2D Electron in a Maghetic Field with Dissipation

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2D Electron in a Maghetic Field with Dissipation

G. Cristofano, D. Giuliano, G. Maiella

To cite this version:

G. Cristofano, D. Giuliano, G. Maiella. 2D Electron in a Maghetic Field with Dissipation. Journal

de Physique I, EDP Sciences, 1996, 6 (7), pp.861-872. �10.1051/jp1:1996100�. �jpa-00247219�

2D Electron in

_a

Magnetic Field with Dissipation _(*)

G. Cristofano (**),

D.

Giuliano (**)

^{and G. Maiella}

Dipartimento

di Scienze

Fisiche,

Universitl di

Napoli,

INFN Sezione di

Napoli,

Mostra

d'oltremare,

Pad. 19, 80125

Napoli, Italy

(Received

20 December 1995, received in final form 3

April

1996,

accepted

9

April 1996)

PACS.73.40.Hm

Quantum

Hall effect

(integer

and

fractional)

PACS.72.15.Rn

Quantum

Iocalization

Abstract. We

give

_a method for

determining

_a

complete

set of constants of motion for the system ^of _a ^2D electron in

an external magnetic field and in the presence of _a dissipative term vi;

the

time-dependent

coherent states _are then

uniquely

determined and the effective Hamiltonian found.

1. Introduction

We

present

an

analysis

of the

Lagrangian

and Hamiltonian formalism for the

problem

of _one electron in 2D in _an external

magnetic

field in the _presence of

dissipation 11,2].

First _{we use} the usual method _[3] to obtain the Caldirola Kani Hamiltonian for _a

system

with

dissipation.

Then _we show that such _an

approach

fails when _one has _a

"giroscopic"

Lorentz force.

We _are then forced to _{use a} different method based _on the construction of _a

complete

set of constants of motion.

After

showing

how this method works for the _~/

= 0 _case [4], we

apply

it to the unresolved

~/

~

0 one; more

explicitly

_we construct the

Heisenberg algebra

based _on the constants of motion

showing

also that it is left invariant under _a

"symmetry group" SU(2)

_x

U(1).

hforeover _we

build,

in the

complete

Hilbert _space

spanned by

the

eigenvalues

of the _constants of

motion,

_a

quadratic

form which _we

interpret

_as

Hamiltonian,

first for the

~/ = 0 _case and

then for the

~/

~

0 _one.

Finally

we

give,

for

~/

~ 0,

a

complete

set of coherent states

,

which

depend

_on two pa-

rameters, (,

_o,

showing

that for _o

~ 0 _we

get

back the _one

parameter

^coherent states which determine the

dynamics

at t »

1/~/ previously

obtained

ill.

At the end _we comment _on the time evolution of _a Landau level with _zero

angular

momentum.

(*)

Work

supported

in part

by

MURST and

by

EEC contract _no. SCI-CT92-0789 (** ^{Authors for} correspondence

(e-mail:

cristofano@na.infn.it and

giuliano@na.infn.it)

@

Les

(ditions

_de

_Physique

₁₉₉₆

2.

Lagrangian

and Hamiltonian Formalism for

Dissipative Systems:

the Problem

of _a 2D Electron in _an External

Magnetic

Field

In this section _we

give

_a brief review of the standard way of

quantizing simple dissipative

_sys-

tems

by introducing

_a viscous term in the

equations. Starting

from _an

explicit time-dependent Lagrangian

^{the so-called} Caldirola-Kani Hamiltonian is obtained

(see

^for

example

Ref.

[3]).

Let _{us start}

by considering

_a ID

particle subject

to _a conservative force

f(q)

and let

V(q)

be the

corresponding potential

such that:

)lq)

=

flq) ^Ii)

The

Lagrangian

of the

system

is

given by:

£(q, q)

₌

]q~ ^{V(q) 12)}

The Euler

Lagrange equation

of motion is written _as:

( _{(q, 4)} ( _{(q, 4)}

= 0

(3)

By introducing expression (2)

for £ into

equation (3)

we obtain:

'~lq "

-$

"

f(q) (4)

which is the correct form of Newton

equation

of motion.

In the _presence of _a viscous force

-~/( equation (4)

assumes the form:

ml

_"

flq) 'f4 (5)

It is

straightforward

to write _an

explicit time-dependent

effective

Lagrangian £e~r(q,(,t)

in order to obtain the above

equation.

In

fact, by rewriting equation (5)

_as:

m)le~~q)

=

-e~~)lq) ⁽⁶⁾

where the

viscosity

coefficient

~/ is

given by

_{~/ =}

~llm,

it follows that:

£e~r(q, (, t)

= e~~

()(~ _V(q)) (7)

From

£e~r(q,(,t)

_we obtain the

corresponding

effective Hamiltonian

7ie~r(q,p,t) by

the usual

Legendre

^{transform. It} turns out that:

7ie~r(q, p,t) ~2

=

e~~~-

₊

e~~V(q) (8)

where the canonical momentum p is

given by:

7ie~r

is

7iKc

is used in reference _[5] to

quantize simple

1D models like the harmonic oscillator and is

easily

extended to the _case in which there _are 2n canonical coordinates

q~,.

,

q~,

_{pi, pn} and do not appear

giroscopical

^forces.

In the

remaining part

^{of this section} we will show

that,

in the _presence of _a

giroscopic

term in the

Lagrangian,

the Caldirola Kani

approach

fails.

Let _us consider the

simple

_case of _a 2D electron in _an external transverse

magnetic

field B and in the _presence of

dissipation.

The

equations

of motion _are

given by:

I +

(ij

+

iwc)d

"

0, I

₊

(ij iwc)2

# 0

(10)

where _z

= x +

iv,

_{wc =}

eB/mc

is the

cyclotron frequency;

from _{now on we} will _use the units

m = c = e = 1. If _we

apply

the _same

procedure

used above in

obtaining 7iKc,

_we

get:

lie~~z)

⁺

^iwcie~~z)

= 0

~

(e~~z) iw~je~~z)

₌ o

(ii)

dt

and _an effective

Lagrangian £e~r

must

obey

the relations:

~li~ ^=e~~z ~li~

=

e~~z i12)

°~~~~ _°~~ ~~~~'

~~~

₌ ^e~~

jzz

11°C

izz zz)) l13)

2 2

and the Euler

Lagrange equation

^is

given by:

I ₊ (~/ +

iwc)I

+

"z

= 0

(14)

which is _not the correct

equation

^{of motion}

(unless

~/ ⁼

0).

Any change

in the

explicit time-dependence

of

£e~r

does not

improve

the result.

These considerations show that it is not _a trivial

problem

to define

7iKc

for _our system.

Since _we cannot derive _an "effective"

time-dependent

Hamiltonian

formalism,

we will _{use a}

completely

different

approach

which will be the _content of Sections 3 and 4.

3. Constants of Motion Linear in the Canonical Variables for the Landau Problem

for the _~

= 0 Case

For _a

dynamical system

described

by

_a

Lagrangian

^formalism it exists _a strict relation between

symmetries

and _constants of

motion, given by

the Noether theorem. However if there is _no

Lagrangian

neither Hamiltonian formalism for the

system,

it is still

possible

to obtain constants of motion

directly

from Newton

equations

^{of motion. As} an

example,

let _us consider _a 2D

electron in _an external

magnetic

field for

~/ ⁼ 0. The

equations

^{of motion} are

given by:

I +

iwci

₌

0, 1- iwci

= 0

(15)

and _{one can}

easily

obtain two

independent

constants of motion

a(z, I, t)

and

b(z, I):

a(z, z, t)

₌

e~"~~zlt), blz, z)

₌

zlt) )zlt) l16)

with their

complex conjugates.

In reference _[4] it has been shown how to use the constants of motion defined above to construct a basis for the Hilbert _space of the

correspondent quantized theory,

where the

starting point

is the canonical

correspondence

between _z,

I,

_pz

given by:

By using

the

correspondence principle

_{pz ~}

-2i)

_one

_gets:

~~

d

~j~

^~

~~ ~~

z

~

~j~

^{~ ~~~~}

and the

operators h,I corresponding

to the constants of motion

(with

their

conjugates)

_are

given by:

16,

b~l

=

Ii, i~l

= 2LGc, 16,

II

= 16,

i~l

= °

12°)

and the "normalized

operators"

_are defined _as:

~ li'~ ~

~~~~

which

satisfy

the

Heisenberg algebra [I, lt]

=

16, fit]

₌ 1.

A basis for the Hilbert _space is obtained

by simultaneously diagonalizing I

_and

fi:

J<ia, i)

=

a<ia, i) fi<ia,i)

=

i<ia,1) ₁₂₂₎

where

#(o, ()

is _a two parameters coherent state

given by:

jja ()

=

e-(lal~+lfl~)/2 ~~~j~fi(~~

^~

^{j~)) ~-Qjzj2 (~~)}

' 2

first introduced in

[6]. By recalling

that the time evolution of the operator

I

_is:

J(t)

=

e~"~~J(0) (24)

while

#

_is

time-independent.

#(a, (, t)

at a time t _can be written _as:

j(a, (, t)

₌

e-(lal~+lfl~l/2 ~~(~fi(~~~w~~

~

j~))~- ^qjzj2 _j~~)

2

Since

#(o, (, t)

is _a

(over)complete

set of _{states at any}

given t,

the time evolution of _{a state}

~fi

is derived

by projecting

it onto the set of coherent state at t ₌ 0 and

by making

such state to evolve. Thus the

quantum dynamics

of the

system

is

completely

determined.

Apparently

we have not used the

explicit

form of the Hamiltonian

operator

or of the

Lagrangian,

_so this

approach

could be used also for the _case in which _we don't know these

quantities. Unfortunately

to obtain

quantum operators

from the

"classically"

conserved

quantities (defined

_as function of _z, d and

t)

_we have to express them

again

in terms of t, z and of the canonical momentum pz. In the

~/ = 0 _{case pz} and _{@z are} defined _as:

obtaining

for _pz and _@z

Pz "

Pz(Z, i),

ji= "

jiz(d, I) ₍₂₇₎

However, equations (27)

may be _more

general

than

(26),

in the _sense that

they

_may exist also if £ is _not known. In the

following

we'will obtain

equations (27)

for the _{case ~/}

~

0

by making simple

and reasonable _ansatz and then _we will find _an "effective"

(non-Hermitian)

Hamiltonian for the system.

4. General "Kinematics" of the Problem in the _~

~

0 Case

We _{use a} different

approach

to _our

problem

based _on the

requirement

that the

"dynamical

evolution" of the

system

for _~/

~

0 does preserve the

angular

momentum if _we

impose

a

constraint _on the Hilbert _space. In this section _we

analyze

the "kinematics" of the

problem.

To

begin with,

let _us recall the

algebra

_g closed

by

the operators

I, lt, #, fit

defined _in

equation (21):

[1, lt]

=

16, fit]

=

1, [1, _fi]

=

[I, _fit]

= 0

(28)

At this level _we don't need to _{use an}

explicit

realization for them.

The

algebra

_{g is} the direct _sum of two

Heisenberg algebras:

g =

7ia

e

7ib (29)

On g viewed as a linear space the

angular

momentum

operator Jif

_{is realized}

as:

Jif

=

#t# ltl ₍₃₀₎

(for

the moment let _us consider it _{as a} definition of

it).

Now,

the basic

question

_we ask is what is the group of

(linear)

transformations defined onto g which leaves 3~t and the

algebra

_g itself

unchanged.

To _answer this

question

we restrict

ourselves to transformations of the form:

1'

=

~l

₊

~#t, fit'

=

~/I

₊

bfit lt'

=

~*lt

₊

~*#, ^#'

=

~/*lt

₊

^6*# ₍₃₁₎

Now _we

apply

the

previous

method to the _{case q}

~

0 which has

peculiar properties,

at least in the limit _{t »} i

/~/,

as it has been shown in references

11,2].

^{In fact such} a limit is

equivalent

to

require

that the "mass" of the

system

goes ^to zero [7] which is

expressed by

_an

explicit

constraint _on the Hilbert _space

[8].

This constraint is

equivalent,

in _{our case,} to

require

^that the

angular

momentum is conserved.

~~

_~~~~~~~~~'

ii',it'j

m

jj', jt'j

m

i, ii', j'j

m

ii', jt'j

m o

(32)

and:

fit'fi' It'l'

=

l~tfi ltl ₍₃₃₎

we obtain:

i -1 _/~ i _~l

ei~

^{- U W}

¹³⁴⁾

with the condition:

det U

= (~(~ (~(~ = l

(35)

The full _group of transformations which leaves

unchanged

the

algebra

_g and the

angular

_mo-

mentum

3i

_{is then}

_{given by:}

G =

su(i, i)

_x

uji) (36)

For

completeness

let _us write the four

I-parameter subgroups

of G. We have two boosts:

1(x)

= cosh

I

_sinh

() fit, fit ix)

= sinh

() ^I

₊ cosh

(37)

1(d)

=

cosh(~)l+isinh(~)l~t, _l~(d)

=

-isinh(~)l+cosh(~)fit ₍₃₈₎

(and

the

corresponding complex conjugates),

and two rotations:

Jil~)

"

e~~J, _l~ti~J)

=

e~~fl~t ₁₃₉₎

Jia)

=

e~iJ, l~tja)

=

e~fl~t ₁₄₀₎

Their

generators

may be realized _as

quadratic polynomials

in

I, lt, _{fl, fit} by requiring that,

if

l~

_{is the}

_generator

of the I-th

subgroup:

~~~

=

i[l~, _b~] (41)

where

b~

is _one of the four operators

I, lt, _{fl, fit.}

_{In this}

way we obtain:

li

=

i(lt fit lfi), 12

=

(fitfi

₊

It1) 13

=

(It fit

₊

16), 14

=

I(fitfi It1)

=

13i (42)

where it has been

explicitly

noted that

3i

_is

proportional

to the

generator

of the _group

U(I)

which commutes with the full _group

G,

_as it must be.

5.

Dynamics

in the _1~

= 0 Case

In the context of _our formalism the definition of _a

dynamics

consists in the identification of _a

"Hamiltonian

operator" acting

on the _{space g,} with the

following requirements:

I)

^{it does} preserve the

angular

momentum in the iull _{space or} where this is not

possible,

in

a

suitably

chosen

subspace;

it)

it

gives,

when

applied

to

I

_and

fit,

the correct

equations

of motion defined

by

the

"classical"

dynamics.

Let _us

present

this

approach

in the _~

= 0 _case.

The first

hypothesis

about the

dynamical

evolution is that it preserves the

angular

_momen-

tum. Due to the

symmetry G, (see Eq. (36)), li

is

expressed

_as:

fi

=

~ili

₊

_~212

₊

_~313

₊

_{~414 (43)}

where

lz

are the

generators

^{of G itself:}

j

~

10

i ^I0

^j

^{~~ 0}-l ¹0

~~

"

~l ^~~

^"

l~~)

From the above

requirements

it turns out that

fi

_{is of the form:}

~ lb

0 ~~~~

By noting

that the vector

(representative

of

Et

_is

an

eigenvector belonging

to the _zero

eigenvalue

of H and that the time evolution for A is _a

simple multiplication by

^the

phase

factor e~~ _we

finally

obtain _a "reduced" form for

fi:

fi

= =

)(13 ⁺¹⁴⁾

⁺

^{flR (46)}

and

correspondingly I, fi

_evolve

as:

~

J11J) ez~Jio)

fi~iiY) fi~i°)

~~~~

Then the

symmetry

_group G

gets

^restricted to its

subgroup

e

U(I)

_x

U(I).

As _a consequence

fiR

_{is in}

_general

modified

by

_a transformation U E G to it. In fact if _we take for U the

general

matrix:

U -

I i ₍₄₈₎

with ~~ (~(~ = l

(~

is

real),

_we

get:

I(I,

_~t)

=

Al

₊

~tfit, fit(I,

_~t)

=

I*fit

₊

~t*1 and,

then:

l~

^~

^l~~)

fiR(U)

~~~~

"

/~i

_(~t(~

~~

det

fiR(U)

=

l~(~(~ l~(~(~

= 0

(50)

That is due to the existence of the constant of motion

fit(1, _~);

_{however the other}

_eigenvalue

~(l, ~)

is _no

longer

_~

= l but has been modified to:

~ll,

_/~)

=

l~

₊

i/~i~

lsl)

In other terms the

frequency

which determines the time evolution of the

operator I _gets

renormalized

by

_a scale factor _p

=

1~

_{+ (/~(~} when

transforming

from

I, fit

_{to the basis}

I(I, _~t), fit(I, _~t).

6. Time Evolution Restricted _{to a}

Subspace

A

(linear) subspace

F of the full _{g space may} be defined

by requiring

that

~l

₊

_~tfit

= 0.

Let _us notice

that,

if _we restrict ourselves to

F,

the

"projected" angular

momentum is

given

by: RF

" 1l~ l~>

~)fit>' ~) ₁₅₂₎

where

fit(I,

_~t)

=

~t~l

₊

l~fit.

From _{now on} for

simplicity

of notation _we will refer to

Ill, _~)

_{and to}

fit _IA, ~) simply

_as

I, fit

By requiring

that time evolution leaves

unchanged

the

subspace

F

itself,

_we find that the

only

admissible candidates _are the two

U(I) subgroups

of G defined

by:

Ji~J)

=

e~~J10), ljti~J)

=

e~~ljt10) 153)

Jjd)

=

e~~J10), l~tid)

=

e~~~l~t10) 154)

However,

^since

I

_and

It

_{do not}

appear in

Ly,

we find _a

larger symmetry

_group ^{which includes} dilatations.

If

I

_{is the}

_generator

of the dilatations

given by:

1

=

(55)

we can write for the Hamiltonian

by:

Hy

=

a3A3

+

a4A4

+ 6A

(56)

in

agreement

^{with the}

previous requirements

_on time evolution.

We will _now show that this is indeed _an admissible form for the~ Hamiltonian in the _case q

~

0.

7. The

Subspace

F in the Case _1~

~

0

The

starting point

^is the

interpretation

of Newton

equations

of motion for the classical

quanti-

ties as

equations

of

dynamical

evolution for the

corresponding quantum operators (see

Sect.

3).

Let

I

_and

fit

_{be the}

_operators corresponding

to the

analytic

_component of

velocity ("chiral velocity")

_uz

= uz

+ivy

and to the "center of orbit coordinate" _z~

respectively.

The

equations

of motion for them _are

given by:

~

=

(~

+

i~u~)I djt

~

^{= 0}

⁽⁵⁷⁾

By reminding

that the action of the dilatation group D _on

I

_and

fit

_{is defined}

as:

lid)

~

ejJ1°)

~~ /(o)

BtliY) Bt1°) Bt1°)

~~~~

where

Ui

E

D,

we find that the time evolution differential

equations (59)

_may be rewritten in the form:

~ ~~~)

^~

~~~~ ~~~~

_~

~c~~ t~~)

~~~~

By comparing equation (59)

with

equation (56),

_we

_get

_a3

= a4 "

1/2

and 6

=

~/~uc.

To

give

_a

physical meaning

to _our

approach

_we remind that the

dependence

of the operators

I

_and

fi

on

1,

_~ and _on the "overall"

phase

_~g is

given by:

Iii,

~,~g) ⁼

d~(11

₊

~fit fl(1,

_~,~g)

=

e~~~(~flt

+

~l) ₍₆₀₎

where the

operators I

_and

fit

are

expressed

in _terms of the coherent coordinate

(,

I.e. the

eigenvalue

of

ic,

_as:

~ £~(~~~~

~ ~~~~/~~ ₍₆₁₎

By substituting equation (61)

into

equation (60)

_we

_get:

Also it has been shown in

ill

that _a

suitably

chosen subset of the Hilbert space does evolve in such _{a way}

that,

_as t _- cc, a basis for the

surviving

space of functions is defined

by

the substitution

(

_-

~f(.

This

subspace, being

the "time evoluted" of the

LLL,

has to be identified with the

subspace

annihilated

by I(1,~,~g).

_This consideration allows us to

determine the

parameters 1,

_~ and _~g

by simply comparing

the _two results.

To do

that,

_we notice that the form of the operators

I(1, _~)

_and

fill, _~)

_in

_{equation (62)}

allows _us to choose _a "vacuum" _state

simultaneously

annihilated

by I(I,

~t,~g) and

fill,

_~t,~g).

First, by looking

at the

expression

of

I

_and

fl, _equation (62),

the

requirement

of the existence

of _{a common} "vacuum state" annihilated

by

them

implies I,

_~t to be real parameters. ^Then

the "vacuum state" is

given by:

lbo _"

Cexp (-fl

^{~ ~}

^(((~j (63)

4 1~ _l~

In

ill

this state _was

explicitly

written _as:

~o

-

CexP li~ ^{/~i~~ <i~l} (64)

By comparing

the two

expressions

_we obtain:

~ _e

cosh(x/2)

= p ⁺

, ~t ^%

sinh(X/2)

=

lp ⁽⁶⁵⁾

2

~

2

~

where the "scale factor" _{p is} defined _as:

pelf(66)

~i~~

The

operators 1(~,

~, ~g) and

11([

_{~, ~g) are} then written _as:

>i~,/~,

~J~ ez~

ii

iP

⁺

ii >

₊

PI ^{1 ~)}

~ i~

ⁱ

ii ^j

~~~,~,~~

- ~z~

p~ ^{At +~} P+p

¹ⁱ

^i~~~

To determine the

phase

_~g, let _us

apply fit

to lbo°

~i

=

l~t~o

- e~~

fi<

exP

/~~l~~~ <i~l ⁶⁸⁾

From

ill

we obtain:

~~ ~~~

_~~~

^{~/~~~ ~~~~}

~~~~

from which _we obtain the final result:

~g = arctan

l~ ⁽⁷⁰⁾

1°c

Then,

if _we indicate with

l~

_and

_ll~

_{the operators defined for}

~

# 0,

_we

get:

~~~i~~~~i~

~~~i~~~i~

~~~~

Equations (71)

_are the solution to _our

previous problem,

since

they give

_us the

explicit

relation between

Jiq (which

_we

identify

with the

"velocity" operator), (

and the canonical momentum.

We notice that the above

expression

is

simply

obtained

by using equations (19)

for the

velocity

and center of orbit

operators

with the normalization

given by equation (21)

if _we make the

replacement:

ldc ~l~

(~

~

(

c

8. Time Evolution of _a

Complete

Set of Coherent States in

the1~ #

0 Case and

"Effective Hamiltonian"

In the

previous

sections _we have found the

operators l~

_and

_ll~

_{to evolve in time}

as follows:

J~lt)

=

J~1°)e~~~+~"~~~, ll~lt)

₌

_11~1°) 172)

where the

operators l~(0)

and

ll~(0)

have been defined in

equation (71).

Let _us notice that since

[l~(0),ll~(0)]

=

0,

_so we can

simultaneously diagonalize l~(0)

and

ll~(0).

^If _oo ^{denotes the}

eigenvalue

of

l~(0)

and _So the _one of

ll~(0),

we

get

^the

following

simultaneous

eigenstate

of the two

operators:

16(ao,fo)

"

~~~P ~/~~~

~~~~

_~~~

~~~~~~

^~

~~~~~~

~~~

~~~~~~

~

~~~~~3)

To construct the

corresponding

state at _a time

t, 16(oo,fo,t),

_{we can} take into account equa- tion

(72)

and

repeat

^the

arguments

^of Section 3

obtaining:

, , =

~~~~

_{~~ ~~}

~~~~ ~~~~~~ _~~~~

~~~

~~$fl~~°

^~

jfldaoe~(~+zW~)tj

~

1

~~~

2~~~0(~e ~~~+jj~j2)

The

subspace

defined

by l~F

= 0 is

spanned by

the coherent _states with _ao

"

0,

which _are

of the form:

16((o

_" C _exp ~ ~ _~°~

z(~

exp

~°~ ~~

z(o

_exp

to

_(~

(75)

41°c

^~

fi

2

Since _{So is an}

explicitly time-independent

constant

of motion,

_we obtain that this

subspace

is left invariant

by

time

evolution,

_as

required.

The

angular

momentum restricted to such _a

subspace

is

given by:

fly

=

11((t)ll~(t)

=

11((0)ll~(0) (76)

which

implies

that

Ly

is _a constant of motion.

For _a

general

state the

angular

momentum

operator:

flit)

=

l~jj0)1~~j0) e~~~~Jjj0)J~j0) j77)

has

expectation

value _on the state

16(cxo,fo)

llblao, &)13ilt)ilblao, &))

=

ifoi~

icYoi~e~~~~

178)

showing

how the _non conservation of

3i

_{is due to the}

dissipative

constant ~.

As t _{- cc}

only

the

subspace

F

survives,

which

gives

a one-dimensional

representation

of the

Heisenberg algebra given by fl~

^and

fl(.

Since the "scale factor" _p which _appears in the

"metric factor"

exp(-uJcp~(z(~/4)

is

given by:

p~ =

~~~ (79)

~-l~

we derive two

important

^facts:

I)

^the

representations

^obtained

by varying

_p are, in

general, "inequivalent";

it)

we can "select" which

representation

will be

asymptotically

obtained

by changing

the definition of

l~

_and

_{fl(, I.e.,} by changing

^the values of

~,

_~ and _~g.

In

ill

it has been shown

that, by selecting

different

inequivalent representation (I.e., by choosing

different values for _~ and

uJc),

^the "conduction

properties"

of the

system change significantly.

Having

now at our

disposal

_a

(complete)

set of coherent states and their detailed time

evolution,

_{we can} define _an

operator 7iea

in such _{a way} that:

1(~(a, ^{i, t)}

m

i~a~(o, _{i, t) (80)}

From

equation (80)

_we

get

as an

explicit

realization of the Hamiltonian:

fi~~

=

~°C

~~l§~t§~ j81)

UJc

Let _us notice that this

operator

^does

generate

^{time evolution for the}

operator I(t),

while the

one for

it It)

is

generated by

the

conjugate

_one:

fit

^{~°C + ~~l}

j~th~ j82)

ea

~

We have verified that

7iea given by equation (81)

^coincides

, as it should be for

consistency

with the

general expression 7iF

for the values of the

parameters

found in Section 7.

We have

explicitly

evaluated how the

generic

Landau state

ILL) projects

_on the _{two param-} eters coherent states

(see Eq. (74) ), I.e.,

the matrix element

lfalt)ILL)

°~~~~~~~~"

t-m

jj2j ~+Ldllzl~j

(f°(t)(LL)

_{~ ~~P}

_~$

_~~~

uJ)

⁴

In other terms the Landau level

ILL)

has _{a non zero}

projection

_on

large

time scales

only

_on

the ~

#

o

ground

state

given by equation (64), extending

the result obtained in reference

ill.

9. Conclusions

In this _{paper we}

study

the

"dynamics"

of _a 2D electron in _a transverse external

magnetic

field in the _presence of

dissipation.

After

constructing

_a

complete

set of constants of motion _we exhibit _a

complete

set of coherent states

depending

on two

parameters (, cx(t).

Then the time evolution of _a

given

Landau level is

explicitly

determined at any t and _we

recover the results obtained in

ill

for t »

1/~.

References

iii

Cristofano

G.,

Giuliano

D.,

Maiella G. and Valente

L.,

Int. J. Mod.

Phys.

B 9

(1995)

707.

[2] Cristofano

G.,

Giuliano

D.,

Maiella G. and Valente

L.,

^Int. J. Mod.

Phys.

B 9

(1995)

3229.

[3] Dekker

H., Phys. Rep.

80

(1981)

1.

[4] Malkin I-A-, Man'ko V-I- and Trifonov

D-A-, Phys.

Rev. D 2

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1371.

[5] Croxon

P., Phys.

Rev. A 49

(1994)

588.

[6] Malkin I.A. and Man'ko

V-I-,

Zh.

Eksperim.

I Tear. Fiz. 55

(1968)

1014; Soviet

Phys.

JBTP 28

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Feldman A. and Kanh

A., Phys.

Rev. B1

(1970)

^4584.

[7]

Colegrave

R.K. and Abdalla

M-S-,

J.

Phys.

A 14

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2269.

[8] ^{Dunne G. and Jackiw}

R.,

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l14.