12 Zero Magnetic

8/10/2019 12 Zero Magnetic

1/8

J . Phys.

A:

Math. Gen.

17

(1984) 1631-1638. Printed in Great Britain

Coherent states

of

an electron in a hom ogeneous constant

magnetic field and the zero magnetic field limit

S Varr6

Central Research Institute

for

Physics, H-1525 Budapest

114,

O B 49, Hungary

Received 20 September

1983

Abstract. Cohere nt states of an electron embedded in a constant homogeneous magnetic

field are constructed. Th e cen tres of the probability distributions belonging to these s tates

gyrate along possible classical trajectories. Suitable packets of such coheren t stat es are

defined which reduce to properly normalised free electronic states in the zero magnetic

field limit. simple example is given to illustrate the dynamics of free electron localisation

due to the presence of a magnetic field.

1. Introduction

The stationary solutions to the Schrodinger equation of an electron in a constant

homogeneous magnetic field, the Landau states, have long been known and can be

found in every textbook on quantum mechanics ( Lan dau and Lifshitz 1 97 5). T hough

the Landau states are very powerful and a commonly used mathematical tool for

describing various physical processes which take place in uniform m agnetic fields, they

have a seriou s insufficiency. Namely, they d o not rep rodu ce any kind of f ree electro nic

states in the zero magnetic field limit.

Recently many physicists have for example worked on the theory

of

potential

scattering in the presence of external electromagnetic fields. In particular, induced

and inverse bremsstrahlung processes in the presence

of

a uniform magnetic field are

being very extensively studied due

to

their importance in fusion research (Seely 1974,

Ferrante

er a1

197 9, Bergou

et a1

19 82 ). T he difficulty with these calculations is tha t

th e scattering cross sections do not t ak e th e field-free form if Landa u states ar e used

as initial and final states (see also Ven tura 19 73 ). T he sou rce of these inconsistencies

is clearly the application of L andau state s, so it is imp ortant to find suitable superposi-

tions of the Landau states which behave properly in the zero magnetic field limit (see

also Faisal 1982).

The aim of this paper is to present a possible solution to the problem of the zero

magnetic field limit. In

2 we give an elem entary m ethod to construct coherent states

of a Schrodinge r electron em bedded in a constant hom ogeneo us magnetic field. O ur

result coincides with tha t obtained by Malkin and M anko (1 96 9) who used a diff ere nt

approach to the problem. In 3 it is proved that a special class of coherent states is

able to reprod uce free electron states in th e field-free limit. In

4

a short summary

completes our analysis.

0305-4470/84/081631+08 02.25

98 4 Th e Institute of Physics 1631


2/8

1632

S

Varrd

2. Coherent states of an electron in a homogeneous constant magnetic field

Coherent s ta tes

of

a char ged partic le in a mag netic field we re first published by M alkin

and M anko (1969) who have developed a general formal theory of such states based

on suitably defined creation and annihilation operators of the quantum excitation

of

the system unde r discussion. The mo re general proble m of co here nt states of charged

particles in a uniform electric field and a m agnetic field has been th orou ghly discussed

by Johnson et a l (1983) . In the present paper we concen trate on the problem of the

zero magnetic field l imit and we d o not nee d th e com plete theory developed by M alkin

and Manko. The explicit form

of

the mentioned coherent states in coordinate rep-

resentation will be used in

3 .

Due to this, we think that it is not completely useless

to

construct them by using

an alternativ e and sim pler (thoug h less gene ral) method which is based simply on th e

well known one-dimensional harmonic oscillator wavefunctions.

Th e coherent s ta tes

to

be presented here correspond

to

gaussian probability d istribu-

tions

of

the electrons posit ion on the

x - y

plane gyrating a long circles with th e usual

cyclotron frequency

w c =

l e / B / M c )

e

and

M

are the electrons charge and mass,

respectively,

c is

th e velocity

of

light in vacuum a nd B is th e magn etic field stren gth) .

So

the centres

of

these distributions move alo ng possible classical trajectories. Th eir

widths are th e sam e, namely the usual magnetic length

y-1 /2= 2 h ~ / l e l B ) ~h

is the

Planck constant divided by

2 ~ ) .

he mentioned coheren t states ar e parametrised by

two complex numbers representing the velocity, the initial phase and the position

vector

of

the guiding cent re of the gyration.

The Schrodinger equation for an electron in a uniform constant magnetic field

directed along the

z

axis reads

M

[(

x

-e)

+( y+E) 4

where

w

=

( / e lB /ZMc)

=

4 2 )

s the Larm or frequency,

L,

=

xpy

p ,

is the

z

com-

ponent

of

the angular momentum opera tor , and p x

= - ih (a /ax) ,

p y

=

- ih(d/dy) are

the canonical mom entum operators. In

(2 .1)

and henceforth we adopt a symmetric

gauge vector potential

(B/2) ( -y , x, 0)

and leave out

of

account the irrelevant z -

dependence

of

the electronic motion. By introducing the polar coordinates

p

and cp

with the definitions x =

p

cos

cp, y = p

sin cp, equation

(2.1)

can be brought to the form

The stationary solutions to

(2 .2)

are the well known L andau states (Landau and Lifshitz

1975)


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1634 S Varrd

after applying the rotation operator exp(-wt

a/dcp)

we get th e following explicit form

for

@ :

4 , @ ( p , p ; t ) = M ~ , / 2 ? r h ) l ~

xp(-$iw,t) exp(-(Mwc/4h)

~ [ p ~ + I a _ l ~ + I a + 1 ~ + 2 a - a +

xp(-iw,t)

2a-p elv exp(-iw,t) - 2 a + p e-]),

2 . 7 )

where

a ,

=

;(a kip) . 2 . 7 ~ )

Of cou rse, the cohe rent state given by 2 .7 ) satisfies the Schrodinger equation

2 .2 )

by construction, but this can also be proved by direct substitution.

Ap art from notational differences

2 .7 )

coincides with the coherent state given by

Malkin and Manko (see equation 3 9 ) in their paper).

To

see the physical meaning of t he co here nt stat e d a B irst

let

us

cite th e classical

description of cyclotronic motion . T h e New tonian equation of an electron in th e

uniform magnetic field under discussion reads

U

= w c ( U X n , where n is a unit vector

along the

z

axis and

U

is the velocity

of

th e electron. Th e solution t o this classical

equation is as follows:

x ,

t )

=

(v/w,)[cos(w,t+x ~ / 2 )COS(^ ~ / 2 ) ] +

o ) ,

2 .8 )

where x O ) , y ( 0 ) ) is the initial position vector and COS

x,

in x

=

v, O),

v, O))

is

the initial velocity on the

x - y

plane at

t

=

0.

given by 2 .7 ) we can easily

realise that if we identify a- and a+ with the following combination of classical

quantities,

y,(t)

=

(v/o,)[sin(

w, t +x ~ / 2 )

in(x-

~ / 2 ) ]

y(O),

Now , if w e calcu late th e modu lus squ ared of

=

( v / w , ) e-l(x-n/*)

a+= u 0 ) - a* , u 0 )

= x 0 ) +iy(O),

2 .9 )

the result for

/ p / 2

can be brought

to

the form

x - ~ ~ ( t ) ) ~ + ( y - y ~ ( t ) ) ~ ]

2: lO)

Equation

2 .10)

tells us that

Id,p12

is a gaussian probability distribution of width

2 h /Mu,)* = Y - ~

whose cen tre follows the classical tra jec tor y 2 .8 )without changing

shape,

so

is really a coh ere nt stat e in th e usual sense. By taking t he classical limit

h-* 0 we get from 2 .10) a two-dimensional Dirac delta distribution describing the

classical motion of a point-like electron.

Th e connection between the co herent states C J,~ and the usual Lan dau states given

by 2 . 7 ) and 2 .3 ) , respectively, can be established by expanding the exponential

into power series and rearranging the triple sum we get after that expansion. The

result is

2 . 1 1 )


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Coherent states and the zero magnetic field limit

1635

where we have introduced the notations

U

=

(Mwc/2h) 2a- , b=(M w,/2h)1 '2a+ , (2 .11 a)

T he probability distribution of L andau states 41ms a two -dimensional Poisson distribu-

tion. It can be proved that

f f

ICfm12= = -m l :/ml,(21abl)) exp(-Iu12-lb12)=1. ( 2 . 1 1 ~ )

m= - m f = O

In (2 .11 ) we have used the power series represe ntatio n of th e modified Bessel function

I m

and the relation

(Gradsh teyn and Ryzhik 1980 ). Of course, it is not very surprising that t he d istribution

ICfmI2

s properly normalised as stated by ( 2 . 1 1 ~ ) ince the coherent state

4oLp

s

normalised to unity and the states 41morm an orthonormal ised se t (see (2 . 3 ~ ) ) .

3.

The

zero

magnetic field limit

In the present section we shall consider only a special class of coherent states given

by (2.7 ). W e set

u 0 )

= 0 in (2.9), so a+= -a in this case. T he special coheren t st ate

4oLporrespon ding to these particular values of a- and a+ can now be characterised

by a two-component real vector K =

M u ( O ) / A ]

=K(cos

x,

sin

x ) ,

where o(0) is the

initial velocity of t he classical mo tion. T he cen tre of t he circle alon g which the elec tron

gyrates is given by the vector (1 /2 y ) fi (1/ 2y )K (-si n

x,

cos

x )

which is perpen dicular

to the vector K as is shown in figure

1.

If the mag netic field goes to zero y 0) the radius of the trajectory goes to infinity

and t he circle is degenerated to a straight line representing a uniform m otion of a f ree

electron with the co nstant velocity

o(0)

= h K / M ) . Since the centr e

of

the probability

distribution 14mp(2ollows th e classical trajectory and the width of th e distribution is

proportional to

B-1 2,

on e may expect that in the zero m agnetic field limit the coh eren t

state

q5ap

will be reduced to a free electronic plane wave with w avevector K. To show

this, first we write down the explicit form of such a coherent state:

exp(

-iw,t)

4Y -iw,t


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Coher ent states and the zero m agnetic field limit

1637

an arbitr ary function of

K ,

d2K Ig(K)( = 1 .

3 . 5 ~ )

In 3 . 4 ) we have introduced the real normalisation factor N,(y) with the definition

N i y )=

/ /{d2 Kg (K )4K 112 . aking into account 3 . 2 ~ )e get

N i y ) =

d 2 K d 2 K g * ( K ) g ( K ) c ? y ( K - K ) .

3 . 5 b )

We note that due to 3 . 5 b ) , 3 . 2 ~ )nd 3 . 5 u ) ,Ng ( y )ends t o unity if y tends to zero.

Now if we ta ke th e limit

y

goes to zero from 3 . 4 ) we have

d 2 K g ( K ) - e x p ( - i - h K 2

27T 2 M

Equation 3 . 6 ) epre sent s the main result of th e present pap er.

4g (x ;

I

y ) is a solutio n

to the Schrodinger equation of the electron in the presence of a constant homoge neous

ma gnetic field and g ( x ; t ) is a free electron wavepacket. Both of th em a re normalised

to unity and & ( x ;

tI y )

reduces to

g ( x ;

) in the zero magnetic field limit. We note

that thoug h coherent states form a comp lete set, they are not orthog onal to each ot her.

Thus , on e may expec t tha t th e applicability of such states in scattering the ory as initial

and final states is questiona ble. This problem

is

the subject of our forthcoming paper

(V a r r6

et a1

1 9 8 4 b ) .

Bef ore concluding this paper we give a simple example fo r the packet solutions of

the type given in 3 . 4 ) . Let us take th e weight factor g ( K ) in the form of a gaussian

distribution of parameter

r

The probability distribution

l~ ,&x; y)12

can be easily calculated, yielding

where we have introduced the time -depend ent width cry t ) by the definition

3 . 7 )

In 3 . 8 ) , x c ( t ) ,yc( t ) ) represents a classical trajectory of the type given in 2 . 8 ) with

x(0)

= y ( 0 )

= O

and initial velocity

~ ~ ( 0 )

h K o / M ) .From

(3.8)

and 3 . 8 ~ )ne can

see that the centre of the packet solution 4, follows the classical trajectory as the

simple coherent states

C J~@

do (see also (2 .10 )), but the distribution has a time-

depen dent width a, ?)which oscillates with the La rm or frequency 4 2 ) . That means

that the maximum value of 14,,12 and the width of it vary periodically in an opposite

m ann er. W e can inter pret this oscillation as a result of th e competition between the

natural spreading-out tendency of a free wavepacket and the contraction effect due

to the presence of the magnetic field which results in the localisation of the electron.


8/8

1638 S Varrd

4.

Summary

The aim of the present paper was to show certain superpositions of Landau states

which reduce to free electron plane waves or normalisable free electron wavepackets

in the zero magnetic field limit.

T he limiting pro ce du re can be easily carried o ut for a classical electron if we solve

the initial value problem of t he Newtonian equ ation of m otion. Th e classical trajec tory

of the w ell known cyclotronic motion is a circle whose rad ius is equal to th e ra tio of

the m odulus of the initial velocity and th e cyclotron frequency. In the z ero magnetic

field limit the circle becomes a straight line describing a uniform m otion of t he el ect ron

with the same initial velocity and position as we had for the gyration along the circle

when the magnetic field was on.

W e have first constructed coherent state s of the electron em bedded in the magnetic

field in the hope that they behave as properly as their classical counterparts in the

zer o ma gnetic field limit. It has been show n in the second part of o ur paper (se e

(3.3)

and (3.6)) hat this expectation was corr ect, so the states & introduced in (3.2) reduce

to fre e electronic plane waves of wavevector K as the m agnetic field goes to zero. We

have also proved that suitable normalised superpositions of c ~ eproduce normalised

fre e elec tron wavepackets with the same norm in the ze ro field limit, thus, for cohe rent

states the limiting procedure does not affect the normalisation.

Finally, we gave an example for gaussian packets of c ohe rent states. Th e centres

of the probability distributions belonging to these states again follow the classical

trajectories, but the width of these distributions oscillates with the Larmor frequency.

Th e latter result can be considered as a possible description of t he dynamics of el ect ron

localisation d ue to th e presence of th e magn etic field.

Acknowledgment

The author is very grateful to Professor F Ehlotzky for many stimulating discussions

on th e cohe rent states and on the problem of t he ze ro magnetic field limit.

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