chapter 9.pdf

1. Introduction

The problem of the dissipative systems from the quantum point of view has been of increasinginterest, but it is far from having a satisfactory solution (1). There are many problems wherethe dissipation has an important role such as in quantum optics, in quantum analysis of elds,in quantum gravity (2). Dissipation can be observed in interactions between two systems, theobserved system and another one often called reservoir or the bath, into which the energyows via an irreversiblemanner (3). The system is embedded in some environmentwhich is inprinciple unknown. For an effective description of such systems, we can use time dependentHamiltonians which in classical physics yield the proper equation of motion. If the frictionis a linear funcion of the velocity with friction constant the Hamiltonian is the well knownCaldirola - Kanai Hamiltonian

H = 12m

e t p2 +V(q)e t

Some special potentials have been studied in reference (4). It should be noted however, thatH is not constant of motion and does not represent the energy of the system.A second method for the dissipative systems, is based upon the procedure of doublingthe phase space dimensions. The new degrees of freedom may assumed to represent theenvironment which absorb the energy dissipated by the dissipative system. H. Bateman (5)has shown that one can double the numbers of degrees of freedom so as to deal with aneffective isolated system (6). In this article we assume that the coordinate operators q1 andq2 , of these two systems respectively do not commute, that is [q1, q2] = i where is a realparameter and plays an analogous role of h in standard quantum mechanics.For a manifold parametrized by the space - time coordinates x, the commutation relationscan be written as

[x, x] = i, , = 0, ..., d

where is a real antisymmetric tensor. This relation gives rise to the following space - timeuncertainty relations

(x) (x) 12||

The space - time non commutativity however violates causality, although could be consistentin string theory (9). Field theories with only space non commutativity, that is 0i = 0 have aunitary S matrix, on the other hand theories with space - time non commutativity 0i = 0 are

Non Commutative Quantum Mechanics in Time-Dependent Backgrounds

Antony Streklas Department of Mathematics, University of Patras, Patras

Greece

9

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2 Will-be-set-by-IN-TECH

not unitary. The non - commutativity of space, in the quantum eld theory, appears throughthe modication of the product of the elds which appear in the action, by the so - calledMoyal or star product (7),(8).The idea of non commutative space - time was presented by Snyder (10) in 1947, with respectto the need to regularize the divergence of the quantum eld theory. The idea was suggestedby Heisenberg in 1930. It was Jon von Neumann , who began studying this "pointlessgeometry". The physical theories of today they hold only in empty space which in realitydoes not exist. The seed of the original idea goes back to ancient Greek Stoik Philosophers,especially to Zeno the Kitieus, who contrary to the followers of Democritus, said that there isnot empty space (11).In the past few years there has been an increasing interest in the non commutative geometry,extensively developed by Connes (12), for the study ofmany physical problems. It has becomeclear that there is a strong connection of these ideas with string theories (13), nding manyapplications in solid state and particles physics. The non commutative geometry arises verynaturally from the Matrix theory (14).There is another immediate motivation of noncommutative theories in quantum gravity.Classical general relativity breaks down at Planck scale lp, where quantum effects becomeimportant.

lp =

hG

c3 1033cm tp =

lp

c 1044sec

Einsteins theory implies that gravity is equivalent to spacetime geometry. Hencequantum gravity should quantize spacetime and spacetime quantization requires to promotecoordinates to hermitian operators which do not commute. The wave function actuallybecomes an operator. The point is replaced by some "cell" and thus the spacetime becomesfuzzy at very short distances. It is apparent that this conicts with Lorentz invariance. Physicsnear Planck dimensions is not yet fully understood. At these dimensions the cone of light actsas if it were fuzzy and we cannot distinguish between the past and the future. A review ofrecent efforts to add a gravitational eld to non commutative models can be found in (15) andreferences therein.Non commutativity is the central mathematical concept expressing the uncertainty. The phasespace of ordinary quantummechanics is a well known example of non commuting space. Themomenta of a system in the presence of a magnetic eld act as non commuting operatorsas well. The canonical commutation relations of quantum mechanics introduce a cellularstructure in phase space. Noncommuting coordinates will introduce a cellular structure inordinary space as well, similar to a lattice structure, which a priory lead to a nonlocal theory.This nonlocality however may be desirable since there are reasons to believe that any theoryof quantum gravity will not be local in the conventional sense. Anyway it is a long - heldbelief that in quantum theories, space - time must change its nature at distances compared tothe Planck scales.The experimental signature of noncommuting spatial coordinates which is currently availableseems to be the approximate noncommutativity appearing in the Landau problem, the lowestenergy levels, for the case of very strong magnetic eld. The study of exactly solvable models,as is for instance the present note, should lead to a better understanding of some issues innoncommutative theories.In recent years there has been increasing interest in quantizing the harmonic oscillator witha variable mass in a time varying crossed electromagnetic eld (16). In this article we willstudy the problem of a two dimensional time dependent harmonic oscillator within non

188 Theoretical Concepts of Quantum Mechanics

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Non Commutative Quantum Mechanics

in Time - Dependent Backgrounds 3

commutative quantum mechanics (17) (18), the parameters of which are also time dependent.All the time dependent factors are exponentials of the form et. These factors have beenchosen so that the resulting nal formulas, uctuate with some external frequencies which donot depend on time.We postulate rst the two dimensional phase space. The momenta commutator showsthat we have a time dependent magnetic eld. With a time dependent, Bopp shift, lineartransformation we reduce the phase space to a new phase space with two independentdimensions. The coordinates and the momentum of the second dimension of this newphase space satisfy a deformed time independent commutation relation. Next we give theHamiltonian of the system which is a two dimensional damped harmonic oscillator. It isactually a linear combination of two Caldirola - Kanai damped harmonic oscillators withfriction terms. Following that, we nd the exact propagator of the system and the timeevolution of the basic operators. In the next section we nd the propagator in the case wherethe deformed parameter vanishes and the system becomes one dimensional. Finally inthe last section we nd the statistical partition function for two distinct particular cases whichdiffer in the number of the dimensions but the nal results depend on one common frequency.

2. The two dimensional phase space

We postulate the following time dependent commutation relations.

[ p1, p2] = ie1 t, [q1, q2] = ie1 t, [qj, pk] = ihjk (1)

Recently non commutative theories with non constant parameters have been found in somereferences such as (19). The commutators of the pairs q1, p2 and q2, p1 are zero andthe system can be described initially by one of the following wave functions (q1, p2, 0) or(q2, p1, 0).The rst relation among momenta operators means that the system is in a time dependentmagnetic eld, ( B), perpendicular to the plane of q1 and q2. The magnetic eld is denedin terms of the symmetric time dependent potential A = e1 t(Bq2/2, Bq1/2, 0), whichindicates that there is also an electric eld present.

B = A = e1 tBk E = A t

= 1 A

The second commutation relation between the coordinates, expresses the non commutativityof space. The parameter has a dimension of (length)2. In lowest Landau levels thecoordinates of the plane are also canonical conjugate operators and so satises the samerelation with (1/B) (20). This commutation relation implies the following uncertainty.

q1q2 (/2)e1 t (2)So the position of the system cannot be localized in space, except for minus innite times. Thecoordinates space becomes now fuzzy and uid. The parameter represents the fuzzinessand the parameter 1 the uidity of the space.The above relation is the known relation of ordinary non commutative geometry except thatthe parameter (t) expands exponentially with the evolution of time. This is the mainmotivation of this paper. The effect of a changing magnetic eld is given by Faradays lawwhich states that the magnetic ux running through a closed loop may change because theeld itself changes or because the loop is moving in space. So if we focus on point one, that is

189Non Commutative Quantum Mechanics in Time-Dependent Backgrounds

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q1 > 0 then the second point becomes fuzzier as time passes, that is q2 . We cansay that the background space is a dynamical two - dimensional fuzzy space (21).We will transform the problem of the non commuting two dimensional space, to a problemof two coupled harmonic oscillators in a more familiar two dimensional quantum mechanicalspace. The two dimensions of the new phase space are now independent of each other, butthe second phase space satises a deformed commutation relation.For this purpose we make the following linear transformations

P1 = p1 Q1 = q1

P2 = p2 + e1 t

hq1 Q2 = q2 e1 t h p1 (3)

The commutators for the basic operators within the capital letters are

[Q1, Q2] = 0 [P1, P2] = 0

[Q1, P1] = ih [Q2, P2] = i(h h

)= ih

(1

h2

)= ih (4)

The basic operators of the second dimension P2, Q2 satisfy a deformed commutation relationwhich it is time independent. We denote the deformed parameter by . We emphasize thatthe time dependence of the commutation relation in equations (1), are chosen so that this

new parameter = 1 (/h2) becomes time independent. If this parameter becomes zero, = 0 that is = h2, the four - dimensional phase space degenerates to a two - dimensionalone and consequently as we will see later, the nal solutions depend only on one frequency . The canonical limit ( 0, 0) does not exist in this case. It is clear that we have tokeep both parameters non zero, = 0 and = 0.Instead of an algebra of commutators, some theoretical physicists (22),(23) consider itsclassical analogon involving Poisson brackets of functions on real variables. But the standardlimit from quantum to classical mechanics that is h 0 has as a result to vanish thethird of the commutators (4), while the last one tends to innity. So the noncommutativequantum mechanics seems to has no classical analogous or we have to treat the second phasespaces with some deferent manner as an extra dimension. It seems reasonable to make thesubstitutions h and h so that the limit h 0 vanish all the commutatorscollectively.We see that the time dependence on the magnetic eld makes the time dependence of thecoordinates commutator unavoidable, rendering the resulting commutator of the capitaloperators P2 and Q2 time independent. The system is now described initially on thecoordinates space or on the momentum space by one of the following wave functions

0(Q1,Q2) = 0(q1, q2 h e1 tp1),

0(P1, P2) = 0(p1, p2 +

he1 tq1) (5)

The time dependence of the initial wave functions is due to the moving phase space of thesecond point (Q2, P2), resulting from the time dependent magnetic eld.


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3. The damped Hamiltonian of the system

We will use the following Hamiltonian

H(p,q, t) = e2(1+2) t p21

2m1+ e2(1+2) t 1

2m1

21 q

21

(e22 t

p222m2

+ e22 t1

2m2

22 q

22

)(6)

which is usually referred to as the Caldirola - Kanai model (24). The coupling constant will take one of the values 1. For = 1 the Hamiltonian is an ordinary two dimensionalHamiltonian. For = 1 we can say that the second Hamiltonian is an harmonic oscillatorwith negative mass m2 < 0 (25).The damped harmonic oscillator in a crossed magnetic eld in ordinary space has beenstudied by many authors (26). We will study this problem in non commutative quantummechanics. Such problems with magnetic elds in noncommutative quantum mechanics hasalso been studied by some authors see for instance (27), (28), and references there in.We can assume that this is a Hamiltonian of two particles, one on the phase point (q1, p1) andthe other on the point (q2, p2). It has been shown, by Bateman, that we can apply the usualcanonical quantization method if we double the numbers of degrees of freedom so as to dealwith an effective isolated system. The new degrees of freedom may be assumed to representthe environment which absorbs the energy dissipated by the dissipative system and the timedependent magnetic elds. The canonical quantization of these dual Betemans type systemshave many problems which have been pointed out in the relevant literature (29). We thinkthat the non vanishing commutator of the coordinate operators q1 and q2 , corrects many ofthese problems. Notice that as it was pointed out earlier, the Caldirola - Kanai Hamiltonian isnot really the energy of the system but rather an operator which generates the motion of thesystem.For the case where 1 = 0 the Hamiltonian becomes symmetric or antisymmetric with thereversal of time. This case has been studied in ref (31). The presence of the 1 parameterobviously breaks down this very important symmetry and indicates the presence of an electriceld.The Hamiltonian (6) describes a system of two particles with varying masses of m1 m1e

2(1+2) t and m2 m2e22 respectivelly. The product of these two masses is obviouslytime dependent

m1m2 m1m2 e21 t (7)This is the consequence of the time varying magnetic eld B(t) Be1 t. The time factorsof this Hamiltonian have been chosen so that the nally oscillators of the system become timeindependent.With the linear transformation (3) the Hamiltonian becomes

H(P, Q, t) = 12e2(1+2) t P21 (1 222) +

1

2e2(1+2) tQ21(

21 2)

1

2e22tP22

1

2e22t22Q

22 e(1+22) t22P1Q2 + e(1+22) tP2Q1 (8)

where we have set h = 1 and m1 = m2 = 1.


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This is a Hamiltonian of two coupled harmonic oscillators in the deformed quantummechanical space. The last two factors of the above Hamiltonian are the coupling terms.Notice that if 1 + 22 = 0 these terms become time independent.In order to simplify the relations we shall use the following symbolism

P1 T1 Q1 T2 P2 T3 Q2 T4 (9)The commutation relations become

[T2, T1] = c21 [T4, T3] = c43, c21 = ih c43 = c21 (10)

while all the others commutators vanish. The last commutation relation goes to innity ash 0 so the problem has no classical analogy (Figure 1).

iC21

iC43

iC43

2

2

0

2

Fig. 1. The commutators i c21 and i c43 of the phase space as functions of h.

The Hamiltonian is written

H(T) = k11T21 + k22T22 + k33T23 + k44T24 + k41T4T1 + k32T3T2 (11)Where we have set

k11 =1

2e2(1+2) t(1 222), k22 = 12 e2(1+2) t(21 2)

k33 = 12 e22t, k44 = 12 e22t22

k32 = e(1+22) t, k41 = e(1+22) t22 (12)

The coupling terms k32 and k41 become zero in the case of commutative quantum mechanics,where 0 and 0.


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4. The exact propagator of the system

We will expand the time evolution operator in an appropriate ordered form so that thepropagator will be calculated easily with a straight manner.The Shrdinger equation of motion of the time evolution operator is:

ih

tU(t) = H(t)U(t) U(0) = 1

We look for the following, normal ordered expansion of the evolution operator:

U(t) = U4U3U2U1

where

U4 = ef44 T

24 e

12 f43(T4 T3+T3 T4)e f42 T4 T2 e f41 T4 T1 U3 = e

f33 T23 e f32 T3 T2 e f31 T3 T1

U2 = ef22 T

22 e

12 f21(T2 T1+T1 T2) U1 = e

f11 T21 (13)

The functions f jk(t) are time dependent and because of the condition U(0) = 1 , they satisfythe initial conditions f jk(0) = 0.

The above solution is always possible if the Hamiltonian takes the following form H =mi=1 a(t)Hi where the operators Hi, i = 1, ...,m forms a closed Lie algebra (30). We have

proved in reference (32) using standard algebraic technics that all the unknown functions f jkcan be written with the help of the functions xjk(t). The functions xjk(t) satisfy the followingclassical differential system

x1j = 2k11x2j + k41x3jx2j = 2k22x1j k32x4jx3j = k32x1j 2 k33x4j

x4j = k41x2j + 2 k44x3j (14)

with the following initial conditionsxjk(0) = jk (15)

The functions xnj give the time evolution of the basic operators Tj. We have

Tn(t) = e ih tH Tj(0)e

ih tH = xnj(t)Tj(0) (16)

The solution of the above system is as follows

x11 = e(1+2) t

[a2 (1 + 2)b2 222 (a1 (1+ 2)b1)

]x22 = e

(1+2) t[a2 + (1 + 2)b2 222 (a1 + (1+ 2)b1)

]x21 = e

(1+2) t[222

2b1 b2 + 222b2]

x12 = e(1+2) t

[21(2222b1 b2) 2b2

]


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x13 = e2t

[22

(22b1 (1(a1 + 2b1) + b2)

)]x14 = e

2t[(a1 2b1) + 222 (a1 (1+ 2)b1)

]x23 = e

2t[22

(21(a1 + 2b1) 2(a1 + (1+ 2)b1)

)]x24 = e

2t[221

22

2b1 + (1(a1 2b1) b2)]

x31 = e(1+2) t

[22122b1 + (1(a1 + 2b1) + b2)

]x32 = e

(1+2) t[(a1 + 2b1) 222 (a1 + (1+ 2)b1)

]x41 = e

(1+2) t[22

(21(a1 2b1) 2(a1 (1+ 2)b1)

)]x42 = e

(1+2) t[22

(2b1 + (1(a1 2b1) b2)

)]

x33 = e2t

[a2 + 2b2 +

(21 + 212 21 + 21222 + 2

)(a1 + 2b1)

222(1 ) (a1 + (1+ 2)b1) ]x44 = e

2t[a2 2b2 +

(21 + 212 21 + 21222 + 2

)(a1 2b1)

222(1 ) (a1 (1+ 2)b1) ]x43 = e

2t[22

(22b1 + ((

21 + 212 21)b1 + b2)

)]x34 = e

2t[221

22

2b1 + ((21 + 212 21)b1 + b2)

]Where

a1 =1

w3(cos (1t) cos (2t)) b1 = 1w3

(sin (1t)

1 sin (2t)

2

)

a2 =1

w3

((22 +

21

)cos (1t)

(22 +

22

)cos (2t)

)

b2 =1

w3

((22 +

21

) sin (1t)1

(22 +

22

) sin (2t)2

)(17)

The frequencies 1 and 2 are the solution of the following algebraic system

w1 =(22 +

21

) (22 +

22

)= 222

(21 1(1+ 22)

)(18)

w2 = (22 +

21) + (

22 +

22) =

21 +

222 (21222 + 2) 1(1 + 22) (19)We have also set

w3 = 21 22 =

w22 4w1 (20)

We nally nd the following solutions which are time independent.

1 =

(w2 + w3)/2 22 2 =

(w2 w3)/2 22 (21)


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The above results are valid for any particular value of the parameters with the exception ofthe parameters and which must be real. The solutions (eqs. 17) have well dened limitswhen 1 0 and 2 0, or for the more interesting limit 1 0 and 2 i2.The results found in this paper, coincide with that of paper (31) when 1 = 0, where wehave a constant magnetic eld. For this zero value of parameter 1 , equation (18) becomessymmetric, which means that it has two solutions with respect to , of opposite signs. Thissymmetry is present again if 2 = 0. This symmetry of the parameter is crucial, sinceit is the deformed parameter of the second phase space (equation 4) and the transformation means that [Q2, P2] [Q2, P2].We can reconstruct the same symmetry also if we set

1+ 22 = 0 =(22 +

21

) (22 +

21

)= 221

22

2 (22)

In this case we nd

1 + 2 =2

2 =h+ hh h

2 (23)

This term which is on the exponential of the rst Hamiltonian in eq. (6), changes sign in theduality h /h. For all the above symmetric cases equation (18) has two solutions withrespect to with opposite signs.In general equation (18) is a parabola with respect to and becomes zero for the followingtwo values of the deformed parameter .

= 0 = 1

and =21221 21

= 1

(1 212

21 21

)(24)

In both cases the frequencies are identical if 21 = 21 + 12.

We will solve the case where = 0 in the next paragraph because it leads to the relation[Q2, P2] = 0 which means that the problem is one dimensional since the commutator of thesecond phase space vanishes.When 1 = 1 equation (18) has only one solution, with respect to , We nd

= (22 +

21

) (22 +

21

)2212

22

(25)

while the second solution tends to minus innity.Next we will calculate the exact propagator of the system.As is well known the action of the time evolution operator on the delta function, produces thepropagator of the system.

G(1, 1, 2,

2, t) = U(t)(1 1)(2 2) (26)

Because of the commutation relations (10) only two quantities can be simultaneouslymeasured. We choose the following observables

T2 = Q1 = q1 1 T4 = Q2 = q2 h e1 tp1 2 (27)

For the calculations we consider the following representation

T1 = c211 T2 = 1 T3 = c432 T4 = 2 (28)


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We can of course choose another couple of commuting observable. The various propagatorsare appropriate Fourier transforms of each other.By the help of the above representation, and after a simple calculation (32) we nd thepropagator:

G(1, 1, 2,

2, t) = (29)

1s0

exp

{ 12s0

[c43(21

1x34 +

21(x14x31 x11x34) + 21 (x24x32 x22x34))+

c21(222x12 +

22(x13x32 x12x33) + 22 (x14x42 x12x44))

2c211(2x14 2(x14x33 x13x34)) 2c431(2x32 + 2(x34x42 x32x44))

]}where

s0 = c21c43 (x12x34 x14x32) (30)If we nd the propagator we can calculate the time evolution of a quantum system whichmust have initially the following form.

(1, 2, t)|t=0 = 0(q1, q2 h e1 tp1, 0) (31)

We can not get rib of the time factor in the initial state except if 1 = 0 or = 0, wherewe have a time independent non commutative space or an ordinary commutative spacerespectively. This is a consequence of the time dependance of the Bopp shift transformations(4), as the whole phase space of the second point is moving with respect to the rst one.The wave function of the system is given by the relation

(1, 2, t) =

G(1, 1, 2,

2, t)(

1,

2, 0)d

12 (32)

The Hamiltonian (6) has the same form as the Hamiltonian in ordinary quantum mechanics.As a consequence the spectrum of H is a linear combination of the following energies

Ej = hj

(nj +

1

2

), nj N (33)

The frequencies 1 and 2 are given by equations (21).

5. One dimensional case

In this section we will examine the case where we have one external frequency.

c43 = 0 = = 0 = h =h

(34)

The system is now one dimensional while the second Hamiltonian is like a potential energy.The functions f jk satisfy a differential system which can be found in ref (32). The solution isas follows:


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f11 = i21 222

cos ( t) (1 + 2) sin ( t)sin ( t)

f21 = i(1 + 2) t i log{cos ( t) (1 + 2) sin ( t)

}

f22 = i2 e2(1+2) t

21 2

cos ( t) (1 + 2) sin ( t)sin ( t)

f31 = ie(1+2) t

1 22222 +

2

{e2t cos ( t) + 2 sin ( t)

}

f32 = ie(1+2) t{sin ( t)

+

122 +

2

(e2t cos ( t) + 2 sin ( t)

)}

f33 = i2 e2 t k

22 +2

{(1 222)

(k2

sin ( t)

21

sinh (2 t)

2

)+

1(1 + 22)sinh (2 t)

2

} i

2f31 f32

f41 = ie(1+2) t22

{sin ( t)

1

22 +2

(e2t cos ( t) + 2 sin ( t)

)}

f42 = ie(1+2) t

22(

21 2)

22 +2

(e2t cos ( t) + 2 sin ( t)

)

f43 = i222

22 +2

{1 t e2 t1 sin ( t)

e2t(1+

21222 +

2

)(e2t cos ( t) + 2 sin ( t)

)}

f44 =i

2e2t

2222 +

2

{(21 2)

(sinh (2 t)

2 222

sin ( t)

)

1(1 + 22)sinh (2 t)

2

} i

2f41 f42

Where the frequency is

=21 +

222 (21222 + 2) (1 + 2)2 (35)So all the formulas involved depend on one common frequency .The same frequencies can be found also for the two dimensional case of the previousparagraph if

21 = 1(1 + 2) or 2 = 1 +211

(36)

For this value of the friction parameter 2 and from equations (18) and (19) we nd

= 2 or = 1

1 = 2 = i2 (37)


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It seems that for both cases, namely

= 1 h2

= 0 and 2 = 1+ h2

= 0 (38)

the nal propagators depend on only one frequency. In these cases, which we will study inthe next section, the commutator of the second phase space is [Q2, P2] = 0 and [Q2, P2] = 2ihrespectively.In the space of the Tj operators, we have now three commutative operators. We choose thefollowing:

q1 1 p2 + h e1 tq1 2 q2 h e

1 tp1 2 (39)

To calculate the propagator we assume the representation

T1 = c211 T2 = 1 T3 = 2 T4 = 2 (40)To nd the propagator we calculate rst the propagator in the rst dimension. After somealgebra, we nd the distribution.

G1(1, 1, t) = U2U1(1 1) =

e 12 (1+2) t2i(1 222)

sin ( t)

exp

{i

2

e(1+2) t

1 222

sin ( t)

[e(1+2) t

(cos ( t) + (1 + 2)

sin ( t)

)21+

e(1+2) t(cos ( t) (1 + 2) sin ( t)

)1

2 211]}

(41)

The nal propagator can be found by the action of the operator U4U3(2 2) on thisdistribution G1(1,

1, t). Because c43 = 0, the operator U4U3(2 2) does not contain any

operators with respect to the second dimension. Consequently a delta function remains in thenal result.After a simple calculation we nd

G(1, 1, 2,

2,2, t) = U4U3(2 2 )G1(1, 1, t) = (42)

e f4422+ f33

22+( f43i f32 f41)22+1( f322+ f422)G1

(1 i f312 i f412, 1, t

)(22)

If we nd the propagator we can calculate the time evolution of a quantum system whichmust initially have the following form.

(1, 2,2, t)|t=0 = 0(q1, q2 h e1 tp1, p2 +

he1 tq1, 0) (43)

The initial wave function depends on time unless 1 = 0 or even 1 = 0 but = 0 and = 0.


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The wave function of the system is given by the following single integral.

(1, 2,2, t) =

G(1, 1, 2,

2,2, t)(

1,

2,2, 0)d

1d

2 = e

f4422+ f33

22 (44)

e( f43i f32 f41)22+1( f322+ f422)G1

(1 i f312 i f412, 1, t

)(1, 2,2, 0)d1

The spectrum of the Hamiltonian H is the following

E = h

(n+

1

2

), n N (45)

which now depend on one frequency of equation (35).

6. Canonical density matrix

As is well known we can nd the statistical distribution function from the propagator. Therelation is

(1, 1, 2,

2, b) = G(1,

1, 2,

2,ihb) (46)

where b = 1/kT , k is the Boltzman constant and T is the temperature. To nd the partitionfunction, we set 1 =

1 and 2 =

2 and then we integrate the distribution (1, 2, b) with

respect to 1 and 2. The partition function is as follows

z(b) =

2G(1, 1, 2, 2,ihb)d1d2 (47)

The problem is two dimensional. The coordinates q1 and q2 are operators which do notcommute and its commutator must be time dependent too ( e1 t). With a particular valueof the parameter = 0, the problem becomes one dimensional and so the resulting formulasdepend on one frequency. We can also nd one frequency for the two dimensional case. Fromequation (18), it is obvious that being = 0 requires that one of the frequencies 1 and 2must be equal to i2. We choose 2 = i2.We will nd the statistical partition function for two district cases which both depend on onlyone common frequency. One of these cases is the one dimensional case where = 0 and theother is the two dimensional case where 2 = i2. To simplify the results we will study theparticular case where 1 = 0 and 1 = 2. For these two cases we have

= +1/ or = 0 one dimensional (48) = 1/ or = 2 two dimensional (49)

where of course = 0 and = 0.The Hamiltonian takes now the most simple form and it is the following:

H( p1, q1, p2, q2) = 12 p21

(e22 t 1

2p22 + e

22 t 1

222 q

22

)(50)

The basic operators satisfy the commutators

[q1, q2] = i(1/) e2t, [ p1, p2] = i e2t, [qj, pk] = ihjk (51)The common frequency of the nal results for both cases, is

=22 2 (52)


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which is independent of the parameter 2.For the case of one dimension ( = 0) the partition function is that of an ordinary oscillatorthat is

z1(b) =

sinh ( b2 )

R

{lim22

G(2,2, b) (2 2)}d2 (53)

where G(2,2, b) is a Gaussian type classical distribution function which we do not write ascan be found easily from equation (42).For the case of two dimensions ( = 2) we nd the following partition function.

z2(b) =2 + 22

16 2 sin (2 b2 )

2

2 cos (2 b2 ) sinh (

b2 ) cosh ( b2 ) sin ( 2 b2 )1

(22 2) cos ( 2 b2 ) sinh ( b2 ) 2 222 cosh ( b2 ) sin ( 2 b2 )(54)

For high temperatures, that is for a large value of the temperature parameters, (b 0),the partition function has singularities on the points b = 2n/2, where n is an integern = 1,2,3, .For low temperatures, that is b , the hyperbolic functions cosh (b/2) and sinh (b/2)are both equal to eb/2/2 . The partition function is multiplied with the factor e(1/2)b andso tends to zero. The nal partition function has singularities for the following values of b:

b =2

2

(n + arctan (

2

))

b =2

2

(n + arctan (

2

22 2222

)

)(55)

In the sequel we will study the case where the parameter becomes zero.If = 1 the common frequency can not be zero unless 2 i. For = 1 and = 2 we nd = 0. The energy of the system now comes exclusively from the varyingelectromagnetic eld while the energy eigenvalues of the Hamiltonian vanish.The rst partition function becomes:

z1(b) =cb2

R

{lim11

(1 1)}d1 (56)

The limit in the above equation, as well as the limit in the equation (53) means that the systemcan not be localized in space.The second partition function has a well dened limit for 0. We nd

lim0

z2(b) =22

16 2 22 sin (b22 )

1sin ( b22 )

1sin ( b22 ) b22 cos ( b22 )

(57)

This last partition function has again singularities for b = 2n/2, where n A and inaddition for some new values of the temperature b = 1/kT which are the solutions of theequation tan (b 2/2) = b 2/2.With the help of the partition function z(b), we can nd all the thermodynamic properties ofthe system. The thermodynamic energy is given by the following formula:

< E >= b

log (z(b)) (58)


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5 10 15

1

2

b

2

2

E

Fig. 2. The thermodynamic energy possesses many zeros and singularities.

After a simple calculation, and under certain conditions, we nd the followingthermodynamic energy

< E >=22

b2 + 2b2 cos (b2) 3 sin (b2)2+ 2 cos (b2) + b2 sin (b2) (59)

which possesses many zeros and singularities. These particular points disappear if theparameter 2 vanishes. (Figure 2). We nd

lim20

< E >= 3/b (60)

Noncommutativity is basically an internal geometric structure of the conguration space,which can not be observed per se. The last simple dynamical systems considered, havedifferent conguration spaces and the same frequency which of course gives the same energyeigenvalues of the Hamiltonian. The thermodynamic singularities appears basically only forthe two dimensional case and manifest the noncommutative phenomena. These singularitiesresults from the varying magnetic eld through the 2 parameter, and disappear for 2 = 0,that is for a constant magnetic eld.

7. Conclusion

In this paper we have found the exact propagator of a two dimensional harmonic oscillatorin non commutative quantum mechanics, where the ordinary non commutative parametersare time dependent. We have not investigated the problems which arise from this peculiardamping of the space, which stems from a time varying magnetic eld. In this paper weare satised by the fact that the nal results oscillate with two frequencies that are constantin time. The calculations have been made in such a way so that we can arrive at the nalresults, taking into consideration every special value of the parameters involved. We haverst expand the time evolution operator in some kind of normal ordered form so that thepropagator results easily by the action of this operator on some delta functions.


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We have three damping mechanisms. The time dependent magnetic eld ( e1 t), the mainharmonic oscillator with varyingmass ( e2(1+2) t) and the secondary harmonic oscillatorwith varying mass (e22 t). The energy emitted from one of these is absorbed by the othersso that the nal results depend on two frequencies 1 and 2 which are time independent.The momenta of the system satisfy a time dependent commutator ( e1 t) which meansthat we have a time dependent magnetic eld and consequently an electric eld present. Thecoordinates of the system satisfy a commutator which is also time dependent ( e1 t). Thecoordinates space is fuzzy and uid with parameters and 1 respectively and the momentais also fuzzy and uid ( and 1).With a linear time dependent transformation, the problem reduces to that of a twodimensional harmonic oscillator on a phase space with two independent phase spaces.The new commutator relations of this new phase space become time independent. Thecommutator between the coordinate and the momentum of the second phase space satisesa deformed commutator relation equal to i(h /h). This factor has no dened limit forh 0, so the problem seems to have no classical analogy.The Hamiltonian of the system is a linear combination of two Caldirola - Kanai Hamiltonianswith friction parameters which differ by the parameter 1. H. Bateman uses a similarHamiltonian and the energy emitted from one Hamiltonian was absorbed by the other one.In this paper the energy emitted from the rst Hamiltonian on the point one and the timedependent magnetic eld is absorbed by the other Hamiltonian on the point two. The timedependence of this second mirror Hamiltonian is the appropriate one, so that the resultingnal formulas depend on two time independent frequencies.The propagator provides the time evolution of the system. The initial wave functions dependon the two commuting variables (q1, p2) or (q2, p1). The rst point particle has a well denedposition and the second well dened momentum. The system can be considered as a massiveobject located at two separate massive points on a fuzzy and uid two dimensional dynamicalspace.This paper is a generalization of paper (31) in the case where the magnetic eld is increasing(or decreasing) exponentially at a rate equal to 1. For 1 = 0 we found the same resultswhile for 1 = 0 we derived some new interesting conclusions.The parameter 1 destroys the time symmetry of the Hamiltonian and the product m1m2( e21 t ) is time increasing (or decreasing) exponentially. This is the main motivation ofthis paper. The propagator, as is well known, can also be used to nd the statistical partitionfunction and thus all the thermodynamical properties of the system. We have investigatedtwo situations where the nal results depend on only one common frequency .The rst one is of course the case where the problem reduces to that of a one dimensionalphase space, where [Q2, P2] = 0. The second is the two dimensional case where one of thefrequencies becomes imaginary and equals to i2. We have investigated the case where[Q2, P2] = 2ih. Because the oscillations give the energy of a harmonic oscillator the two caseshave similar energies eigenvalues while they have different dimensionality. We have alsofound the statistical partition functions of these two systems. We have concluded that, in thetwo - dimensional case the partition function has many interesting zeros and singularitieswhich are not present in the one - dimensional case. These particular points depend on theparameter 2, which is responsible for the time damping of the magnetic eld.Finally we have found the partition function in the resonance where this last frequency becomes zero. The thermodynamic energy of the system, possesses similar zeros andsingularities that disappear when 2 vanish.


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8. References

[1] H. Dekker. Classical and Quantum mechanics of the damped harmonic oscillator.Physics Reports 80, 1. 1-110. (1981).

[2] G. t Hooft. Quantum gravity as a dissipative deterministic systemarXiv:gr-qc/9903084v3 (1999).

[3] A. Leggett, S. Chakravarty, A. Dorsey, M. Fisher, A. Garg, and W. Zwerger, Dynamics ofthe dissipative two - state system, Rev. Mod. Phys., vol. 59, p.1, (1987).

[4] A. Jannussis, G. Brodimas, A. Streklas. Phys. Lett 74A 6 (1979).[5] H. Bateman. Phys. Rev. 38(4), 815, (1931).[6] V. Aldaya, F. Cossio, J. Guerrero and F. Lopez-Ruiz. A symmetry trip from Caldirola to

Bateman damped systems. arXiv:1102.0990v1 [math ph] 4 Feb 2011.[7] M. Douglas, N. Nekrasov. Noncommutative eld theory. Rev. Mod. Phys. Vol. 73,

977-1029 (2001).[8] R. Szabo Quantum eld theory on noncommutative spaces, Phys. Rep. 378, 207-299,

(2003).[9] N. Seiberg, L. Susskind and N. Toumbas J. High Energy Phys. 0006 044 (2000).[10] H. Snyder. Phys. Rev. 71, 38 and 72, 68 (1947).[11] A. Jannussis. Modern Physics and the Greeks. Lecture in the University "Dunarea de Jos"

Galati. Methematica, Fizica, Mecanica Theoretica. Fascicula II, anul XVII(XXII). (1999).[12] A. Connes. Noncommutative Geometry, Academic Press, (1994),

http://www.alainconnes.org/ Alain Connes Ofcial Website.[13] N. Seiberg, E. Witten. String Theory and Noncommutative Geometry.

arXiv:hep-th/9908142v3 (1999).[14] A. Konechny. A. Schwarz. Introduction to Matrix theory and noncommutative geometry.

Patrs I and II. Phys. Rept 360, 353, (2002).[15] J. Madore. Gravity on Fuzzy Space - Time. arXiv:gr-qc/9709002v2 2Sep (1997).[16] A. Dutra, B. Cheng. Feynmans propagator for a charged particle with time - dependent

mass. Phys. Rev. A 3911 (1989) 5897 - 5902.[17] M. Liang, Y. Jiang. Time - dependent harmonic oscillator in a magnetic eld and an

electric eld on the non - commutative plane. Physics Letters A. 375 1-5 (2010).[18] A. Hatzinikitas, I. Smyrnakis. The non commutative harmonic oscillator in more than

one dimensions. J. Math. Phys. 43, 113, (2002).[19] X. Calmet, M. Wohlgenannt. Effective Field Theories on Non - Commutative Space -

Time. arXiv:hep-ph/0305027v2 25 Jun 2003.[20] J. Gamboa, M. Loewe, F. Mendez, J. Rojas. Mod. Phys. Lett. A 16 2075, (2001).[21] M. Buric, L. Madore. ADynamical 2 - Dimensional Fuzzy Space, arXiv:hep-th/0507064v1

(2005).[22] A. Eftekharzadeh, B. Hu. The Classical and Commutative Limits of noncommutative

Quantum Mechanics. arXiv:hep - th/0504150v2 (2005).[23] M. Baldiotti, J. Gazeau, D. Gitman. Semiclassical and quantum description of motion on

noncommutative plane. arXiv:0906.0388v2[quant-ph] 3 Jun 2009.[24] P. Caldirola Nuovo Com. 18, 393, (1941).

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Physics Reports 362 63 - 192, (2002).


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[27] F. Delduc, Q. Duret, F. Gieres, M. Lefrancois Magnetic elds in noncommutativequantum mechanisc, arXiv:0710.2239v1 (2007)

[28] M. Liang, Y. Jiang. Time - dependent harmonic oscillator in a magnetic eld and anelectric eld on the non - commutative plane. Physics Letters A 375 1 - 5 (2010).

[29] M. Blasone and P. Jizba. Batemanss dual system revisited. arXiv:quant - ph/0102128 v2(2001).

[30] C. Cheng, P. Fumg. The evolution operator technique is solvimg Schrdinger equation,......J. Phys. A: Math. Gen. 21 4115 - 4131 (1988).

[31] A. Streklas. Quantum damped harmonic oscillator on non commuting plane.Physica A 385, 124 - 136, (2007).

[32] A. Streklas. Int. J. Mod. Phys. Vol 21 No 32, 5363 - 5380, (2007).


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Theoretical Concepts of Quantum MechanicsEdited by Prof. Mohammad Reza Pahlavani

ISBN 978-953-51-0088-1Hard cover, 598 pagesPublisher InTechPublished online 24, February, 2012Published in print edition February, 2012

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Quantum theory as a scientific revolution profoundly influenced human thought about the universe andgoverned forces of nature. Perhaps the historical development of quantum mechanics mimics the history ofhuman scientific struggles from their beginning. This book, which brought together an international communityof invited authors, represents a rich account of foundation, scientific history of quantum mechanics, relativisticquantum mechanics and field theory, and different methods to solve the Schrodinger equation. We wish forthis collected volume to become an important reference for students and researchers.

How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following:Antony Streklas (2012). Non Commutative Quantum Mechanics in Time - Dependent Backgrounds,Theoretical Concepts of Quantum Mechanics, Prof. Mohammad Reza Pahlavani (Ed.), ISBN: 978-953-51-0088-1, InTech, Available from: http://www.intechopen.com/books/theoretical-concepts-of-quantum-mechanics/non-commutative-quantum-mechanics

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