Hundred Years of the Bohr atom. Part 2: Planck’s Quantum Plank

In the beginning of the last century the gallery of leading scientists who contributed towards the development of principles of physics has been sketched below in the internet. The main players in this blog will necessarily have to include Planck, Einstein and Bohr (placed here along a principal diagonal) if we have to put in proper perspective the Bohr revolution.

During Bohr’s early times, the world of physics was being dominated by two people, Max Planck and Albert Einstein. As a bench-chemist I got it into my head in my early days that Niels Bohr is the genius that made the quantum leap for modelling the structure of atoms. This I learnt from various classrooms. This blog is to set right some of my --- as well as those of similar others --- early perspectives. One cannot get to Bohr without Planck’s quantum.  Planck’s plank was to assert from empirical evidence that radiation from a blackbody is best fitted by assuming that electromagnetic radiation from a black body is emitted by discrete portions or quanta and not continuously as assumed in classical physics. This bold step is now acknowledged to be the beginning of all quantum descriptions including that of Einstein in his famous year of 1905.

The daring of Planck forms the essence of this blog. The quantum postulate quintessentialy means that there are situations when changes are seen to happen by discrete steps rather than those which are described classically, say, by Newtonian mechanics. Bohr’s achievements are discussed later in this light. The blog includes at the end the contribution by S N Bose in understanding Planck’s law and the role of Einstein in interpreting Bose’s understanding that resulted in a prediction and eventual confirmation of a fifthe state of matter, the Bose-Einstein condensate.

Planck dared walk the quantum plank over a sea of classical physics knowing the dangers of making a false step. It is his daring that subsequently allowed Einstein’s first application of quantum physics to the solid state. It was also this that allowed Bohr to consolidate over earlier attempts to use Planck’s quantum of action to explain the positions of spectral lines of the hydrogen atom.

It is, I find, easy to pontificate on what constitutes good science and the way it should be done once the science has been done. It is a different matter to set an example by actually doing the good science

Mea Culpa anche.

I think part of the reason for writing this blog is to explain --- mainly to myself --- the way good science is done. It is not done by pursuing what is considered to be currently important. Rather it is to convince oneself of the veracity of one’s observation ---usually an empirical fit --- and then to assert one’s findings  and its consequences convincingly. Most importantly, perhaps, it is necessary to have a society that recognises the impending importance. In Planck’s case the society recognised the importance of understanding blackbody radiation if only to improve upon the more mundane behaviour of incandescent lamps,

Planck’s Plank. Resonance of a Musician.

At the end of the nineteenth century, classical theories such as the Newtonian laws of mechanics, Maxwell’s theory of electromagnetism and Boltzmann’s theory of statistical mechanics formed the highpoints of classical physics that could be used to explain most natural phenomena. Max Planck, looking for an area of research in the 1870s was told, as all young people are inevitably told, “...  almost everything is already discovered, and all that remains is to fill a few holes." Thus, the First Baron Kelvin, William Thomson, would comment in his famous lecture, Nineteenth-century clouds over the dynamical theory of heat and light, that the main problems in physics would be to measure known quantities to a great degree of precision so that two important clouds would clear. These were the failure of the Michelson-Morley experiment to detect a change in the speed of light in different directions as predicted by the theory of the luminous ether and the failure to understand the effect --- later termed by Ehrenfest as the “ultraviolet catastrophe” in 1911 --- which predicts that black body radiation will have infinite intensity at high (ultraviolet) frequencies. An ideal black body (usually referred to just as a black body) absorbs all radiation incident upon it without reflecting any of it independent of the wavelength or angle of incidence. Einstein, as we all seem to know, resolved the first problem of ether by his general theory of relativity. It was Planck who fitted the black body radiation spectrum over all wavelengths of light.

 

In his younger days Planck looked very much like an artist (Fig 1, left) when he started his research studies.Kuhn’s book on “Black-Body Theory and the Quantum Discontinuity, 1894-1912,” (University of Chicago Press, 1987) has it that Planck was a “… fine musician and used acoustical analogies in his work, may from the start have thought of the resonant response of a stretched string coupled by a spring or other continuous media to a driving source.” In particular, he must have been familiar with the notion of acoustic resonators in musical instruments that are used to produce sound waves of specific tones. The term resonator is most often used for a homogeneous object in which vibrations travel as waves, at an approximately constant velocity, bouncing back and forth between the sides of the resonator. Resonators can be viewed as being made of millions of coupled moving parts (such as atoms). Therefore they can have millions of resonant frequencies, although only a few may be used in practical resonators  

Planck could have been interested in the principles of Helmholtz resonances. The principle of such resonators is that when air is forced into a cavity, such as by blowing on top of a bottle, there will be a high pressure forcing air into the bottle. When this pressure is removed, the air molecules with higher pressure will move out causing the pressure in the bottle to be lower than outside and air will be drawn back in. The process repeats. Planck as a student took notes from Helmholtz’s lectures.

Planck’s interest in acoustic resonators must have drawn him to Maxwell’s electromagnetism and electric resonators in which a cavity radiation is oscillations of electromagnetic field. Such a field can be modelled in terms of a collection of harmonic oscillators. Planck must have also been familiar with Maxwell and his “demon” when conjecturing on the second law of thermodynamics: it is impossible in a system enclosed in an envelope ... to (spontaneously) produce any inequality of temperature or of pressure without the expenditure of work. The demon works as follows:- “… let us suppose that such a vessel is divided into two portions, A and B, by a division in which there is a small hole, and that a being, who can see the individual molecules, opens and closes this hole, so as to allow only the swifter molecules to pass from A to B, and only the slower molecules to pass from B to A. He will thus, without expenditure of work (by this Maxwell means that the observation and opening and closing of the particles does not involve any work) raise the temperature of B and lower that of A, in contradiction to the second law of thermodynamics...”. (from Maxwell’s Theory of Heat (New York: D. Appleton & Co., 1872), pp. 308-9).

From these Maxwell’s demon arguments Planck was interested in showing that “… irreversibility could be derived from consideration of purely conservative effects.” Of concern to Planck was “… that the … conversion of a plane wave, travelling in a single direction, to a spherical wave, travelling in all --- could not be reversed.”. Maxwell further continues “In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus. It would be interesting to enquire how far those ideas about the nature and methods of science which have been derived from examples of scientific investigation in which the dynamical method is followed are applicable to our actual knowledge of concrete things, By this, Maxwell probably admitted to himself that the exact properties of discrete molecules, themselves, were not crucial to the exact understanding of bulk mater. The condition of irreversibility for radiation processes would determine the law for black body radiation. These considerations must have been important for the philosophy of his approach to blackbody radiation. It is Kuhn, I think, who drew attention to the importance of the difference between the ideal and the real. Thus while matter, as in gas particles, may be considered corpuscular, a theory for elasticity considers matter to be a continuum that fills whole of space. 

The study of black body radiation was important in the late nineteenth century in Western countries in order to understand, for illumination purposes, incandescence or luminosity or why a body glows when heated. Planck’ contribution was that the surface of black bodies had oscillators which absorb and emit radiation with energies which were integral multiples of some small universal value. The permitted energies were quantized! There is a distinction between collective, coherent or “ordered” motion and individual, incoherent or “disordered” motion. Boltzmann introduced statistical models based on microscopic disorder into thermodynamics instead of using classical deterministic continuum models that is suitable for coherent systems without disorder.  In particular, one required the Boltzmann factor, exp(-E_i/k_BT) for a given state, i, at energy E_i, where k_B is Boltzmann’s constant and T is the temperature. The Boltzmann statistics is used for degenerate states with more than one state a a particular energy. For a degeneracy g_i for states at energy E_i, the population, P_i, at energy E_i is given by

P_i = g_iexp(-E_i/k_BT).                                                                                 (1)

The partition function, Q_i, between the various states, i, is given by

Q_i = ∑_I g_i exp(-E_i /k_BT)                                                                             (2)                                                                   

The partition function provides a normalization factor so that population P(E_i) at energy E_i is given by P(E_i) = g_iexp(-E_i/k_BT)/Q. For a continuum of states the summation can be replaced by an integral. The average energy <E> is then obtained as

<E> = S_i E_iP(E_i)                                                                                     (3)

The familiar Boltzmann statistics that is applicable to a gas of molecules with the distribution of energy being treated as a continuous variable is obtained by integrating over all states.   

It is Planck who rephrased the second law of thermodynamics in terms of entropy or disorder. The entropy of a system which is in thermal and mechanical isolation, increases as it evolves towards thermodynamic equilibrium, Put more familiarly, entropy is a measure of disorder of the system, and should be a maximum at thermodynamic equilibrium.

The problem that Planck faced was that the blackbody radiation was fitted at short wavelngths by the Wien function and at long wavelengths by the Rayleigh function. Wien noted that the distribution of thermal radiation at various wavelengths is similar to Maxwell’s velocity distribution law. The Wien’s law for the energy, E_l, of a black body radiation at a wavelength l is given by E_l = Al^-5exp(-a/lT) where A and a are constants.  The Wien’s displacement law says that the wavelength, l_max (in angström), at which the blackbody radiation is maximum is inversely proportional to T in kelvin (l_max  = 2.897x10⁷ /T ). The Rayleigh-Jeans law for the radiation, B_l(T), from a black body at temperature T and wavelength l is given by B_l(T)= 2ck_BT/l⁴. The fit to the Wien’s function (in blue) and Rayleigh-Jeans function (in red) in terms of the frequency (inverse of wavelength) and In units of Radiance is shown in Fig 2.

What Planck did was to find a function that fitted (in yellow in Fig 2) the radiance for all wavelengths. This function is known as Planck’s formula. When one considers an oscillator it can have any frequency, v to oscillate in 3D space. The “ultraviolet catastrophe” comes when it is assumed that most oscillators have high v.  Planck’s epochal contribution comes when he suggested that the energy of each oscillator has integral multiples of some small universal value.

We consider, however – this is the most essential point of the whole calculation – E to be composed of a very definite number of equal parts and use thereto the constant of nature h = 6.55×10−27 erg · sec. This constant multiplied by the common frequency v of the resonators gives us the energy element e in erg, and dividing E by e we get the number P of energy elements which must be divided over the N resonators.

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Wien’s law was convincingly demonstrated by Lummer-Pringsheim-Kurlbaum-Reubens to be “not as generally valid, as many supposed to now”. Planck makes the bold assertion in his 1900 paper (M. Planck, Verhandl. Dtsch. phys. Ges., 2, 202 (1900); from “The Old Quantum Theory,” ed. by D. ter Haar, Pergamon Press, 1967, p. 79) that “Since I myself even in this Society have expressed the opinion that Wien’s law must be necessarily true, I may perhaps be permitted to explain briefly the relationship between the electromagnetic theory developed by me and the experimental data.” Using his knowledge of entropy (dS/dU = 1/T; U is vibrational energy) and Wien’s displacement law Planck showed in his 1900 paper that the expression

E = Cl^-5/[exp(c/lT) – 1)                                                                                (4)

which, as far as I can see at the moment, fits the  observational data, published upto now, as satisfactory as the best equations put forward for the spectrum … which I consider to be the simplest possible, apart from Wien’s expression from the point of view of the electromagnetic theory of radiation.

Planck’s 1900 equation above departs from Wien’s distribution only by the subtraction of unity from the denominator. Having found the law Planck’s anguish was to find an explanation for it.

Wien had used scaling-like arguments (say, fot electron localization) to suggest that at long wavelength matter can be considered to be continuous and a single vector may be used, whereas at shorter wavelengths the molecular constitution would be important. Wien was therefore reluctant to believe that a uniform radiation law for all wavelengths could be derived from processes at a molecular or atomic level. Planck who was a reductionist at heart, perhaps because of his prominent right-brain musical leanings, felt that such an attitude would contradict Maxwell’s electromagnetic theory which is applicable to all wavelengths.

In the paper (M. Planck, Verhandl. Dtsch. phys. Ges., 2, 237) in which Planck makes his announcement of his law, Planck writes (see http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Planck%20(1900),%20Distribution%20Law.pdf)

Entropy means disorder, and I thought that one should find this disorder in the irregularity with which even in a completely stationary radiation field the vibrations of the resonator change their amplitude and phase, as long as considers time intervals long compared to the period of one vibration, but short compared to the duration of a measurement.

Planck’s formulation of his equation that gave eqn 4 departed from convention by describing light as a wave phenomenon that is deterministically described by Maxwell’s equations, to a statistical description in terms of partitioning of quanta of energy á la Boltzmann’s new particle statistics. This was like a return to Newton’s corpuscular theory of light which must have caused Planck much agony because of wave-particle conflict. This agony, I think, is evident in Planck’s wild-ish image (Fig 1, middle) at that time.

Planck’s derivation of eqn 4 is simply based on the recognition that the expression for the average energy, <E>, of eqn 3 cannot be evaluated by treating the summation as an integral but as a summation involving a simple geometrical series. It is this summation that gives the [exp(c/lT) – 1] term in the denominator of eqn 4.

Such a description of the partitioning of quanta of energy marks a clear shift in calculating the “ways of distribution” of the energy or finding the equivalent of Boltzmann’s “complexion”s. The “distribution of P energy elements over N resonators can only take place in a finite, well–defined number of ways.”

The Planck function is then given by

B_l = 2phc²l^-5/[exp(hc/klT) – 1]                                                                   (5)

While deriving eqn 5, Planck came up with two parts of a theorem. The first part concerns the definition of the probability of a state so that the entropy is proportional to the logarithm of this probability. The second part of the theorem, which Planck says is the core of the whole theory.

Of this second part, Planck writes “…in the last resort its proof can only be given empirically.” This empirical proof is the heart of the success of his equation because from it he derives, Loschmidt’s number, L (the number of gas molecules in 1 cm³ at 0^o C and 1 atm), the Boltzmann-Drude constant, a (the average kinetic energy of an atom at the absolute temperature 1) and the elementary quantum of electricity e, (the electrical charge of a positive monovalent ion or of an electron).

The theoretical proof of Planck’s law was much less certain than the experimental proof. One theoretical proof of Planck’s equation led to a famous British-Indian sub-continental triumph --- the Bose-Einstein statistics. Part of this effort was due to Einstein. A crucial element in Einstein’s approach is the use of Planck’s lichtquantum (light quantum) in understanding the spectral lines of the hydrogen atom that has been consolidated and baptized as the Bohr model for the hydrogen atom.

Planck, Einstein and Bose

During the year 1905, when Einstein worked in an isolated atmosphere of the patent office in Berne, he had applied the quantum theory to solids, explained the photoelectric effect, worked on the Brownian motion and the theory of relativity in his “miracle” year of 1905.  Because of the disastrous effect of the Nuclear bomb over the unsuspecting people of Japan, Einstein became known as the genius he is thought to be more for his E = mc² formula than for anything else. The popular reputation of Einstein being infallible in all things he worked upon or said is reflected in the xkdc cartoon below.

After 1905, Einstein’s growing reputation forced him to work/comment on problems initiated by others. Unlike his patent office days, he became immersed among scholars in universities and worked on problems that were considered to be important and thereby lacked the original sparkle of his patent office days! They were nevertheless dazzling enough by common (“non-genius”) standards.

Einstein was interested in the way Planck’s formula for radiation distribution could be “obtained from the condition that the internal distribution of the molecules demanded by quantum theory should follow purely from an emission and absorption of radiation? It was basically a question that dealt with the consistency with which Bohr’s quantized description of spectral lines could be consistent with classical descriptions. In his 1917 paper “On the quantum theory of radiation” (Phys. Zs. 18, 121 (1917) Einstein begins with the sentence “The formal similarity between the chromatic distribution curve for thermal radiation and the Maxwell velocity distribution law is too striking to have remained hidden for long. In fact, it was this similarity that led Wien … to his displacement law (for the radiation density, r) r = v³/f(v/T) and the formula r = an³exp(-hv/kT).

The last sentence in this paper states “… a theory can only be regarded as justified when it is able to show that the impulses transmitted by the radiation field to matter lead to motions that are in accordance with the theory of heat.”

Einstein considered the probabilities for three processes A, B, and B’. Process A is the emission of radiation of energy from a state m to a state n. He considered this to be similar to that of a radioactive reaction with a time for decay that is negligibly small compared to the times spent at states m and n. The process A has a probability, dW, for decay with time as dW = A_mⁿdt. Under the influence of radiation density, r, Einstein considered two processes B and B’ that corresponded to a Planck resonator absorbing radiation energy to make a transition from n to m, with a probability dW = B_n^mrdt or to “liberate” energy with a probability dW = B_mⁿrdT. Einstein then looked for the the exchange of energy between radiation and molecules due to processes A, B, and B’ “ … such that the classical thermal distribution between various canonical states…”  W_n = p_nexp(-e_n/kT₎ is maintained. From these consideration Einstein was able to obtain Plank’s distribution law once it was assumed that the number of processes of type B is the same as that of the sum of processes A and B’ taken together. This derivation depended on Wien’s displacement law which gives A_mⁿ/B_mⁿ = an³ so that the distribution was obtained by Einstein as r= an³/{exp[(e_m - e_n)/kT -1} and from which Bohr’s relationship of (e_m - e_n) = hv followed. In the conclusion of the paper Einstein concedes that the “… weakness of the theory lies on the one hand that it does not get any closer to making the connection with wave theory; on the other, that it leaves the duration and direction of the elementary processes to ‘chance’.”

Einstein’s derivation of the {exp[(e_m - e_n)/kT -1} term in the denominator in Planck’s law basically depends on differences in the distinction of processes of emission (via radioactive-like decay) from processes involved in the absorption of radiation.

Our own S N Bose pointed out in his 1924 paper that Einstein’s derivation for Planck’s formula aimed at resolving the quantum/classical contradictions ended up using Wien’s displacement law which is based on classical theory and the high-temperature limit in which quantum theory agrees with classical theory. Bose wrote

… it appears to me that the derivations have insufficient logical foundation. In contrast, the combining of the light quanta hypothesis with statistical mechanics in the form adjusted by Planck to the needs of the quantum theory does appear to be sufficient for the derivation of the law, independent of any classical theory.

Bose invoked the notion of phase-space volume, h³, of a light quanta and the entropy obtained by the possible distribution of all the light quanta in these cells for a macroscopic light radiation. From this quantum condition all thermodynamic quantities can be calculated,

In the way Bose described his statistics he did not specifically mention he was considering statistic of indistinguishable particles of light quantum, which were yet to be known as photons at that time. Einstein, who had translated Bose’s paper into German so as to publish it in the prestigious Z. Physik, realized that The unique feature of the Bose distribution is the implications for the term {exp(-hv/kT) -1} in the denominator. For the blackbody radiation the number of photons emitted would be expected to decrease with temperature. Bose”s model for Planck’s radiation formula helped in describing the impending “catastrophe” when T tends to zero for a system with a finite number of particles. The number of particles occupying a given state could diverge to infinity! This happens for some particles and it is now known as the Bose-Einstein  Condensation. This Bose failed to point out specifically. Nevertheless, there would not have been an interpretation for a condensation, without the insights from Bose’s counting algorithm

It turns out that following Bose’s treatment for the Planck formula, there were several important development in theoretical physics in the years 1925-1927. These include including those by Pauli (spin quantum number, 1925), Fermi (statistics for indistinguishable particles following Pauli exclusion principle) Schrôdinger (wave equation, 1926) and Dirac (spinors, 1926) . These theories would bring out the full implications of the Bose-Einstein statistics. Such theories were not available to Bose and Einstein when they published their work on the Planck formula, so that Bose did not require specifying the nature of the indistinguishable particles or the different statistics when Pauli Exclusion Principle is applicable. Bose/Einstein could have built on their statistics later which they didn’t. Einstein probably had other matters to work on. In Bose’s defence, we may speculate that it is difficult for us Indians to be confined by an adhaar card-like exclusion system which makes one unique --- all gods are the same even if they are deified differently by different castes,

The physics of the Bose-Einstein condensation in a dilute gas of light-quantum-like particles --- now known as bosons --- initially was concerned with the technology of achieving extremely low temperatures (of the order of nano Kelvin or a billionth of one Kelvin). This meant isolating the system being studied. The details of this technology is given in Cornell and Wieman’s 2001 Nobel lecture “Bose-Einstein Condensation in a dilute gas; the first 70 years and some recent experiments”.

In the figures given below from the above article (Fig 3) we have highlighted two features (click on figure to expand). The one on the left shows the growth of the condensate at very low temperatures --- 50nK --- and the very small (~mesoscopic?) size of the condensate. Einstein describes this condensation in very dilute atomic gases by “A separation is effected; one part condenses, the rest remains a ‘saturated ideal gas’ ” at very low temperatures.

A macroscopic example of such a condensation is thought to be liquid helium, ⁴He. The superfluid state of ⁴He is thought to be a Bose condensate. ⁴He particles are bosons. The macroscopic quantum superfluid state of liquid He4 below its superfluid temperature (lambda point) is reflected in the absence of boiling in the superfluid state (see picture below). ³He particles are not bosons and therefore do not form superfluid (do not bose-condense)  in bulk or unrestricted geometries.

The diagram on the right of Fig 3 shows the collective nature of the excitations. A superfluid, being a fluid, is defined by its dynamical behavior. In the superfluid state any excitation is expected to be a collective coherent excitation at a particular frequency. A standing wave excitation of the density profile in the trap (for an angular momentum quantum number, m = 0) is shown after the system is allowed to evolve after some dwell time. These collective excitations suggest that under appropriate conditions one may move a macroscopic object in the same direction.

I am reminded of my reading an early article by Kohn and Sherrington (reviews of Modern Physics, 1970) on there being two kinds of bosons. The type I boson is the normal bose-condensing boson such as photon or He-4. Th type II boson is an elementary excitations such as phonons (vibrational excitation), excitons (fundamental quanta of electrical excitations consisting of an excited electron and hole bound together in a neutral pair), magnons (magnetic excitation such as misaligned spins in a ferromagnet or antiferromagnet). Bose–Einstein condensation of excitations can also happen by increase their density even at relatively high temperatures. There are several reports of bose condensation of magnetic excitations at room temperatures.

How far does such coherent condensation of excitations occur in nature is something one could worry about, I guess. If a thought is an excitation of a mind at rest, can we have a condensation such that several minds are thinking of the same thing. I am reminded of a cartoon by Thurber (Fig 4). An excitation (in the circle in red) makes a coherent mad rush of a pumped mobile population. Cartoon as it may be, it does portend dangerous mind-warping multi-media possibilities.

Epilogue: Bose in the Indian mind.

The phenomenon of Bose-Einstein condensation has been applauded by many of us Indians, some of whom are leading scientists and directors of National laboratories. The world of particles is now known to be divided into two categories of statistics: bosons (after an Indian) and Fermions (after an Italian, Fermi). It may be a proud matter for some of us. Maybe sometimes we overdo it. The cartoon below perhaps reflects this even if the so-called “god particle’ is a boson (Higgs’) and the bosons form the exciting Bose Einstein condensate.

Finally, we must remember that it is the genius of Fermi that was required to distinguish between different kinds of indistinguishable particle; it is the genius of Einstein that was required to obtain the full impact of Bose’s statistics. The differences perhaps lie in the kind of questions that we ask when we do our science.

Perhaps, we Indians get satisfied too easily.