IC/99/27

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THE DISCRETE PLANCK SPECTRUMOF A SPHERICAL CAVITY

Valentin I. Vlad1

Institute of Atomic Physics and University of Bucharest,NILPRP-Laser Department, P.O. Box MG-36, Bucharest, R-76900, Romania

andThe Abdus Salam International Centre for Theoretical Physics,

Trieste, Italy,

and

Nicholas Ionescu-PallasInstitute of Atomic Physics and University of Bucharest,

NILPRP-Laser Department, P.O. Box MG-36, Bucharest, R-76900, Romania.

Abstract

The energy spectrum of black body radiation at the absolute temperature, T, in anideal spherical cavity of radius, R, is studied. The departures from the classical predictionsof Planck's theory, due to the discrete energies of the radiation quanta confined insidethe cavity, depend on the adiabatic invariant RT and are significant for RT < 40 cm •K. Special attention was paid to evidence strong changes in the spectrum intensities,forbidden bands of frequency, as well as major modifications of the total energy for RT <2 cm • K. Similar effects were present in the case of a cubic cavity.

MIRAMARE - TRIESTE

March 1999

1Regular Associate of the Abdus Salam ICTP. E-mail: vlad@ifin.nipne.ro.

1. Introduction

From a ciassical point of view, an ideal spherical cavity may be defined as a closedspherical surface having a rigorously constant curvature and a unitary reflection coefficient onthe interior side, where the discrete absorption and emission of quanta by the atoms areleading to the thermal equilibrium [1,2]. The quantum counterpart of this classical definitionis the concept of a potential well, ensuring a vanishing probability for the photon presenceoutside the spherical surface. So, the classical concepts of inpenetrability, ideal polishing andideal reflection are, to a certain extent, equivalent to the quantum concept of an infinitespherical potential well. However, the equivalence is not complete, because the quantumversion of the photon confinement is actually an eigenvalue problem, the discrete spectrum ofthe photon energies being a direct consequence of the volume finiteness and of the shape oflimiting surface. The history of the problem of the Schroedinger-Helmholtz eigen-valuedistribution (with Dirichlet boundary condition) is very rich [3-5] and a good account of it isgiven by Gutierrez and Yanez [6]. On the other hand, the sufficient conditions to replace thesums, characterizing the discreteness of the quantum processes by integrals in the classicallimit were demonstrated (when calculating the partition function and the correctedthermodynamic functions of an ideal gas in cavities with low adiabatic invariant) [7-9].

Now, if we refer definitely to the black body (Planck) radiation, the effect of thegeometrical confinement upon the frequency spectrum of the radiation stored inside the cavitymay be assigned to an additional quantizing, beyond that considered by Planck and referred asthe discrete absorption and emission of quanta by the atoms of the cavity in view of reachingthe thermal equilibrium. In this case, not only the energy exchanged between atoms andradiation (Planck's quanta) is quantified, but the radiation energy as well, through the agencyof the discrete spatial directions of the allowed wave-vectors (as a result of the radiationconfinement). We named this quantum device as double quantified cavity [9], In the literature,a good number of papers are describing the corresponding two-dimensional quantum devicesunder the name of quantum billiards [10-14] and are relating them to the relatively new fieldof quantum chaos, Gutzwiller [15]. From this point of view, our study refers to a three-dimensional quantum billiard.

The effect of the additional energy discretizing is controlled by the dknensionlessfactor hv/kT (which, in its turn, is inversely proportional to the adiabatic invariant TVm). For(hv/kT) < 1 (this implying for instance small temperatures, in the proximity of about 1° K, andsmall volumes, in the proximity of about 1 cm3 ), the Planck spectrum of the black bodyradiation presents a discrete pattern (of lines with irregular intensities), strongly depending onthe cavity shape. The total energy does not obey any longer the Stefan-Boltzmann relation,but is modified by a corrective factor, which imposes a faster decrease to zero, for 7V1'3—*•().

So far, a complete study of the geometrical confinement effect was elaborated in thecase of a cubic cavity [9]. For revealing the change in spectrum, induced by the change in thecavity shape, we undertook a similar research for a spherical cavity. The main results of thisstudy show that, in spite of some additional complexity in the eigen-value problem, thebehavior of the double quantified spherical cavity (DQSC) resembles that of the double-quantified cubic cavity (DQCC). Some differences are also put in evidence.

2. Laplace operator eigen-values problem

For a free particle of rest mass mo, the relativistic energy equation is written as

•noc2J + (cpf (1)

A free photon is such a particle, but with vanishing rest mass. Its energy equation is,accordingly,

K - — = 0, K = K =•— = — (2)% nc A

Avoiding the spin effects (presumably irrelevant for a free photon) and associating operatorsto mechanical quantities, p -^>(7)/i)V, a Schrcedinger type equation is derived from theenergy equation (2)[4j:

(3)

This equation holds inside the spherical cavity. Now, we adopt spherical coordinates and write

yr{r)=U{<p,#)F{r)

i 1(1 + 1)'d2F ^dF^

dr r dr(4)

where/ = 0, 1, 2Second quantizing condition is a boundary (Neumann) type one:

F(R) = 0 (5)

which, together with the normalization condition for y/, completely determine the eigen-energies and eigen-functions for radiation [3]:

sin , <p)Yiim

(6)0 0

where A = 2n + I, n = 0, 1, 2, 3, ..., 1 = 0,1,2,3,...,m = 0,±l,±2,...±l, N=l,2,3,....

Of course, n and 1 cannot be zero simultaneously. Besides this, not all the other states (N, I)are allowed. Thus, boundary condition (5) introduces a rich discrete behavior, which will leadto differences between the DQSC and the Planck radiation spectra and total energies.

To enlighten these aspects, we examine the asymptotic quantizing condition:

, sinfiTwr-/yJFNi —> FM - (2nR)~li (7)

where (n, /) are the radial and the angular quantum numbers, respectively. To point out that

F$\r) make up an ortho-normalized set of functions, over the discrete set of quantum

numbers N= 1,2, 3, «>.

F(V(r)F$l(r)4xr2dr = 5]W (8)Q

This feature entitles us not only to adopt the asymptotic classifying of states for the real ones,but also to use asymptotic wave functions for calculating energies and weights. This adoptionfails to hold for (hv/kT)<l, a case when a rigorous treatment of the quantizing problem mustbe applied (see section 4).

3. Quantum effects in the classical limit

We shall use the asymptotic quantum states described by the asymptotic ortho-normalized wave-functions (7) to calculate the asymptotic degeneracy and the total energy ofthe Planck radiation.

The asymptotic energy values, according to (2) and (7), are given by the formula

E = hvoN, N= 1.2,3,...°° , (9)

where VQ = c/4R is the lowest frequency in the spherical cavity of radius R (which can beobtained from (9) by putting E = hcK/2jz and N=2KR/K). AS a state (N, /) has a weight (21+ .1), the supplementary degeneracy, due to the asymptotic approximation,m=+l

£ l = (2/ + l), (10)m=—l

leads to a total weight (l/2)(N+l)(N+2), for an energy level N, irrespective of the parity ofN:

.t=0 2

To account for all the states in the phase space, still we need to multiply the previous weightK2

by a factor 2 (polarization states) and by another factor C = — (for going over from6

Cartesian to polar coordinates). The constant C is determined by the asymptotic condition tohave a single photon in each cell of the space phase, when N -» ».

N ^2- — R'-~p3-— p = ^ N t N -^-CN3

1 3 3 hz ' c 2 3

JV1=JV,=*C = ^ (12)1 I s- - J

Now, we may write the total radiation energy inside the spherical cavity at thermalequilibrium (with the respective atoms):

K ~N(N+l)(N+2) hvQ

E~ hvn > —— —, a = - I n )6 °^1 exp(aA0-l kT

The summation in (13) may be performed in a special way to reveal the asymptoticquantum effects:

« ;=] ,V=]

(13')

" 6 ° l ^ {e*-lf h (e*-lj

This degenerated discrete energy spectrum can be approximated by more simple expressionsgiven by the dominant terms, in the classical approximation, a = hvo/kT « 1:

where aP = — ^ S - T T , V = -«R J , 5 = 4mR% ^(3) = 1.202 056 9.15 / re 3

Eq. (14) is compatible with the Weyl-Pleijel formula for the density of states in a cavity [3, 5].It follows that it is possible to introduce a corrective factor for evaluating the departure of thetotal energy yielded by the asymptotic quantified spherical cavity from the classical Stefan-Boltzmann calculation. The correction factor is:

1 +0.06554584(/?D"1!

(14')Thus, as a result of the asymptotic degeneracy, the energy levels are discretized with a singlequantum number, N. This discretization introduces in the total energy two correction terms,which depend on the cavity surface and radius. These correction terms are significant (largerthan .1%), i.e. the asymptotic total energy "feels" the cavity dimension (and shape) andtemperature, up to RT < 40 .

Coming back to the energy spectrum problem, we may point out the following specificfeatures:1) the spectrum is a discrete one, the density of lines, on a frequency unit being inverselyproportional to the radius of spherical cavity;2) the spectrum - analyzed in terms of the radial quantum number, n and orbital quantumnumber, /, in (n, I) states, having a (21 +1) - magnetic type degeneracy - is a succession ofhigher and higher degenerated multiplets (this second degeneracy being however not intrinsicto the radiation problem, but rather determined by the error associated to the replacing of thereal wave function Fm with the asymptotic ones FNi0));

3) owing to the peculiarity that (n, I.) are combined to make up the principal quantum number,N, in a specific way 2n + / = N, (different from the homologous way we came across in theHydrogen atom case n + I = N), two adjacent states, inside a multiplet labeled by N, namely(N, I), (N, I'), are placed at a relative distance I / - V\ = 2. At the same time, N and / havealways the same parity, whence two adjacent states (N, I), (N\ I) are placed at a relativedistance \N-N'\ =2.

These are serious arguments to classify the states not in the (n, I) plane, but rather inthe (N, I) plane, where the energy states cover the region between the vertical axis ON and thebisectrix of the plane NOl, presenting the characteristic aspects of a chess table. (Figure 1).Reading this diagram horizontally, the multiplet structure of the spectrum is revealed: N - Isinglet, N =2 doublet, JV = 3 doublet, N = 4 triplet, N = 5 triplet, N =6 quadruplet, N =1quadruplet etc. On the other hand, all states with the same /, which appear along thecorresponding vertical line, have the same degeneracy, g = 2/ +1.

Asymptotic eigen-values of energy are useful for estimating the quantum correctionsto the Stefan - Boltzmann energy formula in the classical limit (large values of temperatureor/and volume). However, they cannot be used to estimate similar effects in the opposite limit(low temperatures or/and volume), when we are compelled to use the accurate energy states.

By going from the asymptotic radial wave functions to the realistic radial wavefunctions, the following effects are to be expected:1) the energy state positions are displaced;2) the degeneracy of the multiplets is removed;3) some of the asymptotic states (N, I) became forbidden (dark gray cells in Fig. 1).

4. Exact eigen -values of the energy in the double quantified spherical cavity

Equation (4) has a regular solution, which can be expressed in terms of Besselfunctions of half-integer orders:

K -i-^2vi,z/cos z x \r,

4 ^2 .1=1

(15)

where z = KniR = 2nvniR/c. Here, [...] means the integer part of the argument

G(O,z)=l, g ( U ) = I , Q{2,z)=l~,

P{Q,z)=l, P(l,z)=l, P(2,z)=l

NT12

11

10

9

8

7

6

5

4

3

2

1

li-lill

• :&.:;

: • • . - . :

3 I

17*;*"

SiillliiII! i-a

13

15

17™illlillf•Slim i

ii

19

nil;

H3I:

0 1 2 3 4 5 6 7 9 10 11

Fig, 1. - The diagram of the lowest energy states of Planck discrete radiation in the sphericalcavity in the asymptotic approximation. Allowed states are drawn in gray and labeled by the(2 / + 1) - "magnetic" weights. The ground state is the singlet N - 1, (n = 0, / = 1). Readingthe diagram horizontally, the multiplet structure of the spectrum is revealed. The dark graycells correspond to the antiresonances (forbidden states).

It may be brought to a more convenient form, namely,

*'-f (16)

A P

Equation (4) also has an independent (irregular) solution expressed in terms of the Neumann-functions N {{z)'-

(17)

However, this solution must be discarded, being in contradiction with the probabilisticinterpretation of F(r).

So, the spatial quantizing condition, F(R) = 0 , is equivalent only to the conditionJ [ (z )=0. The (transcendent) equation for finding the roots of the radial function,

corresponding to the respective condition is

Z + arctgri(l,z)=N7^, N = 2n + l (18)

Condition (18) leads to states characterized by the same parity assigned to quantum numbers(N, t) and to the ortho-normalized set of radial eigen-functions (6).

To calculate the realistic energy eigen-values, we need to solve the transcendentequations (18). In a general way, this is done in terms of an asymptotic expansion [16]:

• + •

2Z{~ \2ZL 120Z4 6720Z* '" jp , = 7 A - 6 , p 2=83A 2-204A + 180, p3 = 6 949 A3 - 33 252 A2 + 81 180 (19)

Z = N^ (20)

This formula holds for (A/Z) « 1. To extend its application range we resorted to a Pade-typeapproximation for the correction function, K, defined as z = (Z2 - A)}/2-K. SO, we built up theefficient formula

" 6 ° 10(2A-3) ' ' 2100(2A-3)2 '

P4(A)= 3 104A4 - 3 4 704A3 +18 3681A2 -380 565A + 255 150 ,

1Using the previous formula, we remove the degeneracy inside the N - multiplets:

(21)

2KR K ! 4 R

(22)

We would like to point out that expansion (19) as well as formula (21) apply only forroots (states) more remote than the highest pole of the function 77 (I, z). The allowed stateslocated in Fig.l between the axis ON and the zigzag diagonal line are shared in two categoriesby the (parabolic) barrier of the highest poles (which goes upward as a parabola of dark graysquares, in Fig. 2). There are allowed states beyond this barrier, in the region confined

between the parabola: TV = —^^tf + (TT - 2JA., A - /(/ +1) and the zigzag diagonal line.

Two features characterize the states confined between the axis ON and the parabola:

1) fulfil the inequality z < (7t/2)N ;2) are well approximated by formula (21).The states confined between the parabola and the diagonal line fulfil the inequalityz > (K/2)N . They cannot be approximated by formula (21).

5. The spectrum of the Planck radiation in DQSC

The allowed states in DQSC determined by the roots of the radial functions (i.e., bythe spatial quantizing condition) are shown in Annex 1, up to z =20. (The roots marked with astar are forbidden by the violation of the selection rules from (6), which impose N > I). Ourcalculations are extended to a larger range, z < 50, i.e., up to more than 300 ordered roots.From Annex 1, we ascertain a strong variation of the inter-state intervals, which also seems tobe randomly distributed in the frequency scale, as a result of effective dependence of energystates on two quantum numbers (N, I).

According to a practice imposed by the spectrum measurements, it is more convenientto reduce formally the frequency spectrum to an equivalent one, depending solely on a singleinteger parameter, namely the number of the frequency channel of a spectrometric analyzer, k,which has a conventionally adopted width Az = 0.5. All the weights, g js falling inside a certainchannel k are summed up and the resulting weight, gk, is assigned to the fictitious(dimensionless) frequency, z.k (located at the middle of the channel):

£=1,2 ,3 , . . . . (23)

The spectrum, which is "observed" by this spectrometer with limited resolution, can becalculated by the formula:

^(j ^^[p(^)r (24)3 h 3 Az

W^^(4)W (25)

The discrete PJanck spectrum of DQSC (37) was plotted against the dimensionless frequencyz and compared to the continuous Planck spectrum, for different values of the adiabaticinvariant, £ 7 (Fig.2):

3 h 3

It is essentially a discrete spectrum, exhibiting sudden and random variations of intensity andeven forbidden bands of frequency (antiresonances), just as in the case of a cubic cavity.

Considering that the neglected states in the spectra (24) and (25) should yield arelative error less than 10"2 (due to the restricted number of states), we obtain the condition:

zM >48RT, (26)

which is a truncation frequency of the Planck radiation spectrum of DQSC similar to thatobtained for the equivalent cubic cavity. From Fig.2a,b,c, one can see that this relation holds

well; in Fig.2d, for RT =1.5, the frequency interval covered by the 300 Bessel roots (ZM = 50)is not yet enough to ensure an error of 10"2, which proves again that (26) is correct.

One can remark also, from Fig.2a, that the maximal condition, still preserving thevalidity of summing up to the 70th state, could be a sphere of 1 cm diameter kept at anabsolute temperature of 1K. This means that we can expect quantum effects in the black bodyradiation energy for rather macroscopic conditions.

In Fig.2, the low frequency discrete structure of the spectrum is well resolved by thechosen spectrometer bandwidth. At higher frequencies, the individual components are notresolved by this spectrometer (many components are collected in its bandwidth) and anaveraging occurs. This averaging explains the fact that no antiresonances can be observed inthe high frequency part of the spectrum.

6. Antiresonances (locked states) in the spherical cavity

Formula (21) gives the roots, z (n, I), within a good accuracy. However, not all thequantities, z (N, I), for N and / of identical parities, given by the respective formula,correspond to real energy states. Some of the energy states are locked by the proximity ofpoles zp of the function rj (I, z) (are forbidden) and are called by us antiresonances. These

poles are zeros of the function Q (I, z). In the case of the lowest states of the radiation, theimportant role is played by the highest poles only. These are given by the formula

It

M ) ( ) 2 ( ) ^ ( ) J E l (27)A / M , ) ( l i t 2 ) , ? + ( i j t [ ) ( i f c 2 )q 96 V 271152O n 24 V 2/1920

12) _ ( A - 6 X A - 1 2 X A - 2 O X A - 3 O )_(A

' " A(A-2) ' *2 A2(A-2)2

We give hereunder, comparatively, the quantities, zp, delivered by the formula (27) and the

accurate zP (Annex 2).The influence of the lower poles is not relevant if the states locked by them are placed

under the zigzag line in Fig, 1. This is the case of the pole zp = 14.789 638 (/= 11) but not of

the pole Zp - 15.236 301 (/= 10) (Annex 3).

The antiresonances are drawn by dark gray cells in the diagram of Fig. 1.Given a pole zp, we may always find an integer number N verifying the inequalities

(N-l)<-zp{l)<(N + ll (28)n

where JV has the same parity as /. Thus, (TV, I) is a Bessel state, locked by the pole, zP>provided that it is placed in the respective diagrams above the zigzag diagonal.

An antiresonance always fulfils the relations:

* § > (29)

For instance, the pole zp = 9.749 809 4 (I = 5) yields the locking of the Bessel state (TV =7,1 = 5). The parameters fa, j3) have, in this case, the values: a = 0.103 460 7, j8 = 0.896 539 3.

10

Ul fe)RT=0.5

14

12

10

60

50

40

30

20

10

10 15

tORT=0.75

20 25

10 40

Fig.2. The discrete Planck spectaim of DQSC plotted against the dimensionless frequency z(joined solid lines) and compared to the continuous Planck spectrum (dashed line) for severalvalues of the adiabatic invariant, RT: (a) 0.5; (b) 0.75; (c) 1; (d) 1.5 cm.K

11

The general expression of these parameters for an antiresonace (N, I) is

j 8 = + I ( ^ + l ) - l z ot = _ I ( j V _ l ) + I z (30)2 K 2 71

These states are presented numerically in Annex 3, up to (N, I) = (10,10) and appear in thediscrete Planck spectrum as forbidden bands (Fig.2). One cannot establish a completecorrespondence between the frequencies, z, from Annex 3 and the forbidden bands from thediscrete spectrum of Fig.2 due to the special calculation of the channel frequencies mentionedin (23).

7. The total energy of the black body in the double quantified spherical cavity

Using the allowed states in a sphere of radius R, we may write down the total energyof the black body radiation inside the spherical cavity [2]:

(31)6

w i t h , = -Lf * £ \ i _ = I 4 3 8 7 8 8 6 . Q 2289903^)-'. (32)2x{k jRT 271RT

Now, we may write eq.(31) under the form:

4(q), (33)

in order to render evident a corrective factor for evaluating the departure of the total energyyielded by the discrete spectrum in the double quantified spherical cavity from the classicalStefan-Boltzmann calculation. The correction factor is

(34)*(g) ]An

It may be proved that: l imW#)=l. (35)

Indeed,71

1

in the

*

AK^

classical

(1/2)* '

er'N -1

hmit,

*±(JV

AN -3

we obtain

• _|_ |Vjy _|_ 2)

( 4 1 "r fV^^-l

4 I

7T3 g 4

(36)

^ = 4^rJ_15 ~ 15 q4 ^ Z ^ '

(37)The previous demonstration is based on the asymptotic degeneracy of the N - multiplets. Theconvergence of the function % toward 1 also resorts to the fact that the ratio of theantiresonance number to the allowed state number approaches zero for N -» =». The existenceof allowed states beyond the (parabolic) barrier of the highest poles (which goes upward as a

12

parabola of dark gray squares, in Fig. 1) and the zigzag diagonal line, is essential in reachingthe aforementioned convergence.

The summation in Eq.(34) can be taken up to the index of the maximum frequencyfound in (26), zM = 4SRT. Thus, for example, in the correction calculation at RT = 0.5, theupper limit in the sum can be taken as k = 70.

Including the corrective function, %{q), in the definition of the effective Planckradiation constant, (j-W, we can write

It is convenient to adopt as a free variable the adiabatic invariant q = RT. Thus, the function,which was calculated and drawn, is

, (39)

where we have denoted the constants as A = (15/4;r)(/zt7£)4(27r)~4 = 3.282 084 4xlO~3 and

B = {he I k)(2K)~x = 2.289 903 0 x ] 0"1.

In Fig.3a, we have plotted the corrective function %(RT) in the interval 0 < B, < 1.5,which provides acceptable errors, for the limited number of calculated and ordered states(300). One can remark that this "exact" corrective factor imposes a faster decrease of thecavity energy than that predicted by the Stefan-Boltzmann law, as RT —> 0. Then, a specialfeature of DQSC is the growth of the corrective factor over the value 1 corresponding to theStefan-Boltzmann law (from RT ~ 0.4). The calculations and the graphic representation showa maximum of the corrective factor, ^ = 1 . 3 6 , at #7=1.3 .

Taking into account the accuracy limit of exact calculations of X(RT) set by the limitednumber of calculated eigen-values, we have used the asymptotic formula (14'), whichprovides Xasympi(RT) and can extend the dependence of the quantum correction to larger valuesof the adiabatic invariant RT. Thus, in Fig. 3a, it is possible to remark that the two graphs, theexact one and the asymptotic one are intersecting in a point close to RT = 2, which seems tobe the lower limit of applicability of the asymptotic formula (14'). Another remarkable featureof DQSC is that the corrective factor is significant (larger than 1%), i.e. the asymptotic totalenergy "feels" the cavity dimension (and shape) and temperature, up to RT < 40. This featureis illustrated in Fig. 3b, where the asymptotic corrective factor is plotted against the classicalprediction of the Stefan-Boltzmann law; one can observe that the classical limit is reached forvery large values of RT, which is a different behavior from the cubic cavity (where theclassical limit is LT~ 1).

This particular feature of DQSC was used in some numerical evaluation, which can beuseful for the experimental tests. Thus, considering the measurement of the total energy of aspherical cavity, with a diameter of lem, held at two temperatures, T2 = 4K and 7/ = IK, theratio of its respective total energies (measured with highly sensitive bolometers) is 151.046.The graphs of the total energy ratio are plotted in Fig. 3c, for two different ratios T2 / 7/ andfor different values of the sphere radius, R; one can observe that the total energy ratiosaturates in both cases, at very macroscopic radii. Alternatively, one can measure the totalenergy ratio of two spheres with different radii, for example, Rz = lem and R{ = 0.5cm, held atthe liquid He temperature, 7= 4 K. The ratio of the corresponding total energies is 7.26217.

13

1 . 4

1 . 2

1

0 . 8

0 . 6

0 . 4

0 . 2

1 . 4

1 . 2

1

0 . 8

0 . 6

0 . 4

0 . 2

E2

E2

- ;Asympt.

1 2 3 4 5

j,)S-Blaw ;Asympt-

R*T

10 20 30 40R*T

E250

200

150

100

50

1 fc>T4/Tl=4

S/

(i

i

tt

f

;T2/T1=2

„--•""

70605040302010

(d)R4/Rl=4 ;R2/R1=2-

0 .5 1.5 2 . 5

Fig. 3.(a) The corrective factor of the Planck constant, calculated with the exact eigen-values,up to RT = 2 (solid line) and calculated in asymptotic conditions (dashed), with RT E [2,7].(b) The corrective factor of the asymptotic total energy yielded by the DQSC (dashed) and theStefan-Boltzmann calculation (solid line) vs. RT, in the range RTe [2, 40]. (c) The totalenergy ratio for two different ratios T2 / T} and for different values of the sphere radius, R.(d) The ratio of the total energies of two spheres with two ratios of the chosen radii, held atthe temperature, T.

14

In Fig.3d, the dependence of this ratio on the temperature, for two ratios of the chosenradii is shown; one can observe that the total energy ratio saturates in both cases, at lowtemperatures. These measurements can show important quantum effects at macroscopic scale.

8. Final Comments and Conclusions

The Planck spectrum of black body radiation inside an ideal spherical cavity of radius,R, was calculated, for various absolute temperatures, T, in the region of small adiabaticinvariants (RT < 40 cm.K), where a second quantizing condition is imposed by the spatialboundary condition. It is essentially a discrete spectrum, exhibiting sudden and randomvariations of intensity and even forbidden bands of frequency, just as in the case of a cubiccavity. For calculating the frequencies and the intensities of the lines in the double quantifiedspherical cavity (DQSC) spectrum, a relativistic Schroedinger equation, for a photon evolvingwithin the ideal sphere, was solved. The very intricate aspect of the spectrum, visualizedthrough a random distribution of the consecutive frequency intervals, turns out to be theconsequence of the effective dependence of the quantum energies on two quantum numbers(TV, I) and to the overlapping of the domains pertaining to successive multiplets.

To point out the specific pattern of the spectrum, as a structure of non-degeneratedmultiplets, the first energy levels were calculated (up to z = 20) and located in the diagram(N, I). On this occasion, a particular feature of the spectrum was pointed out, namely theappearance of some frequencies which, although allowed by the asymptotic (N, I) assignation,are however forbidden by the presence in their proximity of the poles of a certain r| function,entering the amplitude of the radial wave function (antiresonances).

The asymptotic energy states and wave-functions (corresponding to the degeneratedmultiplets) were used to estimate surface and radius dependent corrections in the formula ofthe total energy of the radiation, for intermediary range of RT (RT ^ [2, 40]). This correctionis compatible to the Weyl-Pleijel one.

Finally, we have evaluated the departure from the Planck's radiation constant of thecontinuous Planck's spectrum of the black body radiation, by summing up (instead ofintegrating) the exact contributions of the eigen-values and their weights, for low values ofRT e [0.1, 2). The asymptotic corrective factor is valid for RT > 2 and shows an unexpectedlarge interval, in which there are significant differences between the Planck radiation"constant" at low values of the adiabatic invariant and the classical one. These are quantumeffects at and beyond the conventional classical limit. This behavior is slightly different fromthat shown by the double-quantified cubic cavity [9], Thus, the Planck spectrum and totalenergy of DQSC show clear quantum effects at the macroscopic scale.

Acknowledgments. This work was done within the framework of the AssociateshipScheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.V.I.V. would like to thank Professors Gallieno Denardo and Giuseppe Furlan for theirsupport in these visits, which have offered the optimum conditions for thinking and writingthis and other papers.

15

9. References

1. M. Planck, Ueber das Gesetz der Energieverteilung im Normalspektrum, Ann. d. Physik,4, 533-563 (1901); The Theory of Heat Radiation (Waermestrahrung, 1913), Dover Publ.,N.Y.,1959

2. JLD.Landau, E.M.Lifschitz and L.P.Pitaevskii, Statistical Physics, 3rd Ed., PergamonPress, 1980.

3. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol.1, J. Wiley, N.Y.,1989,pp.314 and 445.

4. J. D. Jackson, Classical Electrodynamics, 2nd Ed., J.Wiley & Sons, Inc., Vol.1, Ch. 3,Sect.7, 1991

5. N. Ionescu-Pallas, Quantum Mechanics, I.A.P. Press, Bucharest, 19906. G. Gutierrez and J.M.Yanez, Am. J. Phys. 65(8), 739(1997)7. C. Stutz, Am. J. Phys., 36, 826(1968)8. M. I. Molina, Am. J. Phys. 64(4), 503(1996)9. V. I. Vlad and N. Ionescu-Pallas, Ro. Repts. Phys. 48(1), 3(1996); ICTP Preprint No.

IC/97/28, Miramare-Trieste, 1997; Proc. SPJE, 3405, 375(1998)10. V. Berry, Eur. J. Phys., 2, 91(1981)11. R. Aurich, F. Scheffler and F. Steiner, Phys. Rev. E, 51(5), 4173(1995)12. R. Aurich, A. Backer and F. Steiner, Intl. J. Mod. Phys. B, 11(7), 805(7)13. R.W.Robinett, J. Math. Phys. 39, 278(1998)14. D. L. Kaufman, I. Kosztin and K. Schulten, Am. J. Phys., 67, 133(1999)15. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, Berlin,199016. E. Jahnke, F.Emde and F. Losch, Tables of Higher Functions, 6 Ed, Me Graw-Hill &Teubner,N.Y.,1960,p.l52.

16

Annex 1. The allowed states of DQSC in the interval ze (0, 20)

k

1234567891011121314151617181920212223242526272829303132333435363738394041424344

(N,l)

(2,0)

(3,1)(4,2)(4,0)(5,3)(5, 1)(6,4)(6,2)(5,5)(6,0)(7,3)(6,6)(7, 1)(7,7)(8,4)(8,2)(8,0)(8,8)(9,5)(9,3)(9,9)(9,1)(10,6)(10,10)(10,4)(9,7)(10,2)(10,0)(11,11)(11,5)(10,8)(11,3)

(11,1)(10,12)(12,6)(11,9)(12,4)(HJ3)(12,2)(12,0)(13,7)(12, 10)(12,14)(13,11)

gk

135I739511171331595117117193132191551231117732513199275115212923

Zk = 27Wn[ R/c

3.141 592 74.493 409 55.763 459 26.283 185 36.987 932 07.725 251 88.182 561 69.095 011 39.355 812 09.424 778 010.417 119 010.512 836 010.904 122 011.657 033 011.704 908 012.322 941 012.566 371 012.790 718 212.966 531 013.698 023 013.915 823 014.066 194 014.207 393 015.033 469 315.039 665 015.431 288 015.514 603 015.707 963 016.144 741 016.354 710 016.641 003 016.923 622 017.220 755 017.250 454 8*17.647 975 017.838 644 018.301 256 018.351 261 2*18.689 036 018.849 556 018.922 999 219.025 853 519.447 701 5*19.653 152 1

17

Annex 2. The poles of the T| function (antiresonances)

/

234567891011

zp (approx.)

1.706 9523.816 8606.472 8699.712 58713.561 63418.033 74823.135 62628.870 49835.239 98542.244 927

zp (accurate)

1.732 0513.872 9836.521 5979.749 80913.591 39418.059 64423.159 75428.893 95935.238 03042.268 462

Antireso-nances, N

234781114192227

Annex 3. The parameters (a, (3) for the first antiresonances (N, I)

/

1234567

N12347811

zP0.000 000 01.732 050 83.872 983 06.521597 09.749 809 413.591394 018.059 644 0

I1.0000.9480.7670.4240.8960.1730.251

3000671191111539724436

0122398

a0.000 0000.051 3280.232 8080.575 8880.103 4600.826 2750.748 563

0988712

18