Generalized WKB method through an appropriate canonical transformation giving an exact invariant

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Generalized WKB method through an appropriate canonical transformation giving an exact invariant

J. Guyard, A. Nadeau

To cite this version:

J. Guyard, A. Nadeau. Generalized WKB method through an appropriate canonical trans- formation giving an exact invariant. Journal de Physique, 1976, 37 (4), pp.281-284.

�10.1051/jphys:01976003704028100�. �jpa-00208422�

LE JOURNAL ^DE PHYSIQUE

GENERALIZED WKB METHOD THROUGH AN APPROPRIATE CANONICAL

TRANSFORMATION GIVING AN EXACT INVARIANT

J. GUYARD

(*)

and A. NADEAU

Institut des Sciences de

l’Ingénieur,

^{Parc Robert}

Benz,

⁵⁴⁰⁰⁰

Vand0153uvre,

^France

(Repu

^{le 10 novembre} 1975,

accepte

^{le 11 décembre}

1975)

Résumé. ²⁰¹⁴ La résolution d’équations différentielles du type

d2q/d03C42

⁺

03C92(03C4).q

^{= 0 est d’un} grand intérêt ^en Physique. ^{On introduit souvent une fonction} auxiliaire, solution d’une équation

différentielle non linéaire, que l’on peut ^résoudre par ^une méthode de perturbation. ^Cette approche

n’est en fait rien d’autre qu’une extension de la fameuse méthode WKB. Lewis a trouvé un invariant exact du mouvement exprimant ^{une relation} entre cette fonction et les coordonnées. Par le biais d’une transformation canonique originale ^{nous dérivons cet invariant exact de} façon extrêmement

simple ^ce qui peut constituer désormais une introduction naturelle de la méthode WKB

généralisée.

Abstract. ²⁰¹⁴ The solution of differential equations ^{of the} type

d2q/d03C42

⁺

03C92(03C4)q

^{= 0 is of} great interest in Physics. Authors often introduce an

auxiliary

function w, solution of a differential equation

which can be solved by ^a perturbation method. In fact this

approach

^is nothing ^{but an extension of the}

well known WKB method. Lewis has

found

^{an exact} invariant of the motion given in closed form in terms of this w function. Using ^an appropriate canonical transformation this exact invariant

can be derived in a much easier way. This method ^{can now} be used as a natural way of

introducing

the WKB extension.

Classification Physics ^Abstracts

1.320

1. Introduction. ^- When

solving

^the

time-depen-

dent harmonic oscillator

equation [1-5]

with

position

q, time r and

frequency

^{w or the one-}

dimensional

Schr6dinger equation [6-7]

with wave-function

§, position

^{i and where}

W2(t)

^is

a function of the wave number k and the

potential U,

most authors introduce an

auxiliary function w,

solution of the differential

equation

with a a constant.

When

m(i)

^{is a}

slowly varying

function of _{r, eq.}

(3)

can be solved

by

^a

perturbation

method which is not

the case for the initial

equations ;

moreover we have shown

[8]

^{that an} iterative scheme is a very

powerful

way of

solving

^eq.

(3). Using

^{this w}

^function,

^Lewis

[9]

has derived an exact invariant of the motion. In this paper ^we obtain this exact invariant _very

easily by looking

^{at the}

generating

function of an

appropriate

canonical transformation. In fact it can be seen that the

perturbation

^{treatment of} eq.

(3) gives

^the

higher-

order WKB solutions.

Eq. (1)

^and

(2)

^are

formally identical;

^{we can}

restrict our

study,

without loss of

generality,

^{to the}

first and extend the results to the second.

2. Derivation of an exact invariant of the motion. ^- In

analytical dynamics

^{exact constants of the motion}

are

usually

^obtained

by

the canonical formulation : if the Hamiltonian is

independent

^{of a}

given

^coordinate

the

conjugate

^{momentum is an invariant.}

We consider a system

having

^one

degree

^of

freedom,

so that the Hamiltonian _may be written

1

where q

^and p ^are

respectively

^the

conjugate position

and momentum and t ^-

m(t)

^the

time-dependent frequency. Eq. (1)

^{is the}

corresponding equation

^of

motion;

^{such a} system is called a

time-dependent

harmonic oscillator.

We are

looking

^{for a} canonical transformation ^to

change

^the

(p,

q,

t)

^{set into a new one}

(P, Q, 0),

^such

that the new Hamiltonian K is

independent

^{of 0 and}

is of the form :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01976003704028100

282

If now 0 is considered as a new time

variable,

^K represents ^the Hamiltonian of an harmonic oscillator the

frequency

^{of which is constant and}

equal

^to

unity.

Let us now follow the

trajectory

^{in the}

phase

space

corresponding

^{to each set} of variables.

In the _p-q

plane

^we get ^a

trajectory

^which

spirals (phase-space

^{curves cannot cross}

over)

^{as shown in}

figure la ;

^{in the}

P-Q plane (Fig. lb)

^the

trajectory

is a closed curve

(more precisely

^a circle since the

frequency

^is

equal

^to

unity).

FIG. la. - q-1) plane : ^the parameter t gives ^the position ^{on the}

curve.

FIG. lb. ^- Q-P plane : ^the parameter 6 gives ^the position ^{on the}

curve.

The

new

canonical

representation

^{is :}

In an

ordinary

^canonical

transformation,

^{the time t}

is

kept unchanged.

However when t is not an invariant parameter of the motion

(for example

^{in a relavistic}

formulation)

it is then

possible

^{to find a}

generating

function in which t is treated as a new

coordinate;

the Hamiltonian is now the

conjugate

^momen-

tum

[10].

For convenience the t-transformation will not be included in the

generating

function. The entire trans-

formation will be made into two steps :

- a

change

^{of the t} scale into the 0

scale,

- a canonical transformation from _p-g to

P-Q.

New time scale:

If we let

L(P, Q, t)

^{be the} Hamiltonian

expressed

in terms of these

variables,

^{we must have}

but we also have

so that

substituting

ⁱⁿ

(7a-b)

^{we obtain}

Integrating (9a)

^with respect ^to

Q,

^{we obtain}

and now

differentiating

^{with respect to P and iden-}

tifying

^with

(9b)

^{we obtain}

Upon integrating

^{we obtain}

Choosing

^{L = 0 when P =}

Q

^{= 0 it} follows that

f2 -= 0. Finally

Canonical transformation from the coordinate

(q, p)

^{to the variables}

(Q, P) :

We shall choose the

generating

^{function to be of}

the form

We choose the linear transformation of variables

The momentum p ^{and the new variable}

Q

^{are then}

given by

To find F we must express p and

Q in

^terms

of q

and P.

It is easy ^{to see} that

and

Integrating (l5b)

^with respect ^to

P,

then differen-

tiating

^with respect to q ^and

identifying

^with

(15a)

^we

obtain

where

f

⁼

f (q, t)

^{is a function}

of q

and t which is introduced when we

integrate.

^F

being independent

of P we must have in _eq.

(16)

which

yields

Now to determine w we have to

apply

^the

following

relation between the old and new Hamiltonians and the

generating

^function

[10] :

L, X

^and

OF10t being expressed

^{in terms of} q, P and

w, after a little

algebra

^we get

Then it follows

and we can write

This

result,

first derived

by

Courant and

Snyder [ 1 ],

has been

independently

^derived

by

^Lewis

[9] by applying

^Kruskal’s

asymptotic

^method

[11].

Then we can take as an invariant of the motion the Hamiltonian of the new system :

which can be

expressed

^{in terms of p, q and w}

Lewis remarked that _eq.

(20)

^{defines not}

only

^an

invariant of the motion but more

exactly

^{a class of}

invariants if we take account of the fact that all the

particular

solutions of the differential eq.

(19) gives

^an

invariant of the motion.

3. Connection with WKB solutions to all orders. ^- Let us consider

again

eq.

(1)

If

m(i)

^{is now a}

slowly varying

function such that :

that is the

percentage change

per

period

^is

_to

^{be small.}

Such

variations are called adiabatic.

Jo

⁼

Elw,

^where

E

is the mean energy of the

oscillator,

^{is an}

approxi-

mate constant called an adiabatic invariant.

Approxi-

mate solutions of

(1), taking

^{into account}

(21), implies

^an iterative scheme.

Solving (1) by perturba-

tion is not

possible;

^{therefore in the WKB}

approxima- tion,

^{extended to} all orders

by Kulsrud,

the idea is to

look for a solution

Differentiating

^{twice and}

substituting

^into

(1)

^we

obtain :

We equate ^{real and}

imaginary

parts ^to

get

284

We rearrange the second of these

equations

^to

obtain :

which

gives

^after

integration

where a is an

arbitrary

^{constant which can be taken}

equal

^to

unity by correctly choosing

the initial condi- tions on w and _QB.

Substituting (25)

ⁱⁿ

(24a)

^{we obtain :}

which is identical to

(19).

So we have got ^the

interesting

^result that the w function and its differential _eq.

(26), arbitrarily

introduced

by

^{the WKB}

approximation,

^{follow natu-}

rally

^{from an} invariant of the motion. We can say that w is a variable

indicating how q and p

^{must be}

combined

(through (20))

^{to recover a} strict harmonic motion.

Yet it is

possible

^to find solutions of

(26) by

pertur- bation which are

asymptotic

^{to the exact solution.}

In the same way if ^we

inject

^some

approximation

^{of w}

in

(20)

^we shall obtain a

hierarchy

of adiabatic inva-

riants, Jo being

^{the first one.}

There is a

physical

realization of the w function in accelerator

analysis :

^{if we consider a}

periodically

focused proton

bunch,

^{that is a set of}

particles

^the

motion of which is

governed by (1)

^with

m(i) periodic,

then w is the beam

envelope.

^{The invariant I becomes}

the well-known

typical quantity

called emittance.

(See

^for

example the

^review

by

^{Courant and}

Snyder -

ref.

[1].)

4. Conclusion. - We have derived an exact inva- riant of the motion for the time

dependent

^harmonic

oscillator. This has been

performed by looking

^{for a}

canonical transformation for the _p, q, t variables into the P,

Q, o,

^so that the motion with the new variables is

purely

harmonic. The

generating

^function

brings

an

auxiliary

function which is the same as

introduced,

more or less

arbitrarily, by

^{WKB to} all orders.

Acknowledgments.

^{- We are} very indebted to Pro- fessor Feix who introduced us the

topic

^{and for many}

helpful

discussions.

References

[1] COURANT, ^{E. D. and} SNYDER, ^H. S., ^Ann. Phys. ³ (1958) ^1.

[2] KULSRUD, R. M., Phys. ^{Rev. 106} (1957) ^205.

[3] SYMON, K. R., ^J. Math. Phys. 11 (1970) ^1320.

[4] LEWIS, H. R., Phys. ^{Rev. 172} (1968) ^1313.

[5] GUYARD, J., NADEAU, A., BAUMANN, G. and FEIX, ^M. R., J. Math. Phys. ¹² (1971) ^488.

[6] MILNE, W. E., Phys. ^{Rev. 35} (1930) ^863.

[7] DYKHNE, ^A. M., Sov. Phys. J.E.T.P. 11 (1960) ^411.

[8] BITOUN, J., NADEAU, A., GUYARD, ^{J. and} FEIX, ^M. R., J.

Comput. Phys. ¹² (1973) ^315.

[9] LEWIS, H. R., ^J. Math. Phys. ⁹ (1968) 1976.

[10] GOLDSTEIN, H., Classical Mechanics (Addison-Wesley, ^Cam- bridge) ^1956.

[11] KRUSKAL, M., ^{J. Math.} Phys. ³ (1962) ^806.