L]-:'I'T]~]~]~ A:L N~OVO CIME~NTO VOL. 44, ~. 8 16 Dicembre 1985

Classical and Quantal Frictions.

T. SUZUKI

Depart~ent o] Physics, Shizuoka University - Shizuoka, Japan

(riccvuto il 20 Settembre 1985)

PACS. 03.65. - Quantum theory; quantum mechanics.

Sn~v~,~ary. - On the basis of a t ime-dependent and nonlinear SchrSdinger equation quantal friction is discussed in connection with an explicitly t ime-dependent Lagran- gian for classical friction. The relevant quantum Hamiltonian is thereby selected out of two given ones.

As to the classical frictional force proportional to velocity, there exist different forms of the corresponding quantum tIamiltonian, which originates in description of quantM friction in Ehrenfcst 's theorem. Hence the problem is raised how to devise proper forms generating close analogy between classical and quantal frictions (~). In this note we consider two types of quantum representations of friction on the basis of Ehrenfest 's theorem combined with a t ime-dependent and nonlinear SchrSdinger equa- tion, and draw a distinction between them in comparison with classical equations as- sociated with a given explicitly timc-dependent Lagrangian for dissipation. The Lagran- gian will then tm'n out to have a quantum analog.

Let the one-dimensional classical equation of motion subject to conservative and frictional forces be given as

~U (t) m~ m~ ,

y being assumed to be time-independent for simplicity. Then the Lagrangian is here

taken to read

(2) L = exp [yt] ~2_ U(}) ,

(1) See, fo r i n s t a n c e , R . \V. HASSE: J. Math. Phys. (N.Y. ) , 16, 2005 (1975); Rep. Prog. Phys., 41, 1027 (1978), a n d r e fe rences t h e r e i n .

560

561 T. suzuki

which yields the conjugate momentum and Hamiltonian

(3) p = m,$ exp [75],

~2 (4) H = - - exp [-- # ] + U exp [Tq �9

2m

l~ow we will discuss quantum forms of friction in conformity with the definition of/5 in (2). To this end, we employ our formulation (3) consisting of commuting quantities which may serve to exhibit correspondence between classical and quantal approaches. Let us first briefly repeat it for convenience. Suppose that the basic equation reads

0 ) (5) i~ ~ ~(x, t) = ~ + u(x) + ~; ~(~, O,

where 15 = - i~O/~x, ( ~ ] ~ ) = 1 and ~ is Hermitian and discussed afterwards. In- sertion of ~(x, t) = A(x , ~) exp [ iW(x, 0/~] into (5) gives

~W (6) o---/- + ~ = o ,

1 (OW'~ 2 ~2 ~ A (7) "~ = 2--'m \ ~xx / + Uq + U + ~ , Uq = -- 2 rex ax2 �9

The expectation value (D) = (~]~2]~) of an operator ~ is writ ten in such a way that ( Q ) ~ ( Z ) , Z - ~ Z(x, t) being a c-number. For example, for ~ = 15, 153, Z = 8 W / ~ x (~W/~x) ~ + 2MUq, respectively. A general form of Ehrenfest 's theorem is given by

<~Z I ~ W ~Z> (8) ~<z )= - ~ + ~ a x ~ "

By use of (6), (7) and (8), we readily obtain

(91 d-~ ( x ) . . . . (15),

(lO) " C ) C ) d~ < ~ > - ~ - ~ ,

bearing in mind <aUq/Sx> = 0. Note that the choice Z = W as well is possible, and we thereby have

This is an analog of the relationship between the action integral and Lagrangian.

(*) T. Suzu]~i: Lett. Nuovo Gimento, 23, 387 (1978); Nuovo Cimento A, 247 (1978).

5 ~ 2 CLASSICAL AND QL=ANTAL FRICTIONS

I t is now postulatcd that in view of (9) the quantal equation corresponding to (1) is written as

(12) d [SU~ d-t <t6> = -- \ 8x )~ -- y(~) "

In order that eq. (10) coincides with (12), ~ is assumed to be a linear combination of two types of typical expressions in such a way that

The substi tution

(14) W = exp [-- ~t](W'-- Bx) ,

then converts (6), (7) and ( l l ) into

t

B = fat' t~<~> exp [at']

(15) ~W' @ YF' = 0 , 5t

1 [~W' ~2 (16) g ~ ' = - - t ~x - - B ) exp[ - -~ t ] :+- (Uq~- U) exp[~t] 2m,

<17) d <,v'> : e=pE ,l / 1 } d~ \~t77d/-u~ ~ \ ~ / '

The choice a = y or fl ~ 0 leads to close correspondence between these expressions and classical ones such as (2) ~nd (4). I t is then recognized that 5W/Sx is the analog of m~ as expected from (9) and the conjugate momentum is in general given as 5W'/Sx = exp [at] ~W/ex ~ B. In this context the particle energy is <162/2m § U> = = <(~W/ex)~/2m § Uq § U> --- <~'> exp [-- at], which, by means of (8), immediately proves to satisfy

d--t(~+ = - ~ ~ +n\~xl ] where <SUq/St> = 0 is taken into account. As for <~6'> ~ @W'/~x> it follows from (12) and (14) that

d ~ < l S ' > = - - e x p [ a t ] -~x '

which for a = k corresponds to the classical equation obtained by use of (4). The conventional prescription of quantization based on L in (2) results in violation

of the uncertainty relation. However, this aspect is avoided by starting with time- dependent and nonlinear SchrSdinger equations where the uncertainty relation natural ly holds, and by which different at tempts have been made (1). By put t ing fl = 0 in our approach, it is indicated that the correspondence to the classical Lagrangian (2) is quantum-mechanically derived merely through the transformation (14). In this sense the Lagraugian is not entirely irrelevant to the quantal friction described in terms of eqs. (5) and (12) with the choice ~ = yW. The classical representations (2) and (4) would be regarded as having the quantum analogs in preference to the starting points for quantization.