Volume 50B, number 2 PHYSICS LETTERS 27 May 1974

Q U A N T I Z E D F R I C T I O N A N D T H E C O R R E S P O N D E N C E P R I N C I P L E :

S I N G L E P A R T I C L E W I T H F R I C T I O N s*

K.-K. KAN and J.J. GRIFFIN Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742, USA

Received 18 April 1974

A non-linear time-dependent Schr6dinger equation is derived by means of physical arguments and the correspon- dence principle to describe a quantal system acted upon by a frictional force. It involves nonqinear terms of the same type as those already derived by Kostin for the Schr6dinger-Langevin problem. Exact solutions are exhibited in closed form for the case of an harmonic oscillator. They exhibit reasonable properties and support the belief that the equa- tion proposed provides an appropriate physical description.

Eroader context o f quantal friction problem As the need, and the possibility, for describing the dy- namical aspects [2] o f global motions of quantal many-body systems grows, one is moved [3] to seek for simple, even trivial, examples which might exhibit features of pedantic value for more complicated sys- tem. Thus when one considers nuclear viscosity as a possible model description for the process of exciting intrinsic excitations with energy extracted from a col- lective motion, the most trivial model example is per- haps the motion of a single particle moving under an external frictional force. How should such a system be described in the quantum limit? The present note of- fers, we believe, the beginnings of any answer to this question.

Conditions for acceptable quantum descriptior~ We construct here a description in terms of a non-linear time-dependent Hamiltonian-Schr6dinger equation which follows from two physical requirements: (a) that in the classical limit the equation describes a frictional force, proportional to the negative velocity of the particle, and (b) that at any time the total ener- by, equal to the sum of the expectation values of ki- netic and potential energies, shall be equal to the ex- pectation value of the Hamiltonian operator.

Work supported in part by the U.S. Atomic Energy Commis- sion,

* From a dissertation to be submitted to the Graduate School, University of Maryland, by Kit-Keung Karl, in partial fulfill- merit of the requirements for the Ph.D. degree in physics.

Appropriate classical limit. Condition (a) can be satisfied by adding to the frictionless Hamiltonian, H o = T + V o, a term proportional to a coefficient of friction,/2, and a velocity potential, X(r, t), whose neg- ative gradient equals the velocity of the matter proba- bility density at each point in space and time.

Such a term guarantees that any wave packet de- scribing a well-localized particle moving with a well- defined momentum p - - m u will behave as though sub- ject to a force

Ff = -uv(r , t) = +UVX. (1)

Explicit expression for velocity potential, X. More- over, an explicit form for X is readily available from the fluid dynamical interpretation of the SchrSdinger equation [4], recently utilized in another context by Griffin and Kan [5]. I f

q'(r, t) = ~b(r, t) exp( - im/h)S(r , t ) , (2)

where q~ and S are real functions, is a solution of the time-dependent Schr6dinger equation, then S repre- sents a velocity potential for the matter field of densi- ty I~12 = I~l 2. More specifically, in its time-develop- ment, • always obeys the continuity equation for a fluid of density I~O [2 and local velocity o = - VS. In any wave packet description of a classical motion, therefore, the function S will satisfy the correspon- dence principle requirements for the Schr6dinger ve- locity potential, X. One thus identifies the term

Vf = -laS = Oah/2im) In (~/tI,*) (3)

as an appropriate addition to the frictionless Hamiltonian, H o .

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Such a term, as well as counter term derived below, has also been obtained by Kostin [1] for a SchriSdinger-Langevin equation constructed to de- scribe the Brownian motion of a particle in a thermal environment. The present derivation shows that the structure of the frictional term can be obtained with- out reference to the randomly fluctuating forces which characterize the Langevin problem.

Prescription to guarantee correct total energy. There remains the satisfaction of the requirement (b) that (H) be equal to the expectation values of the kinetic and potential energies at all times, t. This can be met by adding a time-dependent counter-term de- fined to cancel at every time the expectation value of

vr:

Wf(t) = + fd3r q,* V f q t • (4)

This result is a non-linear time-dependent Schr6dinger equation,

H q / = IHo + V f - Wf] • = ih(~q~/~O (5)

which describes in the classical limit a particle which obeys the equation of motion

dp/dt = F o - / a o (6)

and whose Hamiltonian is equal at any time to the sum of its kinetic and potential energies

(H) = E(t) = p212m + V o (7)

where in eq. (6) F o = - VV o. It is straightforward to prove that, in spite of its non-linearity, solutions of eq. (5), once normalized to unit probability, remain so normalized for all time, and also that any separable H o will lead to a separable H.

Exact solution available for damped oscillator. It is a remarkable fact that exact solutions of eq. (5) can be obtained in closed form in the case when H o describes a simple harmonic oscillator. Each such solution corre- sponds to a specified initial displacement, X o cos 8 and initial velocity V o = X o [co sin 6 - 3' cos 6 ] and has the form (in one dimension),

• (x, t) = (mcoo/~irr)ll4 exp ( - (mcoo[2h) (x -X( t ) ) 2 }

X exp ( i[xP(t)/h-g(t)] } (8a)

X(t) = X ° exp ( -7 t ) cos (cot-6) (8b)

P(t) = - m X o exp (- 'y t ) [to sin (cot-6) +7 cos (cot-6)] (8c)

and

g(t) = (coot/2)

(8d) + f (dt ' /h)(p2/2m-mco2X2/2-11PX/m)

0 and where the oscillatory frequency co, is related to the unperturbed frequency, co o, o f H o by

co2 = co2 _ 72 (8e) o

and the damping constant, 7, to the coefficient of friction, ~t, as follows

/a = 2m% (80

Properties o f the damped solutions to the non-lin- ear Schr6dinger equation. One sees that these solu- tions exhibit two eminently reasonable properties. The first was guaranteed by construction; namely, (a) that for large amplitude oscillations the expecta- tion values of position and momentum approach the values of the corresponding damped classical oscillator. In addition, (b) the amplitude of the oscillations di- minishes monotonically with time until the wave pack- et transforms exponentially into the zero-point ampli- tude for the oscillator ground state. During this pro- cess, the energy also diminishes with time

E(t) = (~coo/2) + (mco2/2)X 2 + (1/2m)e 2 (9)

approaching for t ~ oo (according to eqs..(8b) and (8c)) the quantum zero point value, (hcoo/2).

One thus sees how the quantum ground state arises as the limit of the frictional process: as the amplitude of the oscillation diminishes, so does the coefficient o fx in the imaginary exponential of the wave packet. Therefore, the velocity potential, S in (3) approaches a constant value montonically with time and the fric- tional force described by its gradient approaches zero. The dependence of Hf upon xI, (i.e. the non-linearity of the Schr6dinger equation) thus emerges as an essen- tial element in the description.

Undamped solutions. In addition to the solutions (8) any wave function

~n = ~n (x) exp(-ico ot)(n+l[2) ( I0)

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where ~b n is a stationary state solution of H o, is also a solution of eq. (5): for such a 4 , the term Hf is a con- stant, independent of position, which is exactly can- celed by the counter term, Wf(t). Although such wave functions solve the eq. (5) they exhibit no damping whatsoever.

This seems at first sight to be a paradoxical result: a theory of damped oscillations which exhibits at cer- tain discrete energies completely undamped solutions.

This paradox demands further study, which we feel is most appropriately directed towards the systems of many (conservatively) coupled degrees of freedom, the projection of whose solutions into a one dimensional subspace constitutes the microscopic physical basis for the phenomonological eq. (5). In particular, statis- tical averages in such systems exhibit solutions [6] which describe damped oscillations analogous to (8): as time increases the energy flows from the degree of freedom of interest into the systems many other de- grees of freedom. However, such many-body systems may also exhibit oscillations of their individual normal modes in which the energy in any degree of freedom, including the one selected for explicit description in eq. (5), remains constant in time. We are studying the conjecture that the projection into the one-dimension- al subspace of solutions describing such normal oscilla- tions may correspond to the stationary undamped solu-

tions, (10), of the eq. (5). Summary. A non-linear time dependent Schr6dinger

equation describing a particle acted upon by friction is derived on physical grounds. Exact solutions are ex- hibited in closed form for the specific case of the har- monic oscillator. Their reasonable properties provide support for the belief that the equation, although non- linear, may nevertheless provide an appropriate general description of frictional dissipation of energy in the quantum regime.

References

[1] M.D. Kostin, Journ. Chem. Phys. 57 (1972) 3589. [2] J. Griffin, Proc. of the Heavy Ion Summer Study, Oak

Ridge Tennessee, June, 1972. (USAEC No. CONF-720669, National Technical Information Service, U.S. Dept. of Commerce, Springfield, Virginia).

[3] W.J. Swiatecki, especially, has emphasized the lack of a quantum description for even the simplest dissipative sys- tems (private communication).

[4] E. Madelung, Zeit. f. Phys. 40 (1926) 332. [5] J.J. Griffin and K.-K. Kan, Proc. third Intern. Symp. on

the Chemistry and physics of fission, Rochester, N.Y., 1973, SM-174/58. (I.A.E.A., Vienna, 1974).

[6] G.W. Ford, M. Kae and P. Mazur, Journ. Math. Phys. 6 (1965) 504.

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