a general kinetic theory of liquids. iv. quantum mechanics ... · iv. quantum mechanics of fluids...

168Hinshelwood & W illiamson 1934 The reaction between hydrogen and oxygen, p . 46. Kassel & Storch 1935 J. Amer. Chem. Soc. 57, 672.Kassel 1937 Chem. Rev. 21, 331.K istiakowsky 1930 J . Amer. Chem. Soc. 52 , 1868.Lewis & von Elbe 1942 J. Chem. Phys. 10, 366.Napravnik & Smith 1940 J. Amer. Chem. Soc. 62, 385.Pease 1930 J. Amer. Chem. Soc. 52 , 5106.Pease 1931 J. Amer. Chem. Soc. 53, 3188.Poljakow et al. 1934a C.R. Acad. Sci. U.R.S.S. 4 , 454.Poljakow et al. 19346 Acta Phys. Chem. U.R.S.S. 1, 551.Poljakow et al. 1934c Acta Phys. Chem. U.R.S.S. 1, 821.Poljakow et al. 1935 Acta Phys. Chem. U.R.S.S. 2 , 397.Poljakow et al. 1938 Acta Phys. Chem. U.R.S.S. 9 , 517.Poljakow et al. 1939a Acta Phys. Chem. U.R.S.S. 10, 441.Poljakow et al. 19396 Acta Phys. Chem. U.R.S.S. 11, 453.Rodebush 1937 J. Amer. Chem. Soc. 59, 1924.Rosen 1933 J . Chem. Phys. 1, 319.Semenov 1929 Z. phys. Chem. B, 2, 169.Waran 1931 Phil. Mag. (7), 11, 397.

Sir Alfred C. Egerton and G. J. Minkoff

A general kinetic theory of liquids IV. Quantum mechanics of fluids

B y M. B o r n , F.R.S.,* a n d H. S. G r e e n

{Received 7 March 1947— Read 26 June 1947)

In this paper the classical theory of liquids developed in the first three parts of this series is translated into the quantum formalism. After the fundamental equations have been reformulated, it is shown that in the classical limit h = 0, they go over into the corresponding classical equations. A quantized proof of the Boltzmann distribution law is given which is simpler and more direct than that of the Darwin-Fowler method. Then the equation of state is derived in a form which exhibits clearly the deviations from the classical law at very low temperatures.

An approximate method of solution of the fundamental equations is developed in a form suitable for practical application. Finally, the quantum equations of motion and energy transport are obtained, and it is shown that they are formally identical with the classical ‘hydrodynamical equations’. This enables a discussion of the viscosity and thermal conductivity of quantum liquids to be given which exposes clearly the alternative explanations of the abnormal properties of liquid He n.

1 . I n t r o d u c t io n

In the first three parts of this series the theory of liquids was treated with the help of classical mechanics. In the present paper we propose to apply instead quantum mechanics. This is necessary because there are physical phenomena which indicate quantum effects, namely, the superfluidity of He n, and the superconductivity of certain metals; further, it seems very likely that atomic nuclei must be treated as

* I have signed this paper, as it is part of the programme with which we started this series. My contribution consists o f some general suggestions, such as the use of the density matrix as the proper tool, and m any critical remarks. The work itself is due to Mr Green.— M. B o r n .

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A general kinetic theory of liquids 169

droplets of a very dense liquid consisting of neutrons and protons. Superfluidity is the simplest case, in so far as only one kind of particle is involved; the main phenomenon consists in an abnormally small viscosity. We shall develop the theory in this paper far enough to be able to discuss the possible explanations of this effect. Superconductivity is more involved, as it presents a diffusion problem concerning two kinds of particles, the mobile electrons and the almost fixed ions. The rigorous theory of diffusion in liquids is complicated; the classical treatment is now in preparation. The case of diffusing electrons is still more difficult, as the Coulomb forces between them and the ions do not have a short range. Therefore we have to postpone this problem.

The main result of the present paper is that the hydrodynamico-thermal equations are not changed by the introduction of quantum mechanics, but only their interpretation in terms of microscopic phenomena and therefore the values of the physical constants in terms of atomic data. This result is at variance with the only previous attempt to develop a quantum hydrodynamics, namely that made by Landau (1941). Landau transcribes the hydrodynamical equations themselves into operator equations; the meaning of these is obscure, and Landau himself does not make any real application of them, apart from a very general contention that the vortex motion is quantized (in ‘ rotons ’), and has therefore discrete lowest states of rotation. Landau’s results concerning the behaviour of H en have nothing to do with his general quantum theory of fluids, but are based on the idea of rotons and some other independent assumptions. We do not think that he has solved the problem of superfluidity. We believe, on the other hand, that our paper contains the correct basis for an explanation, and we shall discuss the different possibilities, while postponing the solution itself to the following publication.

According to the general principles of quantum mechanics, it is impossible to measure simultaneously the position and velocity of a molecule; so that the classical theory of the early parts of this series is invalidated by the assumption that it is possible to define a function / g(£,x(1) ...x (a), ... which determines the probability of a group of molecules having simultaneous positions and velocities. Thus the theory of parts I to III must be re-examined in the light of quantum theory to ascertain its range of validity.

The transition from classical to quantum theory is effected by replacing every function ag(£,x(1) ...x (a), £(1)... !*(9)) of co-ordinates and velocities by a corresponding operator represented by the ‘matrix’ <xq(t,x(1) ...x (3), x(1)' ... x(3)'), often abbreviated to aa(x,x') or aq. The sum (aq + fiq) and the product 0Lq(lq of two operators ocq and are operators defined by the matrix equations

2 . T h e q u a n t u m fo r m a lism

K +/?<*) (*>x') = aq( 2 - 1)

VoL 191. A 12



170 M. Born and H. S. Green

The operators corresponding to the position vector x(i) and the momentum vector ptf) = ni%(i) of the molecule (i) are represented by the matrices

x«(x,x ') = x<*> n 8(xU)- x (iy),3 = 1

pw(x, x') = - i h n 8(x(i) ~ x°T).( 2-2 )

With the aid of (2*1) the operator aq(x,x') corresponding to any function a3(x, £) of the positions and velocities of the molecules can be constructed. As special cases, one has for a function y3(x) of positions alone,

y3(x,x') = y3(x) n £(xtf>-x«>'),3=1

(p(flraHx,x') = na(x® -x® ').(2-3)

The trace of an operator aq will be denoted by y3(a3), and is defined by

Xq(aQ) = JJ a3(x, x') £(x<«> - (2-4)

for q> 1 it is an operator of the type a3_l5 and must be distinguished from the‘complete trace’ X{ocq), defined by

X{oiq) = XliXzi-• • X*K)}], (2-5)

which is a c-number. Xq(aq) is the quantum analogue of the classical expression

J J a 3(x, QdxfddEfd;b y comparison with (2-4) it is seen that the classical operation

/ is replaced by dx^' S(xiq) — x (qy) in the quantum theory.

I t is clear from what has been said that the classical distribution function / 3(x, \)is replaced by an Hermitian matrix operator pq{x,x'). pq is normalized accordingto the relation . , , , r . /e% ..

Xq+iiPq+i) = (N - q ) p q:, (2-6)

which corresponds to J J f Q+1 dxi q + 1 ) = (N — q)fq

[(2*4) of part I] in the classical theory. The formula

r(a) r «J-J/,n[(2-1) of part I] now reads

f (Q) f QJ •••\pqll S(x{i)—x (iy) dx(iy = nq>

or p3(x,x) = n3(x), (2-7)



171

expressing the important property of the matrix that its diagonal elementis the number-density function. I t will now be clear that is the density matrix introduced into quantum mechanics by von Neumann (1932) and Dirac (1935), the equilibrium properties of which have been carefully examined by Husimi (1940).

To formulate the equation satisfied by consider first the operator pN. Like f N, this may be given an arbitrary initial value, subject only to the normalizing condition and the Hermitian property; its subsequent variation is then given by the Heisenberg equation ^

^ = [W(2-8)

A general kinetic theory of liquids

where JTV is the Hamiltonian operator for the system of N molecules, and, in conformity with Dirac’s notation, [a, /?] means simply — — ficc)jh. The HamiltonianWq for a system of q molecules is given by

wt = 1 + l£ +1 r°, (2-9)2Wj=1 2{j=i i=iwhere i/d-P is the potential energy at x(i) due to the external forces, which are assumed to constitute a conservative system, so that P(l) = — di/P^/dx^.

Taking the trace of (2-8), one obtains

% = * = (2-10) Ot 1

since y,v[p(Ar)2, pN] is reducible to

- J qJ f J (P(N)Pn +PnV(N)) (x , x') 8(xW) - # ) ' ) d x ^ 'd x W ,

which in turn may be transformed to a vanishing surface integral. Repeating this operation indefinitely, and cancelling the factor which appears, one obtains

(2-n)Ot i = 1

Now, the well-known classical analogue of [a3,/?a] is the ‘Poisson bracket’

v l M s M s _ Ms.ifiVSxW'ap<0 0p(i)‘0x(i7 ’

so that [Wq, pQ]becomes

' ( 1 / v MMl-pn)]\ \m \ j t i d x w ;*a^>

.£(0 M l)^ ’ dxU)f

and XQ+i[0(ia+1)>P<i+i] becomes

—ff ¥ " +1) dIs+ldx(Q+i)d S+1)m j ) ax<« a^> *

in the transition to classical theory. Thus (2-11) goes over into the classical form

M l3x(i)+ 1 (p «>- £ = Im \ j=i 8x(1)/ 3§MJ mJJ 8x(,) 8£(t)




in the limit #->0, arid it may be inferred that in the region of high temperatures, where the number of energy states is so large that one may substitute a continuous spectrum without error (thus effectively writing = 0), the classical theory will not give rise to any appreciable error.

3. T h e o r y o f t h e r m o d y n a m ic e q u il ib r iu m

In order to calculate the density matrix by means of the equation (2-8), it must first be specified at an initial time t0. For this purpose it is necessary to choose any commuting set of operators A(r) (r — 1 , 3 and to specify the probability aN(l) = ax (l(1)... /(3Ar)) that at time t0 the molecular assembly is in the state l in which the A(r) have the simultaneous eigenvalues l{r). Let x) be the correspondingnormalized eigenfunction, satisfying

^ N(l, x) — x).Then at time £0, pN is given by (cf. Dirac 1935)

Pn = &(Z,x').1

(3-1)

(3-2)

One possible set of A is that set of independent operators which commute with the Hamiltonian energy WN, and are therefore constants of the motion; it is convenient, however, to choose the A in a slightly different way, as follows. Suppose the fluid is divided into a number of parts containing qx, g2, ..., molecules (q1 + q2+ ... + qw = N) in such a way that the interaction between molecules in different parts is very weak; then the A are chosen to be the Ax, A2, ..., A which commute with Wqi ..., Wqwrespectively. Write w,I+w1,+...+w^+v (3-3)

so that V is very small. Let EN be the eigenvalue of WQi + WQi + which isa function of the A. Then, in the /-representation, (2-8) reads

)}, (3-4)

where, according to (3-2), at trine t0,

px {l, l') = (3*S)The condition for the equilibrium between the various parts of the fluid is expressed by the vanishing of the last term in (3*4), or

V(l,l'){aN( l ') -a N(l)} = 0. (3-6)

I t is not difficult to show further that starting from any initial state the fluid will approach asymptotically a state in which (3-6) is satisfied, apart from small random fluctuations; a detailed discussion of the H-theorem, of which this is a special case, is being given by the authors elsewhere.

(3-6) means that in equilibrium aN(l) will have the same value for all states l between which transitions are not forbidden by the vanishing of the matrix elements V(l, V).




In general, however, the only transitions thus forbidden will be those from one energy state to another, which violate the principle of conservation of energy. Thus, in equilibrium, ^ . „(*„), (3-7)

though in certain circumstances, namely, when motion in a certain direction or rotation about a given axis is entirely uninhibited, the momentum and angular momentum must be included among the ‘ absolute ’ constants of the motion, and aN(l) will depend also on these.

Now let aQl(li) be the probability that the first group of qx molecules is in theeigenstate lx a t time tQ, etc. Then, since the different groups are virtually independentof one another, v n n n . . .

a (-®Ar) — ••• (3*8)also En = Ex + E2 + ...+ E w(3-9)

and depend only on the lx, Z2, ...

(3-10)

where Ex, E2,... are the eigenvalues of WQi, WQi,respectively. I t follows rigorously from (3-8) and (3-9) that

a(EN) = e-i»+fiaql(E1) = e-(*i+^x), etc.,

where a = a x + a 2 + ... + aw. As is well known, (3T0) summarizes the whole of statistical thermodynamics. I t is unnecessary to give here the argument, stated concisely by Schrodinger (1946), by which /? is identified with ( )-1, where T isthe thermodynamic temperature, and the entropy expressed in the form

S = -iE«jv(01ogOjvW = (3-11)

The internal energy U is given by

U = 2 ajv(0 En = (3T2)

and the free energy A = U — TS by

e-AikT = _L ^ e-EiikT = _L X(e-w kT). (3-13)

The constant a in (3-10) is determined by the condition ^ a N(l) = 1 in the form1

e<x — n \ e~AlkT, and it follows that pN may be expressed in the operator formp N = eU-W(3 -J4 )

Using (2-6), one obtains

(N -q ) \p q = XQ+i[XaA - X N ( e u -w»)lkT)}]- (3-15)To calculate the thermodynamic pressure from (3-13) with the aid of the

formula p = —dA/dV, it is convenient to assume that vanishes. WritingcrN = e~w*lkT, and x(i) = 60(i), where 63 = V, one has

r ( N) r n

X(aN) = b ^ \ . . . \ a N(b» ,b f))U d^\J J 1=1

(3-16)




so that

or a / ’ = 3 F / - / s gx«,.{x<*><r„(x,x)} nyxO). (3-17)

This expression can be simplified in two ways. First, writing (3’13) in the form N\e~AlkT = X(orN), and making use of (3*14) and (2*6), one obtains

_ 1cTeAlkT dX(crN)V ~ N\dV

= n' kT+ W ^ ) S i ( * 1K4o:>+**K

= nlk T + ~ ~ j j m f c ) d X ' U X ‘>, (3-18)

where r = x(2)—x(1). Because of the classical equation

(X_M !!Vx(3) = od x ^ k T d x ™ JkT dxf*>

this reduces classically to the formula

p = nxkT — ^n ^r) (r) r dr.

Next, the integrand of (3-17) may be written in the form

t2 (P(i) .x(%.v - <rvp<*> (x, x );"> i = l

to evaluate this expression one may make use of the lemma that if ju, and v are any two operators, and j{v) is any power series in v, then

= v f (v)- ¥ . f "{v)+v. f ’ W + -

= {ihf'W i + W 'V ) } + ... - 1«*{11[ft, /"(>’)] + j j [ft, / <lv)('’)] + •••}.

(3-19)

where {a/?} stands for |(a/?-f/?a), [ix = fiv — v/i and / = (k = 1 ,2,...).This is easily proved by induction for f(v) — vn, after which the generalization is

Ntrivial. Putting v = WN, / = 2 p(t).x(i), and f(v) = e~vlkT = one obtains, for the

i=1integrand in (3-17), the expression

i f ( k T ) - 1, , (kT)~3( , “I l((^T )-2r n _ )- ^I ■- J j — { / W + 3! { /W v } + • •.J + 2 1 2 ! ' + - T f - l> 4» + • • • J •



A general k in e tic theory o f liq u id s 175

The terms involving p 2k vanish on insertion in (3*16), but terms in giving a power series in (h/kT)2, effectively, for the pressure. Since

Puc-i survive,

i=i *j=i )(3-20)

the first term alone givesx r

Pi = nxkT (3-21)

where Tx is defined by § NkTx = p(1)2Pij(3-22)

in terms of the mean kinetic energy of the molecules. This is in quantum theory quite distinct from the thermodynamic temperature T which appears in (3-18). Quite apart from this, the neglect of squares and higher powers of shows that (3-21) is only an approximation valid for high temperatures.

In view of this deviation from the classical formula which occurs under quantum conditions at low temperatures, it is worth noting the physical meaning of the thermodynamic pressure, namely the force per unit area on the wall of the vessel as measured, for example, by the work done on a movable piston. I t does not follow in quantum mechanics that this will be the same as the diagonal elements of the pressure tensor px which appears in the equation of motion

0

and it will be seen from the calculations of § 5 that the two quantities are indeed different, inasmuch as p x is given exactly by the equation (3*21) in equilibrium. Thus in quantum mechanical fluids the situation is that both pressure and temperature may be defined either thermodynamically or in terms of dynamical properties, but the definitions will not be equivalent. In problems where equilibrium obtains, the thermodynamic definitions have the obvious advantage of describing what is actually measured; but in non-equilibrium problems the dynamical quantities may play an important role. Thus in the process of thermal conduction, the energy flux is proportional to the gradient of the ‘ dynamical ’ temperature, and it is the pressure tensor, as defined in § 5, which plays an important part in the phenomenon of viscosity. These considerations may help to elucidate some of the puzzling properties of liquid He n.

4. T h e b e h a v io u r o f m o l e c u l a r c l u st e r s

The exact equilibrium theory of the previous section is quite unsuited to the practical investigation of the properties of condensed systems. For this purpose the study of the density matrices of small clusters of molecules is quite sufficient, since all macroscopic quantities can be obtained in terms of and In this section




the equation (2*11) will therefore be examined with small values of it may be written in the form _

= wtP t- P twt +r,pa- p t v j, (4-i)

where Vq= S Xq+MiQ+1)Pq+l \ (4-2)i = 1

and V\ denotes the Hermitian conjugate, which is generally different from Vq, though both quantities correspond classically to the average potential energy of the cluster of q molecules due to the oth' molecules. As in § 3, one may take an l-representation in which WQ is diagonal with eigenvalue Eq, when, after writing

VQ(l,n =

, t dPi(4-1) takes the form ^qPq Pq^Q’

(4-3)

(4-4)

If t — t0is small, this equation may be solved by writing

Pq +P(1) +/>(2) + ...,where p{0) is the value a t time t0, so that

Lh^tP VqP(s)~P(s)Vq-

(4-5)

(4-6)

If a(l) is the probability of finding the cluster of q molecules in the state l at time t0, one may take p(0) = a(l)Slv; also, if

vQ = u,one finds

U = [ v qdt-J t o

iftp{l) = Up{0)-p(0) W,

h2P(2) = j \u u p (o )+ p { o ) m .\

(4-7)

(4-8)

Taking the diagonal elements of these equations, it becomes clear that the probability pn of a transition from the state V to the state l in the small time t — t0 is

— | U(I, V)|2, but that this is generally different from since Vq is not Hermitian.

The condition for equilibrium is now

a(l) Pi = ) Pit,

where Pj = s C {u (i, v) u(i'i)+VJ to

(4-9)

(4-10)

Whether one has equilibrium or not, if there is to be no increase in the sum of the probabilities of the occupation of all states, one must have

X a(l) Pj — 2 ) Vvi)(4*11) i i v



and it follows from this, since the a(l) are arbitrary, that

Pj — 2 Pir- r

A general kinetic theory of liquids

(4-12)

(4-9), together with the normalizing condition 2 — 1, suffices to determine thei

equilibrium values of the a(l) when Vq is regarded as known. The calculation of however, requires a knowledge of Pq+iPq\ and although certain properties of this matrix are known, for example,

Xq+liPq+lPq1) = N ~1, (4-13)

its exact evaluation is as difficult as that of As in the classical theory, therefore, it is necessary to make some approximative assumption to render the problem in a tractable form. Probably the simplest assumption to make is that is approximately Hermitian, which amounts to replacing the field of the other molecules by a corresponding conservative field imposed from outside; this procedure is similar to Hartree’s method of the self-consistent field. Then, as was seen in § 3, the precise form of Vq is unimportant, and only its vanishing elements must be known. In equilibrium, these will include those which represent transitions from one energy state to another; and we find that this condition, together with (4-13), makes the problem quite determinate.

5. Q u a n t u m h y d r o d y n a m ic s

From the fundamental equation (2T1), the hydrodynamical equations will now be derived in a manner corresponding to the classical theory of part III. For this purpose, the quantum definitions of the number density function the mean velocity u^, and the generalized temperature T (ft, relating to a cluster of molecules whose positions are known, are first required. The definitions ought to conform to the condition that in passing from classical to quantum theory, the operation

becomes S(x{i> — x®'). and a product aft is replaced by {aft} = \{aft+ ft a),

which is Hermitian when a and ft are Hermitian. Then, in agreement with (2-7), nqiX) — Pq(x ’x )> an(i mw4u^(x) = {/3gPW} (x,x),

whilst 3 mnqkTf{x) = {{/>3v®}. v^}},

where vj*} is defined by

v^(x, x') = — p(i)(x, x') — u( (x) £(x—x'). (5-3)

(5-1)

(5-2)

m

Since (p(i)2pq - pqPm ) (x, x') 0X®2 0XW'2

- + al®') • (p‘% + A,P®) <x >x '). (5-4)



178 M. Born and, H. S. Green

one may write (2-11) in the form

A | pq(x, x')+ £ i (— + ) ■. foP®} (x, x')

l i( # » - 9i<«>')p3( x , x ' ) + i ( ^ i» -^ » ')p a(x)x')Zi,j = 1 i = l

Q+ si J JJ($(iq+1) ~ <i>ii(1+lY) Pa+ i(x, x') S(x(q+1) — * * * ) dx(q+1)dx(a+iy. (5-5)

On writing x = x ' in this equation, all terms on the right-hand side vanish, and one ^as p)n q d

(5*6)which is the quantum expression of the equation of continuity, and its generalization for q> 1.

Next, multiplying (5-4) before and after by p(i), and taking half the sum, one obtains

* I fop®} <x ’ x '>+ f + - ( f op^p®} <x ' x '>

- i , i y m - fop®) <x- x'> - * I , f S + 3S ) p°<x’ x<>

+ s (if® - r v) fop®} <x, x ')+ f (P«>+ p®') p„(x, x')i= i 1

a+ s

i] JJ*(0(,9+1) — <jpq+iy) {pa+1p(i)} (x, x') £(xto+1)—xto+1)') dx(s+1)dx(9+1)'

+ 8^ ^ ~ ) ^ +l(x’X,) ^(x(a+1) - x to+1)/) dxfe+1)dx(3+1)'. (5*7)

Again writing x = x', one has

+ g|yj.{{paP(,)}P(i)}(x,x') =

where m r)^a 0^(u)

i=J!0X «jS S 3 5 + P ®

(5*8)

(5-9)

As in the case of the corresponding classical equation [(2-6) of part III], (5*8) is easily transformed to ^ i d

mna ~Jiua} + S ZTi)• kQi} = (5-10)

d d 9 dwhere -r means + my. =-73; , anddt dt i=i 0x(t)

ri 9x°'}'

k f } = m{{pqy f} v®} (x, x). (5-11)Equation (5-10) is the quantum equation of motion, differing only from the classical equation by the definition in atomistic terms of those quantities which classically involve the velocities.



To obtain the corresponding equation for take the scalar product of (5-6) with pw before and after, and half the sum of the resulting expressions, afterwards putting x = x \ The result is


1 « 3% (3 nqk T f + mnQu f 2) + —2_£ ^ .{{{/>,P°')} ^ )}-Pw} (x,x')

Q tyiti) •'a

3 = 12 ^ 0X<*>uj'} + 2w3ptf>.u j}- 2 Jw3+1 g gx(.} . u j l i (5-12)

This is also readily transformed to

3 7 d T f £ ui 1.0x0') ,m<™ + / kHi) _JL\ „«>]

\ 9 0x0)/ 3 1

■ Jwg+10| ^ r > .(uf+1- u0)) dx^+v, (5*13)

where m<f} = |m{{{/93 vj)} y j ) } . y0>} (x , x). (5-14)

(5T3) differs from the classical equation of energy transport only in the atomistic definition of quantities which classically involve the velocities.

At first it is rather surprising to find that quantum mechanics involves no change of the equations of continuity, motion, and energy transport from those obtained through the classical theory, but reflexion shows that this must be so: the macroscopic, visible changes of a fluid in motion give no indication of the existence of quantum phenomena. This is true even in the case of liquid He i i , where the abnormal viscosity, thermal conductivity, etc., in no way conflict with the macroscopic description of matter, though they lead one to suspect that quantum phenomena are responsible. This may indeed be the case, since the evaluation of nQ, u®, T (f , etc., from the atomistic standpoint is quite different in the quantum formalism from the classical procedure, and one therefore expects quantum complications at low temperatures.

6. Discussion

The fact that we have been able, in the last section, to derive hydrodynamical equations, identical in form with those resulting from the classical theory of part III, has far-reaching consequences. One conclusion which may be drawn immediately is that the theory of viscosity and thermal conductivity developed from these hydrodynamical equations in part III can be taken over into the quantum theory unchanged, provided one interprets the classical temperature and pressure as the ‘dynamical’ temperature and pressure of quantum theory. The formulae for the pressure tensor and thermal flux are unaltered, and so, therefore, are those for the coefficients of viscosity and thermal conduction. This enables us without further analysis to examine the possible causes of the abnormal values found experimentally for these constants in liquid He n.




First consider the expression for the viscosity, namely

( 6*1)

where <fi2(v) v . b . v is the distortion of the velocity (momentum) distribution function by the motion, and v(r) r . b . r the corresponding distortion of the radial distribution function, b representing the symmetrical non-divergent part of the velocity gradient tensor. I t is clear that an abnormally small value of ju, can arise a t low temperatures from three main causes. In the first place, it is possible that the second, or ‘ kinetic ’ term, which is usually positive but entirely negligible for the liquid phase, by becoming negative and of appreciable magnitude, might annul the first, or ‘ potential ’ term, which usually has a considerable positive value. This has to be considered because under quantum conditions there is no longer any obvious reason why <fi2(v) should be negative; but in view of the entirely different nature of the two terms, the likelihood of their cancelling in this way over any substantial range of temperature is small enough to be dismissed.

We therefore turn to the alternative hypothesis, that both terms appearing in (6T) are very small for H en. That the kinetic term should be small is quite acceptable: there is no positive reason to suppose that the ordinary formula of the kinetic theory of gases does not describe its behaviour quite well, even down to the lowest temperatures. The potential term, which contains the force on one molecule due to others at distance r in the integrand, may almost vanish for two distinct reasons. In the first place, the function v(r) may almost vanish around the minimum of potential between the two molecules. In principle, this can be decided on theoretical grounds; one of the authors has undertaken the calculation of v(r) for the helium atom, and his results so far appear to confirm this hypothesis in a remarkable way. I t should, however, be possible to decide this point also experimentally by the study of the scattering of X-rays by H en; the experiments ofKeesom& Taconis (1938) and Reekie (1940) indicate that the assumption is correct; yet it is desirable that this important experimental determination of the structure of H en should be effected with the highest possible accuracy.

A third and obvious way of accounting for the smallness of the potential term in (6-1) is by supposing that the interaction between the molecules represented by the potential function (j>(r) breaks down at low temperatures. This question is beyond the scope of the methods developed in our theory of liquids, and must, if it appears to be useful, be treated by the investigation of the means of interaction between the elementary particles.

I t appears that nearly all the properties of He 11 can be explained in terms of its low viscosity. For example, the apparently high thermal conductivity is caused by the transference of energy by the visible motion of the liquid, and the true thermal conductivity is thereby completely masked. For this reason we believe that our formula for the thermal conductivity corresponding to (6-1) has no great interest in the application to He 11. The driving force for the convexion currents transporting



heat is, as was pointed out at the end of §3, not the hydrostatic pressure, but the dynamical pressure, which differs from it at low temperatures. From this standpoint, some strange features of the behaviour of He n become less puzzling. These questions, and the problem of the discontinuity between the normal and superfluid states, may be discussed in a subsequent paper.

Some previous attempts to explain the abnormal properties of He n have been based on the application of Bose-Einstein statistics to the molecular assembly; these are not successful because they omit to take into consideration the interaction between the molecules. I t is worth examining, however, how far the present theory is in accord with the new statistics. For Bose-Einstein statistics the density matrices pq are unchanged for any permutation of either the x(1), x (2), ... or the x (Di, x(2)l, ... while for Fermi-Dirac statistics pq changes sign for odd permutations of either of these two sets, In both events pq remains unaltered if both the xd), -x(2>, ... and the x(1)l, x(2)1, ... are subjected to the same permutation. The general conclusion seems to be that Bose-Einstein statistics do not play a very important part, even at low temperatures, in determining the properties of a fluid; this conclusion was reached also by Kahn (1938). Much more important than this effect of ‘second quantization’ is that of the low temperatures themselves on the occupation of the lowest energy states not forbidden by the exclusion principle.


References

Born, M. & Green, H. S. 1946 Proc. Roy. Soc. A, 188, 10.B om , M. & Green, H. S. 1947 Proc. Roy. Soc. A. 190, 455- 474,Dirac, P. A. M. 1935 The principles of quantum mechanics. Oxford Univ. Press. Husimi, K. 1940 Proc. Phys.-Math. Soc. Japan, Ser. 3, 22, 264.Kahn, B. 1938 Dissertation. Utrecht.Keesom, W. H. & Taconis, K. W. 1939 Physica, 5 , 270.Landau, L. 1941 J. Phys. 5 , 71.v. Neumann, J. 1932 Math. Orundl. d. Qu Ch. 4 . Springer: Berlin. Reekie, J. 1940 Proc. Camb. Phil. Soc. 36, 236.Schrodinger, E. 1946 Statistical thermodynamics. Cambridge Univ. Press.



a general kinetic theory of liquids. iv. quantum mechanics ... · iv. quantum mechanics of fluids...

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