Approximate Time Evolution Operator of an Anharmonic Oscillator

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Approximate Time Evolution Operator of an Anharmonic Oscillator

L. de Falco, R. Mignani, R. Scipioni

To cite this version:

L. de Falco, R. Mignani, R. Scipioni. Approximate Time Evolution Operator of an Anharmonic Oscillator. Journal de Physique I, EDP Sciences, 1995, 5 (5), pp.535-538. �10.1051/jp1:1995148�.

�jpa-00247078�

Classification

Physics

Abstracts

02.60Gf 05.90+m 33.10Cs

Short Communication

Approxilnate Time Evolution Operator of

_an

Anharmonic Oscillator

L. De Falco

(~),

R.

Mignani

_(~>~) ^and R.

Scipioni (~)

(~)

Dipartimento

di Fisica "G.

Marconi",

Università

degh

Studi di Roma "La

Sapienza",

P. Le A. More 2, 00185 Roma,

Italy

(~)

INFN Sez. di Roma I, Università "La

Sapienza",

P. Le Aldo Moro 2, 00185

Roma, Italy

(~)

Dipartimento

di Fisica "E. Amaldi", ^III Università

degli

Studi di Roma, via C.

Segre

2,

00146 Roma,

Italy

(Received

21

February1995,

received in final form 14 March 1995,

accepted

20 March

1995)

Abstract. We _{use an}

approximation procedure

to calculate trie form of trie time evolution operator ^of an anharmonic oscillator in trie _region where perturbation theory _{is no}

longer

vahd.

Suppose

_we bave _a

system

described

by

trie Hamiltonian

iii

p2

A

H = +

-X~

₊

BX~ (1)

with B _> o and

great enough

_so that _we cannot _use

perturbation theory.

Let

lY(o)

^{be trie} wave function at time t

= o. We

know,

from

elementary quantum mechanics,

that at an

arbitrary

time trie _wave function

describing

trie system can be obtained

applying

trie time evolution

operator T(t, _o):

qt(t)

=

T(t, o)qt(o). (2)

By decomposing il(0)

_{as a} linear combination of

eigenfunctions

of

H,

_{un, we} bave:

il(t)

=

T(t, 0)il(0)

=

T(t, 0) ~j

_cnun

=

~j cnT(t, 0)un. (3)

The form of trie

operator T(t, 0)

is _as usual

T(t, 0)

=

e~~~~ (4)

so that for

equation (3)

we can write

il(t)

=

~j cne~~~"~un (5)

n

©

Les Editions de

Phvsique

1995

536 JOURNAL DE

PHYSIQUE

I N°5

Up

to this

point,

^there is

nothing special, but,

when

considering

_{a region} in which trie pertur- bation

theory

is not

valid,

it is in

general

diflicult to

get

^trie

expression

of trie time evolution

operator,

so we cannot write trie function at _a

generic

time t.

In this _{paper we} wish to

apply

trie

optimization procedure

of Burrows et ai.

[2,

3] ^{to find} trie approximate ^form of trie time evolution

operator

in trie _case of _a

system

^described

by

trie Hamiltonian

(1).

To this

aim,

_{we use a} ^scale

trick,

which consists in

performing

trie

following

scale transformation:

~ _~

~yl/2~jl/6~

^~~~

with _{a a} suitable

parameter

to be determined.

Then Hamiltonian

(1)

takes trie

following

form

H(A, B, a)

=

aB~/~(Ho

₊

V), (7)

"'~~~~'

l

d~ x~

(8a)

fÎ0 ~j

_{~~2 2}

~

~

~2

~2/3)

~~ _~

~3'

^~~~~

By

_a little

algebra

_we

get

~ ~~~~

j~~

'~~~

~

~

~~~

i

where

~3

A

1-j

_~y2~f2/3

(io)

Following

reference _{[2] we} obtain _a lower bound for trie

eigenvalues En

of

equation (1) (whose

exact form _{we are} unable to

find)

En (a)

<

En (II)

The

approximate eigenvalues En(a)

_can be calculated

by

trie variational

equation:

àE~(a)

~ ~~~

do

givfg

En(«)

=

aB~/~

n +

~~ (13)

~ ~

"

It is worth

observing that,

while trie method used in reference _{[2] is} valid for _a

generic sign

of

A,

_in this letter _we restrict ourselves to consider

only

trie

quartic

anharmonic

oscillator,

excluding

the _pure

quartic (A

₌

o)

and the double well

potentials (A

<

0).

In such _{a case,}

in Hamiltonian

(1),

A _can be identified with the square of the

frequency.

We want to stress that _every

eigenvalue

has its _own

optimization, independently from

the others.

Solving equation (12)

amounts to

finding

trie real root of trie third-order

algebraic equation

~~ jl13

^{~ ~}

^~

^~

^~

^{~ ~~~~}

whose

general

^solution _can ^be written

using

trie Cardano formula:

i ²

A3

^~/~

~~~

o*

= 2

n

+

-)

+ 4

(il+ -)

_q +

2 2 278

~

(15)

lj _lj~ ^{A~ ~~}

~ ~ _~ ~

2

~ _~ ~

2 2782

Taking

trie limits A _- 0 and A _{- oo we}

get

for trie

eigenvalues, respectively:

4/3

En(ajow)

=

2~/~B~/~

n +

,

(16a)

~

En(ohigh)

%

A~/~ (n

^{+ +}

(B ⁿ

₊

_(16b)

2 2

~

Then,

for trie time evolution operator

applied

to trie n~~

eigenvector,

_we obtain

~~~ ~~~

i ^4/3

e

~~

~

~~ ~ ~un

(17a)

~~~'~~~~

lj

²

_lj~)

~~ ~~~~ _"~

2 ^~

A~

~ ~

2

e un

(17b)

in trie low and

high frequency limits, respectively. Equations (17) provide

_an

approximation

to trie time evolution

operator

^for an anharmonic oscillator. For _a value of trie

coupling

constant B fixed but

great enough

_so that _we cannot _use

perturbation theory,

_{we may assume} in first

approximation

that:

T(t, 0)

m

T(a*,t, 0). (18)

Thus, remembering

that A _may be identified with trie _square of trie

frequency,

_we

_get:

T(a(~~)

= e~~~~~~~~~~~

(19a)

T(cyj,

~~ WHO+

~

H2j

igh ^-e

w2 ^{0 t}

~~~

In

conclusion, exploiting

trie

scaling procedure

introduced

by Burrows,

Cohen and

Feldman,

we can derive _an approximate

expression

of trie time evolution operator ^of an anharmonic oscillator. It is worth

noting

that in

correspondence

to a* _we

get

trie closest

approximation

^for

such _an operator.

Moreover,

trie

approximation

of _every matrix element is

independent

from trie others. Trie

possible application

of these

ideas,

for

example

to some statistical mortels of

quantum

gravity,

is

currently

under

investigation

_[4].

538 JOURNAL DE

PHYSIQUE

I N°5

References

iii

See _e-g-, Base S. and

Tripathy

D., Fortschr.

Phys.

31

(1983)

131.

[2] Burrows

B.L.,

^{Cohen M. and Feldman}

T.,

J. Math.

Phys.

34

(1993)

1.

[3] De Falco

L., Mignani

R. and

Scipiom R., Europhys.

LeU. 29

(1995)

659.

(4] ^{De Falco}

L.,

_m

preparation.