A relation between the schrödinger and radiative transfer equations

J. Quant. Spectrosc. Radiat. Transfer Vol. 41, No. 2, pp. 121-126, 1992 Printed in Great Britain.

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0022-4073/92 $5.00 + 0.00 0 1992 Pergamon Press plc

A RELATION BETWEEN THE SCHRODINGER RADIATIVE TRANSFER EQUATIONS

AND

J. F. GEURDES Universityof Leiden,Wassenaarseweg 52, 2333AK L&den, The Netherlands (Received 28 December

1990; received for publication 1 I September

1991)

Abstract-It is demonstrated in this paper that the quantum-mechanical Schriidinger equation can be reformulated into a radiative transfer equation for the probability density of the wave function. The physical relevance of the formalism is established by applying it to hydrogen. In addition, it is demonstrated that the uncertainty relation for momentum and position can be derived from the Stokes equation. An extension of the derived one-particle radiative transfer equation to an N-particle equation is also discussed.

INTRODUCTION Inspired by the successful use of linear integral equations in soliton theory,] a similar procedure is sought for the quantum-mechanical Schr6dinger equation. Using a method which resembles the use of linear integral equations in soliton theory,* we derive the solution of the general stationary Schriidinger problem. The solution has the form of a linear integral equation in which the integration runs over the complex spectra1 parameters. This equation is subsequently used to derive the radiative transfer equation 3*4for the probability density. Our procedure is an attempt to transform the solutions of a subsystem of stationary Schriidinger equations by using a weighted summation of a plane wave and a free-particle solution. As an example, the general formalism is applied to the hydrogenic atom. The explicit calculations demonstrate the physical relevance of the formalism and serve to evaluate the assumptions made in the derivation of the atomic radiative transfer equation. The physical relevance of the atomic radiative transfer equation is stressed further by deriving the Heisenberg uncertainty relation for momentum and position from the Stokes equation. The probability densities related to variances in position and momentum are identified as separate components of an elliptically polarized beam. In view of the physical relevance of the one-particle atomic transfer equation, we generalize it to the N-particle atomic radiative transfer equation. Some consequences of our approach are presented. Closely related to our topic are the photon transport equation and the Boltzmann equation. These two equations can be derived from quantum-mechanical principles5%6by making use of a position-momentum distribution function’.* which comes close to a classical distribution function. In this connection, the study of Watson9 is also relevant. However, in the description of radiative transfer, the gains and losses of energy in a pencil of radiation3 can be given without explicit reference to the momentum of the particles. Hence, a position-momentum distribution is not necessary in this case. GENERAL

THEORY

We start with the equation ICI(X;k,I)={p(x;k)+~(X;k,I)}~o(x;k,I)exp(-iiE,x4),

(1)

where the four-vector x represents the space-time continuum (x, x4) with x4 equal to time and the vector x equal to the spatial coordinates (x,, x2, x3). The variables k and 1 in Eq. (1) and the following equations, represent complex spectra1 parameters. In addition, 4(x; k, I) is a solution of the time-dependent free-particle Schrcdinger equation $div*grad$(x;k,1)+i8,+(~;k,I)=O, 121

(2)

J. F. GEURDES

122

in which ~3~represents time differentiation

and i = (- l)@. In EQ. (l), p(x; k) is the plane wave

p(x; k) = exp(ik(2/3)“2(x, +x2 + xg) - ik2x4). As noted, the function J/,,(x; k, 1) represents the solution Schradinger equation with energy E,. We have

(3)

of a subsystem of the stationary

$ div - grad $,,(x; k, I) + (E, - V(x))$,(x; k, I) = 0.

(4)

The system in Eq. (4) is under the influence of a general potential energy function V(x), which depends only on the spatial coordinates. Next, we define the sum ‘Z’(x) =

dv(k) dl(l)K(k, I)J/(x; k, I). (9 sD sC In Eq. (5), integrations are performed over two spectral parameters by using two different measures and contours in the complex plane. The kernel in Eq. (5) is sufficiently well behaved for our purpose and depends only on the spectral parameters k and 1. In order to arrive at a SchrGdinger equation for Y(x), we differentiate Eq. (1) with respect to the space and time coordinates. It follows from Eqs. (l)-(4) that idiv.gradY(x)+id,Y(xJprovided the following conditional

sD

dv(k)

V(x)Y(&)=O

(6)

equation holds:

da. (I )K(k, I ) exp( - iE, x4) sC x

{ik(2/3)“2ep(x; k) + grad 4(x; k, I)} grad tjO(x; k, 1) = 0

(7)

with e = (1, 1, 1). Hence, with the use of Eq. (1) and the definition of Y(x) in Eq. (5), it follows that dv(k) dJU)K(k, I) exp(-&x&(x; k) + 4(x; k, I)}$&; k, 1) (8) sD sC solves the Schrijdinger equation (6). We will now demonstrate that a formal analogy exists between the distribution of microphysical probability density and radiation through an atmosphere. We first inspect the stationary equivalent of Eq. (6) with Y = exp( - iEx,)@, i.e., Y(x) =

i div. grad G(x) + (E - V(x))@(x) = 0.

(9)

Here, E represents the total energy of the stationary summed system. Subsequently, two additional conditions are introduced. The first is

s

dL(l)K(k, I) exp( - iE, x~)$~(x; k, 1) = exp( - iEx4)Go(x)

(10)

c

with QO(x) a function related to the ground-state

s

dL(I)K(k, 1) exp(-X,x,)$,(x;

wave function. The second condition is

k, l)c$(x: k, I) = exp(-iZ?x,)@,,(x)$(x;

k)

(11)

C

with c#J(&;k) a function which like +(g; k, 1) is also a solution of the free-particle Schrbdinger equation. Hence, Eq. (8) can be rewritten for the solution of the SchrGdinger equation in Eq. (9). We have

Q(x) =

s D

Wk){dx; k) + 4(x; k))%(x).

(12)

The integration in Eq. (12) should be such that the time dependence drops out. It is shown in the subsequent example that this can be the case. However, before presenting the example the general radiative transfer equation for the probability density Z = CD@ * is derived. For convenience,

123

Schriidinger and radiative transfer equations

the kernel function R(x; k) = p(&: k) + 4(x; k) is introduced. With this shorthand, Z = Z(x) can be written according to Eq. (12) as dv*(k’)R(x; k)R*(x; k’). dv(k) (13) sD s D' Here, D * represents the complex conjugate of the contour D. Further, a function rc(x) is introduced which is similar to the mass-absorption function in macroscopic theory, viz. Z(x) = Z,,(x)

K(X) = (s . grad) log Z,(x).

(14)

In Eq. (14), the vector of directional cosines s = (x,/r, x2/r, x3/r) is used in which r = (xf + x$ + x:)“* is equal to the radius. In addition, a functionj(x) resembling the macroscopic emission function is defined as dv *(k’) (s . grad)R(x; k)R *(x; k’). (15) dv(k) sD s D* Using the definitions in Eqs. (14) and (15), we are able to show from Eq. (13), by taking the s * grad of this equation, that, Z = Z(x) obeys an equation similar to the radiative transfer equation, viz. j(x) = Z,(x)

(s * grad)Z(x) = ~(x)Z(x) +j(x).

(16)

The source function J(x) is equal to the ratio of emission and mass absorption j(x)/rc(x), which is similar to macroscopic theory.3 THE

HYDROGENIC

ATOM

EXAMPLE

In order to solve the hydrogenic Schrbdinger problem with the aid of the formalism developed in the previous section, the following two reductions are introduced in Eq. (7):

s

dl(Z)K(k, I) exp( - iE, x4) grad &(x; k, 1) = aeH(x)

(17)

C

and

dl(Z)K(k, I) exp( -iE,x,) grad 4(x; k, 1)grad &(x; k, I) = be * grad 4(x; k)H(x). (18) fC The a and b in Eqs. (17) and (18) are rational constants. The function H(x) is a well-behaved arbitrary function. We choose for contour D the infinitesimal circle in the complex plane around the origin. The measure is defined as dv (k) = (n - l)!k + dk, with n a positive integer > 1. This selection of measure and contour leads us to the calculation of a Cauchy residue in k-space. It will now be demonstrated that the hydrogenic problem fdiv*grad@,,(x)+

(

En+:

>

@“(x)=0,

(19)

with Z the number of positive charges in the nucleus, is solved by a suitable adaptation of Eq. (12) invoking the previously mentioned measure and contour. Hence, if we substitute for a,,(x) = exp(-Z?r) in Eq. (12), we obtain @n (x) = @ -

‘I!2ni exp(Br) &k s

-“{p (x; k) + 4 (x. 9k)}

in which B = Z/n. Equation (20) is a solution to the hydrogenic problem in Eq. (19), provided the previously mentioned restrictions in Eqs. (17) and (18), are obeyed. Hence, from Eqs. (17) and (18), we can deduce $

I

dk-“(e *grad)4(x; k) = -3i(2/3)“*(a[d-*p(x;

k))k=,,

with A = A, = b/u, which depends on n (the principal quantum number).

(21)

J. F. GEURDES

124

Next, the first four solutions of the hydrogenic Schriidinger problem in Eq. (19) are computed. Starting with rr = 2, it can be shown that 4(x; k) = - ik(2/3)“2(x, + x2 + x3)/A2

(22)

is a solution of Eq. (21). In addition, Eq. (22) is also a solution of the free-particle Schrbdinger equation. If, furthermore, the constant A2 is not equal to unity then the first excited state of the hydrogenic atom can be derived from Eq. (20) with the use of Cauchy’s residue theorem, i.e., Q2(x)=C2(x,+x2+x3)exp(-Zr/2)

(23)

with C, a constant of normalization. It is easy to verify that the eigenvalue or energy associated with Eq. (23) is equal to E2 = -f(Z/2)’ which is the second Bohr term. In a similar manner, we derive for n = 3 the following relation: 4(x; k) = (3/2)k2(ix, + r’/3)/A,. Application

(24)

of Eq. (19) and the Cauchy residue theorem yields G3(x) = C,( -2ix,

- (2/3)(x, + x2 + x3)2 + 3(ix, + r2/3)/AJ)exp( -Zr/3).

(25)

Because cP3(x) is independent of time x,, it follows from Eq. (25) that A3 = 3/2, which leads to the following expression: $(x)

= C3(r2 - (x, + x2 + x3)2) exp( -Zr/3).

(26)

Here, C, is a normalization constant. As can be verified, the associated energy is equal to E3 = -i(Z/3)2, which is the third Bohr term. For n = 4, it can be shown that 4(x: k) = (1/6)k3{ -2~~(2/3)“~(x, +x2 + x3’) + i(2/3)“‘{x:(x,

+ x3) + x:(x, + x3) + x:(x, + ~,)}}/a,.

(27)

Taking A, = l/3, it follows that the wave function is Q3(x) = C, exp(-Zr/4)(3/2)1x:(x,

+x3) +x:(x,+x3)+x:(x,+x,)-(2/9)(x,+x2+x,)}.

(28)

The associated energy is equal to E4 = - 1/2(Z/4)2. Higher order excited states can also be readily computed. For completeness, it is noted that the harmonic oscillator problem can be solved in a similar manner by substituting Q,(x) = exp(Br2) to Eq. (12). The same set of kernel functions that allowed solution of the hydrogenic problem can also be applied to the harmonic oscillator problem. The presented method is different from the usual textbook spherical harmonic treatment.13 UNCERTAINTY

RELATION

FOR

MOMENTUM

AND POSITION

In order to determine the uncertainty relations, we need a time-dependent atomic radiative transfer equation. The foregoing derivation, which led to the stationary atomic radiative transfer equation, shows that Eq. (12) is of crucial importance. However, for a time-dependent atomic radiative transfer problem, Eq. (12) is not suitable. Hence, a more general analysis is needed. We will only indicate the necessary steps that yield a suitable equivalent of Eq. (12). We introduce an adaptation of Eq. (l), viz. It/(x; k, I) = {p(x; k) + 4(x; k, l)}&,(x; k, I).

(29)

In Eq. (29), the wave function &(x; k, I) is a solution of the Schrtidinger equation with a spaceand time-dependent potential Y = V(x). Next, reductions are used similar to those in Eqs. (10) and (11) in which the expression exp( - iE, x4)&,(x; k, 1) is replaced by l(lO(x;k, I) and the expression exp( - iEx,)@,(x) is replaced by Y,(x). This procedure gives

Y’(x) =

s

dv(k){p(x; k) + d(z; W)%(x).

D

(30)

Schradinger and radiative transfer equations

Using Eq. (30), we are then able to derive the time-dependent for Z(x) = Y(x) Y *(& viz. (s * grad)&)

+ 4W

= dx)W

125

atomic radiative transfer equation +Ax).

(31)

In Eq. (31), the mass-absorption and emission functions are similar to those of Eqs. (14) and (15). The only difference is that in the time-dependent case, the s * grad operator is replaced by the (s . grad + a,) operator. Without loss of generality, we assume here that the mean position M, and the mean momentum MP are both zero. Hence, the variances of position (denoted by 0:) and momentum (denoted by 0;) can be calculated by integrating the functions Z,(z) = x*Z(s) and Z*(x) = Re( - Y*(x) div . grad Y (CC))over the configuration volume. It can be demonstrated after straightforward but tedious algebra that both of the functions Z,(T) and Z,(s) follow an atomic radiative transfer equation. We will now demonstrate that the Heisenberg uncertainty relation for position and momentum can be derived as a consequence of the Stokes equation for the components of a polarized beam. As in the macroscopic transfer theory of polarized beams,3 we assume here that the atomic Z, and Zz, prior to measurements of position and/or momentum, are the components of an ellipticallypolarized beam. This beam can be characterized3 by the ellipticity #I and a plane of polarization x. We note that the ellipticity and the plane of polarization will generally depend on the space-time coordinates of the particle. For convenience, the explicit dependence will, however, be suppressed. We are now able to introduce the Stokes parameters for an elliptically-polarized beam, viz. Zs(x) = Z,(x) + Z*(x),

(32)

Q, (xl= 4 (xl - 4 (4,

(33)

G(x) = Qs (xl W&

1,

(34)

and I’s (x) = Qs(x) tanCV 1 W2x 1. The parameters

defined in Eqs. (32x35)

(35)

follow the Stokes equation, which is3

Zf=Q;+

Vi+ I’;.

(36)

The special character of the radiation that, in a sense, replaces the particle in the atomic radiative transfer theory developed thus far may be expressed in terms of an integral equation in the Stokes parameters Us and Vs. This equation restricts the ellipticity and the plane of polarization such that ;

d3x{(Us(x))* + (I’s(x))*} N 1

(37)

sW

in atomic units. In Eq. (37), integration is performed over the spatial volume W. It is clear from Eq. (36) and Eqs. (32) and (33) together with the condition in Eq. (37) and Schwarz’s inequality, that the variances in position measurement 02x =

s s

d% (xl

(38)

d3x’Z2(x’)

(39)

W

and in momentum measurement rJ* =

P

W

follow the uncertainty

relation

r7,0p > 1.

(40)

Hence, the elementary particle behaves on measurement, i.e., on transfer of the quantum to the measuring apparatus, as if it were some sort of plane-polarized radiation, the polarization of which is disclosed in the measurement of non-commuting operators.

126

J. F. GEURDE~

CONCLUSIONS

We have demonstrated that the theory of radiative transfer can be used successfully in the atomic domain. On the conceptual side, the theory is a first step in the direction of a semi-classical interpretation of (parts of) the quantum theory. Because our theory makes use of the quantummechanical probability density Z, our theory is in agreement with the predictions of quantummechanics. Hence, our theory gets round the difficulties that local hidden variable interpretations of quantum-mechanics encounter with Bell’s inequality.‘0~“*‘2 In addition, we may extend the transfer equation (16) or its time-dependent variant, Eq. (31), to more than one particle. To this end, we define a 3N vector of directional cosines s=(S’,S*, . . . , s,.,) and a 3N gradient grad = (grad,, grad,, . . . , grad,) with component threevectors that refer to the respective particles. If we further take x = (xl, x2, . . . , x,.,, x4 = t), Eqs. (16) and (31) become the respective N-particle atomic radiative transfer equations. As an example, we inspect a dilute hydrogen gas. In this case, we may suppress the influence of the neighboring atoms on the n th electron bound to the n th nucleus. From the N-particle equation, it follows that the mass-absorption function takes the general form JC(X)= -2b,-2b2-...-2b,.

(41)

The constants 6, in Eq. (41) are related to the excitation of the particular system (atom + electron). The vector s, is equal to grad,@,) with Y, the distance between the nth electron (coordinate vector x, and the nth nucleus (coordinate vector X,). We note that, as in the hydrogenic example, the nuclei are supposed to be stationary whereas there is motion of the electrons. Given large N, the N-particle atomic radiative transfer equation becomes of macroscopic scale. Because we have demonstrated that the atomic radiative transfer equation does not violate the uncertainty relations, the expansion to N-particles starting from an atomic equation is allowed. In a subsequent paper, the relation between Wigner’s position-momentum distribution function5-g,‘4 and the present formalism will be examined more closely. Acknowledgements-The author wishes to express his thanks to an unknown referee of JQSRT for valuable suggestions. Further, the author would like to thank D. Wolpert (Los Alamos, NM 87545) for stimulating remarks.

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