- Email: [email protected]
Quantum mechanical formalism in classical wave propagation problems
M I En@i~ Set. Vat 20. No. 2. pp. R-280, Ptwted in Greal B&in
00?0-7~S/82/M0271-10.~/~ Pergamon Press Ltd
1982
QUANTUM MECHANICAL FORMALISM IN CLASSICAL WAVE PROPAGATION PROBLEMS I. A. KUNIN apartment
of Mechanical Engineering, University of Houston, Houston, TX 77004,U.S.A.
Abstract-The linear wave equations of continuum mechanics describe wave fields which are regarded as the basic physical quantities. We develop a slightly cruder model in which wave fields are considered up to their phase. The evolution of a wave packet is described in terms of energy characteristics which are functionals of the energy density. Examples of such characteristics are the energy itself, the coordinate of “the center of mass” of the energy density, as well as its mean velocity. width, etc. The advantages of such an approach are as follows: (I) In applications, it is sufficient to consider only a few of these characteristics, and these satisfy equations simpler than the original wave equations (a consequence of laws of conservation). (2) When developed systematically, the corresponding mathematical framework is analogous to quantum mechanical formalism, thus enabling one to draw on this well developed technique. I. INTRODUCTION
THEREIS A well developed technique dealing with the scattering and propagation of waves in continua and crystal lattices. This classical approach is based on exact or approximate solutions of the corresponding wave equations. We discuss here an alternative approach to describing the evolution of wave packets. Rather than the wave fields themselves, we are interested in the evolution of energy characteristics of wave packets. These characteristics are functionals of spatial or spectral energy densities and define a wave field up to its phase only. Such an approach for the l-dimensional case and related applications were considered in[l] where it was called the energy method. Here we provide a more general description of the method. We show that the method has its own advantages in certain theoretical and applied problems and when developed systematically, leads to a far-reaching analogy with quantum mechanical formalism. 2. WAVEEQUATION Let us consider a generalized wave equation of the type
a:u+@4 =o
(2.1)
where u(x, t) is a real vector function of time t and point x = (x,, . . . , x,) in Euclidean n-space E,,, and Cpis a linear operator. For example, let
a=$,
cP=--aca,
(2.2)
where C(x) is a tensor function defined by the material properties of a medium. The eqn (2.1) will be understood in the sense of generaIized functions (distributions). We, thus, regard all fields as defined on the entirety of E, x ( - x, ~1, if necessarv bv setting them equal to zero outside their original domains of definition. This permits one to incorporate the boundary conditions into eqn (2.1) by viewing C(x) as a piecewise continuous function[2]. More generally, Q, is an integral operator with kernel @(x,x’), i.e. (@u)(x) =
@(x,x’)u(x’) dx’.
(2.3)
In particular, for (2.2), the kernel @(x,x’) is expressed through the second derivatives of the S-function. We shall denote by u(k, t) and ufx, w) the Fourier transforms of u(x, t) with respect to x and t, respectively. It is convenient to consider u(x, t) and u(k, t) as x-and ~-representations of a vector u 271
272
I. A. KUNIN
in the Hilbert spacet X with the scalar product
(ulv) = j u(x)v(x) dx
I
i(k)u(k)dk,u,v E X
(2.4)
For A an operator in %‘,denote by (u~A~v>the associated bilinear form
(ulAl4= [I
u(x)A(x, x’)v(x’) dx dx’
=&II
u(k)A(k, k’)v(k’) dk dk’
(2.5)
where A(k, k’) is the Fourier transform of A(x,x’) with respect to x and inverse Fourier transform with respect to x’. We assume the operator @ in (2.1) to be self adjoint (@= @*) and positive, that is
(ul@ju)2 0
(2.6)
for every admissible u E X In order to describe the main ideas of the energy method in the most transparent manner and to avoid excessive technical detail, we will first consider a simplified model. We assume that u(x, t) in (2.1) is a scalar function and hence that 0 is a scalar operator. The positivity (2.6) of @ enables us to express it as @=
v,
(2.7)
where R = R* is a positive (square root) operator. For example, let C in (2.2) be a constant tensor of order 2. Then, in (k, o)-representation, eqn (2.1) has the form [to2 - @(k)]u(k,
o) = 0
(2.8)
where Q(k) = kCk,
(2.9)
o is the frequency, and k is the wave vector. We, thus, have the dispersion relation o =0(k),
a(k)
= v(kCk).
(2.10)
In this case, it is clear that the operator R in k-representation reduces to algebraic multiplication by a(k) and in x-representation is an integral operator with a difference-type kernel (h)(x)
=
j-0(x- x’)u(x’)
dx’,
(2.11)
where n(x) is the inverse Fourier transform of Q(k). In the general case (2.7), we shall also call fl the frequency operator. Although it is a non-trivial problem to find fI in explicit form, the use of this operator substantially clarifies the principal scheme of the energy method. An alternative approach-which does not use the factorization (2.7~is outlined in the Section 9. We associate with eqn (2.1) the energy Wore generally, u belongs to the rigged Hilbert space [2].
Quantum mechanical formalism
E=T-tV,
273
(2.12)
where T=;
I
(2.13)
$dx,
and (2.14) are, respectively, the kinetic and potential energies. It is readily confirmed that the total energy E is conserved provided Q, does not depend upon time. The quantity e(x, tf defined by
e=;li2+;(i-lu)2
(2.15)
can be interpreted as an energy density function. Note that besides the spatial energy densities just introduced, one can use spectra1 energy densities in the (k, w)-representation (or in other representations). Both spatial and spectral densities fail to be unique. However, the main results of the method do not depend on the particular choice of energy density. The form (2.15) is used in this presentation for convenience,
3.THE MAIN IDEA OFTHE
ENERGY METHOD
In many applications, it is sufficient to consider the integral (energy) characteristics of wave packets which can be represented as functionals of the energy density (2.15). It can be shown that the totality of these functionals determine u(x, t) up to its sign (phase in the complex case). Let us introduce some examples of such energy characteristics. The energy E is the simplest (and most important) example. As another quantity of interest, consider the coordinate of the “centre of mass”, (x), of the energy density e(.x, t)
(x) = ;
1xe(x, t) dx,
(3.1)
which may also be interpreted as a “mean value” of the physical quantity “coordinate”. In an analogous way, we can consider such quantities as the mean values of the velocity, frequency, wave vector, and so on for a given wave packet. All of these characteristics occur as linear functionals of an energy density or quadratic functionals of U(X,t). As examples of more complex characteristics, we mention the mean width dx of a packet and quantities which describe the asymmetry of the packet. These characteristics are non-linear functionals of the energy density. The main idea of the method is to describe the evolution of a packet in terms of such energy characteristics as opposed to an approach which requires a knowledge of the exact solution u(x, t). The advantages of such an approach are as follows: (1) For appIications we may restrict our consideration to just a few of these characteristics which may be found by solving equations simpler than the original wave eqn (2.1). In particular, as will be shown, some of the key characteristics satisfy conservation laws. (2) When developed systematically, the corresponding mathematical formalism is to be analogous to quantum mechanical formalism. This permits one to use the well developed mathematical techniques and physical notions from quantum mechanics for application to classical wave propagation problems,
214
I.A.KUNIN 4.ENERGYCHARACTERISTICSOFTHEWAVEPACKETAS HERMITIANOPERATORS
It is convenient to express energy characteristics of the wave packet u(x, t) in terms of the normalized complex function $(x, t) defined by I) =
d(;E)(/.i- ii-h).
It is readily seen that the function 4 satisfies the Schrodinger-type equation ia,Q!l = nl+b,
(4.2)
which has the formal solution * =
e-‘nf+o,h(x) = (I/(x,0).
(4.3)
The corresponding normalized space energy density is equal to e(x, t) =
+i(x,t)t+%(x, t)
(I 4x, t) dx = 1).
(4.4)
The expression for the mean coordinate (x) now assumes the form
cd = h4xl~L
(4.5)
where we use notation analogous to (1.5) but for the complex Hilbert space. 5.QUANTUMMECHANICALANALOGY
The interpretation of energy characteristics of wave packets as Hermitian operators enables one to develop a far reaching analogy with the formalism of quantum mechanics. The state of the system is described by a normalized complex function 4(x, t) which satisfies the Schrddinger eqn (4.2), when 0 plays the role of a Hamiltonian. The normalized spatial or spectral energy densities correspond to quantum mechanical probability densities. The operator characteristics of the packet correspond to quantum mechanical observables [3,4]. This analogy requires careful consideration. First, in classical models there are no characteristic parameters having the dimension of action, such as Plank’s constant h. This leads to correspondence between quantities of different dimension (energy-probability, frequencyenergy, wave vector-momentum, etc.). Second, there are classical wave propagation problems (for example, connected with external forces) which have no counterparts in quantum mechanics. On the other hand, there are physical notions in quantum mechanics for which it would be difficult to indicate a corresponding classical counterpart. A brief sketch of this correspondence between classical and quantum mechanical quantities is given below.
Continuum mechanics
Quantum mechanics
States (vectors of a Hilbert space) *-function Spatial and spectral energy densities
1 Spatial and spectral probability densities
Observables (Hermitian operators) Energy characteristics of wave packets
Physical quantities
215
Quantum mechanical formalism
Continuum mechanics
1
Quantum mechanics Hamiltonian (energy) H External potential
Frequency R Inhomogeneous medium
H=Ho+H,
i-k=n,+n, Wave vector
Momentum kj,
--$,
ih j$
Pjv -
J
J
Group velocity
Coordinate
Coordinate .
a
xi, 1q
Xi,
Generator of the group of rotations Ljk =
i
Xi $
k
ih a
JPi
Angular momentum M/k = ih
- Xk $ J>
Xi $
-
Xk $ J>
k
Dispersion laws String
Photon, acoustical phonon I
44
- lkl
Transverse vibration of beams
I
Non-relativistic particle
o(k) - k2
Reinforced string
Relativistic particle, optical phonon
o(k) - w. + ak2
Equations and solutions
iiia,t+h = H*
ia,$= all, I) = exp
( - itQ)ijo
$=exp
(
-LH h*
>
rLo
Following the analogy with quantum mechanics, we can transfer the time dependence from the states 4 to the operators, in a manner which leaves the mean values invariant. Indeed, we have for an arbitrary Hermitian operator y (Y>= (dYl44 = ($olYtl~o)= (YA
(5.1)
216
I. A.KUNIN
where Yt
=
ein’y
e-ii%,
4d.e = 44%0).
(5.2)
This corresponds in quantum mechanics to the transition from the Schriidinger representation to the Heisenberg representation [3,4]. In the Heisenberg representation, the operator y, satisfies the equation
$ Yt =my,- YAN = w, Y,I
(5.3)
which is equivalent to eqn (4.2) in Schr~dinger representation. If an operator y, commutes with $I then it follows from (5.3) that yI is consent, and as a result (yJ= (y)= const. Such conservatjon laws will have a fundamental significance for applications of the method. 6.ENERGYCHARACTERISTICSOFHIGHERORDER For an arbitrary operator y, let us put sY=Y-tY)
(6.1)
and introduce the correlator (yz) of two possibly nou-commuting operators y and z 2(yz) = 6y& -I-6.&y.
(6.2)
The correlation function K(yt) is defined141 by WYZ)
= ((Yd.
(6.3)
In particular, the quantity Ax = d(K(xx))
(6.4)
gives the effective width of the packet. Similarly, for an arbitrary operator y, the dispersion Ay, defined by tAy12 = K:(YY)>
(6.3
characterizes the deviation of the corresponding quantity from its mean value. In quantum mechanics, Ay is interpreted as the uncertainty of the observable y in the state 4. The following inequality (uncertainty principle) is valid @Y)W
2 ; I([Y, ~1%
(6.6)
In particular, for components xi and xi of the coordinate and wave vector operators we have AXiAkj3 Sii.
(6.7)
The triple correlator (xyz) and the correlation function K(xyz) are defined as (xyz) = sMy&,
K(xyz) = ((xyz))
(6.8)
where S is the symmetry operator. In particular, K(xxx) characterizes the degree of asymmetry of the packet. Analogously, correlators and correlation functions of order it can be introduced. It is
Quantum mechanical formalism
271
that for many applications it is sufficient to know only a few correlation functions of low order. essential
7. CONSERVATION
LAWS
Let us consider first a homogeneous medium. In this case, the operators @ and 0 possess translational invariance. In particular, for the differential operator (2.2), this means that C = const. and @ and R are given by expressions (2.9) and (2.10) in k-representation. As was indicated above, the group velocity operator v is defined as i&&)/dk. It follows that v commutes with R and from (5.3) we have %=O
or
v, = v =const.
(7.1)
As a result we immediately obtain that d2 dt2 x, = 0 or
xI = vt + x0.
(7.2)
In terms of mean values (v) = const.,
(x) = (v)t +(x)0.
(7.3)
We see that the “center of mass” of the packet moves as a free particle. From the basic conservation laws (7.1) and (7.3) we obtain the laws for correlators
$ (xv), = 0, -$(xx), = 0,
(7.4)
and generally d”+l
-(x...x dt”+l
n
v...v)=O. m
(7.5)
In terms of correlation functions K(xv) = t&(u, v) + Ko(xv),
(7.6)
(Ax)’ = K(xx) = t2&,(uv) + 2tK,,(xv) + &(x.x).
(7.7)
Here the subscript zero stands for t = 0, but instead of &(vv) we can also write K(vv) because v, = v. Supposing in addition that the medium is isotropic we obtain also the conservation laws for the angular momentum and its associated correlators. Thus, we can assign to the “free particle” a set of inner degrees-of-freedom which are polynomials in time, the coefficients being calculated from the initial data. If we restrict ourselves to the first few characteristics we obtain a description of the evolution of the packet without having to solve the wave equation. Let us consider now the case of an inhomogeneous medium and let R=R,+R,,
(7.8)
where Q, corresponds to the homogeneous medium and 0, is a pertubation. Taking by definition d UY = z x,,
(7.9)
I.
278
A.KUNIN
we obtain
where the “force” operator
d dt at = f,?
(7.10)
w,,~11~ &I.
(7.11)
fi is given by ft =
In an analogous way we can calculate the “generalized forces” for eqns (7.4) and (7.5). The corresponding equations are the most convenient for approximate solutions. 8.SCATTERINGPROBLEM
Let us consider a medium with a localized inhomogeneity R,. Let 4_%(t) be the given asymptotics as t -+ - 3~of the solution 4(t) of the wave equation with the operator s2 = Q,,+ R,, i.e. (Ir_Jf)+o(l)
$(t)=
as
t+--.
(8.1)
The scattering problem is to find the asymptotics t++=(t) as t + t x [5,6]. For convenience, usually two auxiliary functions $‘“(t) and p’(t) are introduced as the solutions of the wave equation with the operator a,, and the asymptotic conditions
t+@“(t) = t+L(t) + o(1) as P’(t)
= &(t) t o(1)
as
t + - 5. t + t =.
(8.2)
The scattering operator (matrix) S is defined through qP’( t) = S$‘“(l).
(8.3)
Because of energy conservation, S is a unitary operator, i.e. S* = S-l. In terms of the energy method we are interested in the asymptotics at t -+ t x of the mean value (y) with $-Jt) given. Using (8.3) we have for the asymptotics of (y) (y) = (I)‘“IS+~S~$‘“)
as
t--f t a.
(8.4)
The problem is thus equivalent to finding the scattering matrix S. It can be shown that in the case of systems of local defects, the scattering matrix S can be found explicitly[l]. This furnishes the foundation for possible applications of the energy method. 9,ELIMINATIONOFTHESQUARE-ROOTOPERATOR
Let us consider the wave equation
a:~- acau =0
(9.1)
where C = C(x) is a tensor field describing the material properties. We suppose that C(x) is defined by a non-negative potential energy density
This implies that there exists a unique non-negative square root tensor field C’(x), i.e. C(x) = C’(x)C’(x),
C’ = CT.
Note that the explicit calculation of C’(x) is a simple algebraic operation.
(9.2)
279
Quantum mechanical formalism
The wave eqn (9.1) can be rewritten now in the form a:u+ D*Du =O,
(9.3)
where
D=C'd, D*=-XT'.
(9.4)
The expression for the total spatial energy density takes the form e(x, t) = ; Ii2+; (0~)'.
(9.5)
Note that this is the usual local form for the spatial energy density in contrast to the nonlocal expression (2.15). Of course, the difference between the two expressions has a divergence form and gives zero contribution to the total energy E. Let us introduce the normalized vector-field $(x, t) with components $1 = &)
‘%u,
$2
=
,,(;E)
Du.
(9.6)
Then we have f? =
1$12 =
$:
+ 14Q212.
bw
=
1.
(9.7)
It is readily seen that 4(x, t) satisfies the Schrodinger-type eqn (4.2) id,$ = L&G
(9.8)
where (9.9) Now for an arbitrary matrix-operator j the mean value (9) has the form (9.10)
(9) = (ri+W = ME&J where j, = etidy emtiti, & = $(x, t)
(9.11)
and (9.12) Thus we can express all energy characteristics of a wave packet u(x, t) in terms of symmetric operators j. For example, for the coordinate operator x we have f=xZ,
,.
1
z=
1 0
( > o
1
.
(9.13)
It is essential that this procedure involves only the simple operation of square root from the tensor-function C(X). On the contrary, the procedure considered earlier needs a non-trivial square root R for the operator @.
280
I. A. KUNIN IO. CONCLUDING
REMARKS
In continuum mechanics, linear wave equations describe wave fields which are regarded as the basic physical quantities. As we have seen a slightly cruder model-wave fields are considered up to their phase-is mathematically isomorphic to a quantum mechanical model. This permits one to transfer quantum mechanical formalism and notions to continuum mechanics. At the same time, these two models have absolutely different physical background, and the analogy requires careful consideration. In particular, the following problems require special investigation in connection with this approach: (a) use of the creation and annihilation operator technique; (b) factorization of wave equations; (c) dissipative media; (d) stochastic media. Ac~~o~~edge~en~-The author is grateful to L. T. Wheeler for valuable discussions and su~estions. REFERENCES [1] I. A. KUNIN, Theory of Media with ~~c~~tructure,Nauka, Moscow (1975). [2] I. M. GEL’FAND. Generalized Functions, Vol. 1. 4. Academic Press, New York (1964). 131L. D. LANDAU, E. M. LIFSHITZ, QuantumMechanics: Non-relatioistic Theory. Pergamon Press, Oxford (1%5). [4j D. BOHM, QuantumTheory. Prentic&Hall, New York (1951). [S] P. D. LAX, R. S. PHILLIPS, Scattering Theory. Academic Press, New York (1967). [6] R. G. NEWTON, Scattering Theory of Waoes and Park/es. McGraw-Hill, New York (1966). (Receioed 23 April 1981)
00?0-7~S/82/M0271-10.~/~ Pergamon Press Ltd
1982
QUANTUM MECHANICAL FORMALISM IN CLASSICAL WAVE PROPAGATION PROBLEMS I. A. KUNIN apartment
of Mechanical Engineering, University of Houston, Houston, TX 77004,U.S.A.
Abstract-The linear wave equations of continuum mechanics describe wave fields which are regarded as the basic physical quantities. We develop a slightly cruder model in which wave fields are considered up to their phase. The evolution of a wave packet is described in terms of energy characteristics which are functionals of the energy density. Examples of such characteristics are the energy itself, the coordinate of “the center of mass” of the energy density, as well as its mean velocity. width, etc. The advantages of such an approach are as follows: (I) In applications, it is sufficient to consider only a few of these characteristics, and these satisfy equations simpler than the original wave equations (a consequence of laws of conservation). (2) When developed systematically, the corresponding mathematical framework is analogous to quantum mechanical formalism, thus enabling one to draw on this well developed technique. I. INTRODUCTION
THEREIS A well developed technique dealing with the scattering and propagation of waves in continua and crystal lattices. This classical approach is based on exact or approximate solutions of the corresponding wave equations. We discuss here an alternative approach to describing the evolution of wave packets. Rather than the wave fields themselves, we are interested in the evolution of energy characteristics of wave packets. These characteristics are functionals of spatial or spectral energy densities and define a wave field up to its phase only. Such an approach for the l-dimensional case and related applications were considered in[l] where it was called the energy method. Here we provide a more general description of the method. We show that the method has its own advantages in certain theoretical and applied problems and when developed systematically, leads to a far-reaching analogy with quantum mechanical formalism. 2. WAVEEQUATION Let us consider a generalized wave equation of the type
a:u+@4 =o
(2.1)
where u(x, t) is a real vector function of time t and point x = (x,, . . . , x,) in Euclidean n-space E,,, and Cpis a linear operator. For example, let
a=$,
cP=--aca,
(2.2)
where C(x) is a tensor function defined by the material properties of a medium. The eqn (2.1) will be understood in the sense of generaIized functions (distributions). We, thus, regard all fields as defined on the entirety of E, x ( - x, ~1, if necessarv bv setting them equal to zero outside their original domains of definition. This permits one to incorporate the boundary conditions into eqn (2.1) by viewing C(x) as a piecewise continuous function[2]. More generally, Q, is an integral operator with kernel @(x,x’), i.e. (@u)(x) =
@(x,x’)u(x’) dx’.
(2.3)
In particular, for (2.2), the kernel @(x,x’) is expressed through the second derivatives of the S-function. We shall denote by u(k, t) and ufx, w) the Fourier transforms of u(x, t) with respect to x and t, respectively. It is convenient to consider u(x, t) and u(k, t) as x-and ~-representations of a vector u 271
272
I. A. KUNIN
in the Hilbert spacet X with the scalar product
(ulv) = j u(x)v(x) dx
I
i(k)u(k)dk,u,v E X
(2.4)
For A an operator in %‘,denote by (u~A~v>the associated bilinear form
(ulAl4= [I
u(x)A(x, x’)v(x’) dx dx’
=&II
u(k)A(k, k’)v(k’) dk dk’
(2.5)
where A(k, k’) is the Fourier transform of A(x,x’) with respect to x and inverse Fourier transform with respect to x’. We assume the operator @ in (2.1) to be self adjoint (@= @*) and positive, that is
(ul@ju)2 0
(2.6)
for every admissible u E X In order to describe the main ideas of the energy method in the most transparent manner and to avoid excessive technical detail, we will first consider a simplified model. We assume that u(x, t) in (2.1) is a scalar function and hence that 0 is a scalar operator. The positivity (2.6) of @ enables us to express it as @=
v,
(2.7)
where R = R* is a positive (square root) operator. For example, let C in (2.2) be a constant tensor of order 2. Then, in (k, o)-representation, eqn (2.1) has the form [to2 - @(k)]u(k,
o) = 0
(2.8)
where Q(k) = kCk,
(2.9)
o is the frequency, and k is the wave vector. We, thus, have the dispersion relation o =0(k),
a(k)
= v(kCk).
(2.10)
In this case, it is clear that the operator R in k-representation reduces to algebraic multiplication by a(k) and in x-representation is an integral operator with a difference-type kernel (h)(x)
=
j-0(x- x’)u(x’)
dx’,
(2.11)
where n(x) is the inverse Fourier transform of Q(k). In the general case (2.7), we shall also call fl the frequency operator. Although it is a non-trivial problem to find fI in explicit form, the use of this operator substantially clarifies the principal scheme of the energy method. An alternative approach-which does not use the factorization (2.7~is outlined in the Section 9. We associate with eqn (2.1) the energy Wore generally, u belongs to the rigged Hilbert space [2].
Quantum mechanical formalism
E=T-tV,
273
(2.12)
where T=;
I
(2.13)
$dx,
and (2.14) are, respectively, the kinetic and potential energies. It is readily confirmed that the total energy E is conserved provided Q, does not depend upon time. The quantity e(x, tf defined by
e=;li2+;(i-lu)2
(2.15)
can be interpreted as an energy density function. Note that besides the spatial energy densities just introduced, one can use spectra1 energy densities in the (k, w)-representation (or in other representations). Both spatial and spectral densities fail to be unique. However, the main results of the method do not depend on the particular choice of energy density. The form (2.15) is used in this presentation for convenience,
3.THE MAIN IDEA OFTHE
ENERGY METHOD
In many applications, it is sufficient to consider the integral (energy) characteristics of wave packets which can be represented as functionals of the energy density (2.15). It can be shown that the totality of these functionals determine u(x, t) up to its sign (phase in the complex case). Let us introduce some examples of such energy characteristics. The energy E is the simplest (and most important) example. As another quantity of interest, consider the coordinate of the “centre of mass”, (x), of the energy density e(.x, t)
(x) = ;
1xe(x, t) dx,
(3.1)
which may also be interpreted as a “mean value” of the physical quantity “coordinate”. In an analogous way, we can consider such quantities as the mean values of the velocity, frequency, wave vector, and so on for a given wave packet. All of these characteristics occur as linear functionals of an energy density or quadratic functionals of U(X,t). As examples of more complex characteristics, we mention the mean width dx of a packet and quantities which describe the asymmetry of the packet. These characteristics are non-linear functionals of the energy density. The main idea of the method is to describe the evolution of a packet in terms of such energy characteristics as opposed to an approach which requires a knowledge of the exact solution u(x, t). The advantages of such an approach are as follows: (1) For appIications we may restrict our consideration to just a few of these characteristics which may be found by solving equations simpler than the original wave eqn (2.1). In particular, as will be shown, some of the key characteristics satisfy conservation laws. (2) When developed systematically, the corresponding mathematical formalism is to be analogous to quantum mechanical formalism. This permits one to use the well developed mathematical techniques and physical notions from quantum mechanics for application to classical wave propagation problems,
214
I.A.KUNIN 4.ENERGYCHARACTERISTICSOFTHEWAVEPACKETAS HERMITIANOPERATORS
It is convenient to express energy characteristics of the wave packet u(x, t) in terms of the normalized complex function $(x, t) defined by I) =
d(;E)(/.i- ii-h).
It is readily seen that the function 4 satisfies the Schrodinger-type equation ia,Q!l = nl+b,
(4.2)
which has the formal solution * =
e-‘nf+o,h(x) = (I/(x,0).
(4.3)
The corresponding normalized space energy density is equal to e(x, t) =
+i(x,t)t+%(x, t)
(I 4x, t) dx = 1).
(4.4)
The expression for the mean coordinate (x) now assumes the form
cd = h4xl~L
(4.5)
where we use notation analogous to (1.5) but for the complex Hilbert space. 5.QUANTUMMECHANICALANALOGY
The interpretation of energy characteristics of wave packets as Hermitian operators enables one to develop a far reaching analogy with the formalism of quantum mechanics. The state of the system is described by a normalized complex function 4(x, t) which satisfies the Schrddinger eqn (4.2), when 0 plays the role of a Hamiltonian. The normalized spatial or spectral energy densities correspond to quantum mechanical probability densities. The operator characteristics of the packet correspond to quantum mechanical observables [3,4]. This analogy requires careful consideration. First, in classical models there are no characteristic parameters having the dimension of action, such as Plank’s constant h. This leads to correspondence between quantities of different dimension (energy-probability, frequencyenergy, wave vector-momentum, etc.). Second, there are classical wave propagation problems (for example, connected with external forces) which have no counterparts in quantum mechanics. On the other hand, there are physical notions in quantum mechanics for which it would be difficult to indicate a corresponding classical counterpart. A brief sketch of this correspondence between classical and quantum mechanical quantities is given below.
Continuum mechanics
Quantum mechanics
States (vectors of a Hilbert space) *-function Spatial and spectral energy densities
1 Spatial and spectral probability densities
Observables (Hermitian operators) Energy characteristics of wave packets
Physical quantities
215
Quantum mechanical formalism
Continuum mechanics
1
Quantum mechanics Hamiltonian (energy) H External potential
Frequency R Inhomogeneous medium
H=Ho+H,
i-k=n,+n, Wave vector
Momentum kj,
--$,
ih j$
Pjv -
J
J
Group velocity
Coordinate
Coordinate .
a
xi, 1q
Xi,
Generator of the group of rotations Ljk =
i
Xi $
k
ih a
JPi
Angular momentum M/k = ih
- Xk $ J>
Xi $
-
Xk $ J>
k
Dispersion laws String
Photon, acoustical phonon I
44
- lkl
Transverse vibration of beams
I
Non-relativistic particle
o(k) - k2
Reinforced string
Relativistic particle, optical phonon
o(k) - w. + ak2
Equations and solutions
iiia,t+h = H*
ia,$= all, I) = exp
( - itQ)ijo
$=exp
(
-LH h*
>
rLo
Following the analogy with quantum mechanics, we can transfer the time dependence from the states 4 to the operators, in a manner which leaves the mean values invariant. Indeed, we have for an arbitrary Hermitian operator y (Y>= (dYl44 = ($olYtl~o)= (YA
(5.1)
216
I. A.KUNIN
where Yt
=
ein’y
e-ii%,
4d.e = 44%0).
(5.2)
This corresponds in quantum mechanics to the transition from the Schriidinger representation to the Heisenberg representation [3,4]. In the Heisenberg representation, the operator y, satisfies the equation
$ Yt =my,- YAN = w, Y,I
(5.3)
which is equivalent to eqn (4.2) in Schr~dinger representation. If an operator y, commutes with $I then it follows from (5.3) that yI is consent, and as a result (yJ= (y)= const. Such conservatjon laws will have a fundamental significance for applications of the method. 6.ENERGYCHARACTERISTICSOFHIGHERORDER For an arbitrary operator y, let us put sY=Y-tY)
(6.1)
and introduce the correlator (yz) of two possibly nou-commuting operators y and z 2(yz) = 6y& -I-6.&y.
(6.2)
The correlation function K(yt) is defined141 by WYZ)
= ((Yd.
(6.3)
In particular, the quantity Ax = d(K(xx))
(6.4)
gives the effective width of the packet. Similarly, for an arbitrary operator y, the dispersion Ay, defined by tAy12 = K:(YY)>
(6.3
characterizes the deviation of the corresponding quantity from its mean value. In quantum mechanics, Ay is interpreted as the uncertainty of the observable y in the state 4. The following inequality (uncertainty principle) is valid @Y)W
2 ; I([Y, ~1%
(6.6)
In particular, for components xi and xi of the coordinate and wave vector operators we have AXiAkj3 Sii.
(6.7)
The triple correlator (xyz) and the correlation function K(xyz) are defined as (xyz) = sMy&,
K(xyz) = ((xyz))
(6.8)
where S is the symmetry operator. In particular, K(xxx) characterizes the degree of asymmetry of the packet. Analogously, correlators and correlation functions of order it can be introduced. It is
Quantum mechanical formalism
271
that for many applications it is sufficient to know only a few correlation functions of low order. essential
7. CONSERVATION
LAWS
Let us consider first a homogeneous medium. In this case, the operators @ and 0 possess translational invariance. In particular, for the differential operator (2.2), this means that C = const. and @ and R are given by expressions (2.9) and (2.10) in k-representation. As was indicated above, the group velocity operator v is defined as i&&)/dk. It follows that v commutes with R and from (5.3) we have %=O
or
v, = v =const.
(7.1)
As a result we immediately obtain that d2 dt2 x, = 0 or
xI = vt + x0.
(7.2)
In terms of mean values (v) = const.,
(x) = (v)t +(x)0.
(7.3)
We see that the “center of mass” of the packet moves as a free particle. From the basic conservation laws (7.1) and (7.3) we obtain the laws for correlators
$ (xv), = 0, -$(xx), = 0,
(7.4)
and generally d”+l
-(x...x dt”+l
n
v...v)=O. m
(7.5)
In terms of correlation functions K(xv) = t&(u, v) + Ko(xv),
(7.6)
(Ax)’ = K(xx) = t2&,(uv) + 2tK,,(xv) + &(x.x).
(7.7)
Here the subscript zero stands for t = 0, but instead of &(vv) we can also write K(vv) because v, = v. Supposing in addition that the medium is isotropic we obtain also the conservation laws for the angular momentum and its associated correlators. Thus, we can assign to the “free particle” a set of inner degrees-of-freedom which are polynomials in time, the coefficients being calculated from the initial data. If we restrict ourselves to the first few characteristics we obtain a description of the evolution of the packet without having to solve the wave equation. Let us consider now the case of an inhomogeneous medium and let R=R,+R,,
(7.8)
where Q, corresponds to the homogeneous medium and 0, is a pertubation. Taking by definition d UY = z x,,
(7.9)
I.
278
A.KUNIN
we obtain
where the “force” operator
d dt at = f,?
(7.10)
w,,~11~ &I.
(7.11)
fi is given by ft =
In an analogous way we can calculate the “generalized forces” for eqns (7.4) and (7.5). The corresponding equations are the most convenient for approximate solutions. 8.SCATTERINGPROBLEM
Let us consider a medium with a localized inhomogeneity R,. Let 4_%(t) be the given asymptotics as t -+ - 3~of the solution 4(t) of the wave equation with the operator s2 = Q,,+ R,, i.e. (Ir_Jf)+o(l)
$(t)=
as
t+--.
(8.1)
The scattering problem is to find the asymptotics t++=(t) as t + t x [5,6]. For convenience, usually two auxiliary functions $‘“(t) and p’(t) are introduced as the solutions of the wave equation with the operator a,, and the asymptotic conditions
t+@“(t) = t+L(t) + o(1) as P’(t)
= &(t) t o(1)
as
t + - 5. t + t =.
(8.2)
The scattering operator (matrix) S is defined through qP’( t) = S$‘“(l).
(8.3)
Because of energy conservation, S is a unitary operator, i.e. S* = S-l. In terms of the energy method we are interested in the asymptotics at t -+ t x of the mean value (y) with $-Jt) given. Using (8.3) we have for the asymptotics of (y) (y) = (I)‘“IS+~S~$‘“)
as
t--f t a.
(8.4)
The problem is thus equivalent to finding the scattering matrix S. It can be shown that in the case of systems of local defects, the scattering matrix S can be found explicitly[l]. This furnishes the foundation for possible applications of the energy method. 9,ELIMINATIONOFTHESQUARE-ROOTOPERATOR
Let us consider the wave equation
a:~- acau =0
(9.1)
where C = C(x) is a tensor field describing the material properties. We suppose that C(x) is defined by a non-negative potential energy density
This implies that there exists a unique non-negative square root tensor field C’(x), i.e. C(x) = C’(x)C’(x),
C’ = CT.
Note that the explicit calculation of C’(x) is a simple algebraic operation.
(9.2)
279
Quantum mechanical formalism
The wave eqn (9.1) can be rewritten now in the form a:u+ D*Du =O,
(9.3)
where
D=C'd, D*=-XT'.
(9.4)
The expression for the total spatial energy density takes the form e(x, t) = ; Ii2+; (0~)'.
(9.5)
Note that this is the usual local form for the spatial energy density in contrast to the nonlocal expression (2.15). Of course, the difference between the two expressions has a divergence form and gives zero contribution to the total energy E. Let us introduce the normalized vector-field $(x, t) with components $1 = &)
‘%u,
$2
=
,,(;E)
Du.
(9.6)
Then we have f? =
1$12 =
$:
+ 14Q212.
bw
=
1.
(9.7)
It is readily seen that 4(x, t) satisfies the Schrodinger-type eqn (4.2) id,$ = L&G
(9.8)
where (9.9) Now for an arbitrary matrix-operator j the mean value (9) has the form (9.10)
(9) = (ri+W = ME&J where j, = etidy emtiti, & = $(x, t)
(9.11)
and (9.12) Thus we can express all energy characteristics of a wave packet u(x, t) in terms of symmetric operators j. For example, for the coordinate operator x we have f=xZ,
,.
1
z=
1 0
( > o
1
.
(9.13)
It is essential that this procedure involves only the simple operation of square root from the tensor-function C(X). On the contrary, the procedure considered earlier needs a non-trivial square root R for the operator @.
280
I. A. KUNIN IO. CONCLUDING
REMARKS
In continuum mechanics, linear wave equations describe wave fields which are regarded as the basic physical quantities. As we have seen a slightly cruder model-wave fields are considered up to their phase-is mathematically isomorphic to a quantum mechanical model. This permits one to transfer quantum mechanical formalism and notions to continuum mechanics. At the same time, these two models have absolutely different physical background, and the analogy requires careful consideration. In particular, the following problems require special investigation in connection with this approach: (a) use of the creation and annihilation operator technique; (b) factorization of wave equations; (c) dissipative media; (d) stochastic media. Ac~~o~~edge~en~-The author is grateful to L. T. Wheeler for valuable discussions and su~estions. REFERENCES [1] I. A. KUNIN, Theory of Media with ~~c~~tructure,Nauka, Moscow (1975). [2] I. M. GEL’FAND. Generalized Functions, Vol. 1. 4. Academic Press, New York (1964). 131L. D. LANDAU, E. M. LIFSHITZ, QuantumMechanics: Non-relatioistic Theory. Pergamon Press, Oxford (1%5). [4j D. BOHM, QuantumTheory. Prentic&Hall, New York (1951). [S] P. D. LAX, R. S. PHILLIPS, Scattering Theory. Academic Press, New York (1967). [6] R. G. NEWTON, Scattering Theory of Waoes and Park/es. McGraw-Hill, New York (1966). (Receioed 23 April 1981)