Eq. (3) is determined for each above characteristic ( and ), and the following constants of motion are gotten

 K(1)α(x,v)=−m2g2αln(1−αmgv2)+mgx , (6)

and

 K(2)α(x,v)=m22α(−g+αmv2)e−2αx/m+m2g2α . (7)

Note that the following limit is gotten

 limα→0K(i)α=12mv2+mgx   i=1,2 . (8)

3 Lagrangians and Hamiltonians

Using the know expression (Kobussen 1979, Leuber 1987, Yan 1981, and López 1996),

 L(x,v)=v∫K(x,v)v2dv , (9)

to get the Lagrangian, the Lagrangian associated to Eq. (6) and Eq. (7) are

 L(1)α(x,v)=m√mgα v arctanh(√αmg v)+m2g2αln(1−αmgv2)−mgx (10)

and

 (11)

Their generalized linear momenta () are

 p(1)α=m√mgα arctanh(√αmg v) (12)

and

 p(2)α=mve−2αx/m . (13)

Thus, their associated Hamiltonians, , are given by

 H(1)α(x,p)=−m2g2αln[1−tanh2(√αmg pm)]+mgx (14)

and

 H(2)α(x,p)=m22α(−g+αp2m3e4αx/m)e−2αx/m+m2g2α , (15)

where one has made the substitution of and by just . One must note that the following limits are gotten

 limα→0L(i)α(x,v)=12mv2−mgx , (16)
 limα→0p(i)α=mv , (17)

and

 (18)

At first order in the dissipation parameter , the Hamiltonians are given by

 H(1)(x,p)=p22m+mgx−α12m4gp4 (19)

and

 H(2)(x,p)=p22m+mgx+α(xp2m2−gx2) . (20)

4 Quantization at first order in perturbation theory

Our Hamiltonians Eq. (19) and Eq. (20) can be written as

 (21)

where represents the non dissipative part of the Hamiltonian,

 H0(x,p)=p22m+mgx , (22)

and represents the contribution of the dissipation at first order in ,

 W(1)(x,p)=−αp412gm4 (23)

and

 W(2)(x,p)=α(xp2m2−gx2) . (24)

The Schrödinger equation, represents an stationary problem. Therefore, in order to get the quantization of the system, one just need to solve the following eigenvalue problem

 ^H(i)Φn(x)=E(i)nΦn(x) , (25)

where is the associated Hermitian operator of Eq. (21). Of course, one must not allow the particle to go beyond down the surface level. Thus, Eq. (22) is representing the Hamiltonian of the quantum bouncer () (Gean-Banacloche 1999), where the eigenvalue problem

 ^H0(^x,^p)ψ(0)n(x)=E(0)nψ(0)n(x) (26)

has the eigenvectors and eigenvalues solution given by

 ψ(0)n(x)=Ai(z−zn)|Ai′(−zn)| (27)

and

 E(0)n=mglgzn . (28)

The functions and are the Airy function and its differentiation respect to . The variable is defined as , where is given by , and is the nth-zero of the Airy function (). In fact, the bouncing problem has already been studied for linear and quadratic dissipation (López 2004). For the later, the correction given to the eigenvalue problem using Eq. (24) at first order in perturbation theory is

 ⟨n|^W(2)|n⟩=α4gl2gz2n15 . (29)

Now, using the relation , the correction at first order in perturbation due to Eq. (23) is given by

 ⟨n|^W(1)|n⟩=−αℏ4z2n60gm4l4g . (30)

Therefore, for the same classical dynamical system we have two different associated quantum systems which have completely different quantum dynamics, which is shown through the eigenvalues

 E(1)n=E(0)n+α4gl2gz2n15 (31)

and

 E(2)n=E(0)n−αℏ4z2n60gm4l4g . (32)

5 Classical Statistical model for dissipation

Consider a system of particles, where particles are small of mass , and particles are are big of mass (). The small particles move under the action of an external force with components and suffer collisions with the walls of the container which consists in a narrow-square shape pipe of cross sectional area . In addition, each small particle can have occasional (stochastic) collision with the big particles, when they are added, establishes the dissipative medium where the big particles will move. The big particles move in this dissipative medium, and it is assumed that, since this type of collision does not occur frequently, its average effect may have neglected contribution on the dynamical macroscopic variables of the system. Newton’s equations of motion for this system can be written as

 m1¨q1ij=0j=1,…,N1; i=x,y (33)
 m1¨q1zj=−m1gj=1,…,N1 (34)
 m2¨q2ik=α(˙q2ik)2k=1,…,N2; i=x,y (35)
 (36)

where , and are the generalized coordinates, velocities and accelerations of the light-small () and heavy-gross () particles, and the parameter characterizes the dissipative medium. The Hamiltonian associated the the motion of 1-particle, Eq. (33) and Eq. (34), is given by (López et al 1997)

 H1;x,y,z=N1∑j=1[3∑i=1p21ij2m1+m1gq1zj] . (37)

The Hamiltonian associated to Eq. (35) is written as

 H2;x,y=N2∑k=12∑i=1p22ik2m2exp(2αq2ikm2) , (38)

and, as we have seen in section 3, there are at least two Hamiltonians associated to Eq. (36) which are given by

 H(1)2;z=N2∑k=1{−gm222αln[1−tanh2(√αm2g p2zkm2)]+m2gq2zk} (39)

and

 (40)

Therefore, one has two different Hamiltonians to describe the same system, and , which are written as

 H(1)=N1∑j=1[3∑i=1p21ij2m1+m1gq1zj]+N2∑k=13∑i=1p22ik2m2exp(2αq2ikm2)
 (41)

and

 H(2)=N1∑j=1[3∑i=1p21ij2m1+m1gq1zj]+N2∑k=13∑i=1p22ik2m2exp(2αq2ikm2) +N2∑k=1{−gm222αln[1−tanh2(√αm2g p2zkm2)]+m2gq2zk} .
 (42)

Then, one can calculate for each Hamiltonian the canonical partition function (Toda et al 1998) which is associated to the same statistical system,

 Z(i)=1N1!N2!h3N∫exp(−βH(i)) dqdp  i=1,2 , (43)

where is defined as with being the Boltzman’s constant and T being the temperature, and the integration is carried out over all the coordinates and linear momenta of the two particles. The integration of momenta is carried out in the intervale . The integration on the transverse coordinates () is carried out in the intervale , and the integration of the vertical coordinate is carried out in the intervale . The partition functions for both cases are given by

 Z(1)=L2N1N1!N2!h3N(2πm1β)3N1/2(1−e−βm1gzβm1g)N1(2πm2β)3N2/2(m2α)2N2×
 (44)

and

 Z(2) = L2N1N1!N2!h3N(2πm1β)3N1/2(1−e−βm1gzβm1g)N1(2πm2β)N2(m2α)2N2 ×(e−αLm2−1)2N2(1−e−βm2gzβm2g)N2⎡⎢ ⎢ ⎢ ⎢⎣√παm32g Γ(βm22g2α)Γ(βm22g2α+12)⎤⎥ ⎥ ⎥ ⎥⎦N2 .
 (45)

The system has two internal energies, ,

 U(1)=(5N12+2N2)1β−N1m1gze−βm1gz1−e−βm1gz−N2m22g2α−N2f′(β)f(β) (46)

and

 U(2) = (5N12+2N2)1β−N1m1gze−βm1gz1−e−βm1gz−N2m2gze−βm2gz1−e−βm2gz −N2m22g2α[ψ(βm22g2α)−ψ(βm22g2α+12)] ,
 (47)

where the function () has been defined as

 f(β) = Erfi⎛⎝√βgm222α e−αz/m2⎞⎠−Erfi⎛⎝√βgm222α⎞⎠ ,
 (48)

and is the complex error function which can be expressed in the form of the Dawson’s integral, , and is the digamma function, . Thus, one can have two heat capacity expressions for the system, ,

 C(1)V=(5N12+2N2)k−N1k(m1gzβ)2e−m1gzβ(1−e−m1gzβ)2+N2kβ2(f′′(β)f(β)−(f′(β))2(f(β))2) (49)

and

 C(2)V=(5N12+2N2)k−N1k(m1gzβ)2e−m1gzβ(1−e−m1gzβ)2−N2k(m2gzβ)2e−m2gzβ(1−e−m2gzβ)2 +N2k(m22gβ2α)2[ψ(1)(m22gβ2α)−ψ(1)(m22gβ2α+12)] ,
 (50)

where is the trigamma function, . Figure 1 shows the difference as a function of . As one can see, this difference is not small at low temperatures (high values). From lower than about 2100, is higher than , and the situation is reversed for higher values. This difference seems to have an important implication related with the ergodic hypothesis (Toda et al 1998). Assuming the validity of the hypothesis, one would expect not difference at all on the calculated heat capacities (or internal energies) since averaging over the time variable must bring about the same value for both Hamiltonians (they represent the same dynamical system). However, averaging over the canonical ensemble must be different if the Hamiltonians are different. This ambiguity will remain when quantum canonical ensemble is considered (using Eq. (31) and Eq. (32)) for quantum statical analysis of the system.

6 Conclusions

We have shown two constants of motion, two Lagrangians, and two Hamiltonians for a free falling particle moving in a media with quadratic velocity dissipative force. These quantities describe the same dynamics of the classical system, but their quantization and the associated statistical mechanics (for an ensemble of particles) describe two different quantum and statistical dynamics. We have showed this at first order in the dissipative parameter and at first order in perturbation theory. There is still a point which reamins to to study and has to deal with quasi-classical limit. The question is whether or not both quantum Hamiltonians, Eq. (21), describes the same quasi-classical dynamics () and coincides with the classical dynamics in this limit. We will deal with this problem and hope to report some results soon.

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Figure Captions
Difference of the heat capacities as a function of for , , and

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