The Origin and Development of the Quantum Theory

"When I recall the days of twenty years ago, when the conception of the physical quantum of 'action' was first beginning to disentangle itself from the surrounding mass of available experimental facts, and when I look back upon the long and tortuous road which finally led to its disclosure, this development strikes me at times as a new illustration of Goethe's saying, that 'man errs, so long as he is striving'."

"The pursuit of a goal, the brightness of which is undimmed by initial failure, is an indispensable condition, though by no means a guarantee, of final success. In my own case, such a goal has been for many years the solution of the question of the distribution of energy in the normal spectrum of radiant heat."

"The discovery by Gustav Kirchhoff that the quality of the heat radiation produced in an enclosure surrounded by any emitting or absorbing bodies whatsoever, all at the same temperature, is entirely independent of the nature of such bodies, established the existence of a universal function, which depends only upon the and the , and is entirely independent of the particular properties of the substance. ...[T]his remarkable function promised a deeper insight into the relation between energy and temperature, which is the principal problem of thermodynamics and therefore also of the entire field of ."

"A most suitable body... seemed H. Hertz's rectilinear oscillator (dipole) whose laws of emission for a given frequency he had just... developed. If a number of such oscillators be distributed in an enclosure surrounded by reflecting walls, there would take place, in analogy with sources and resonators in... sound, an exchange of energy by means of the emission and reception of electro-magnetic waves, and... corresponding to Kirchoff's law should establish itself in the vacuum-enclosure."

"I expected, in a [naive] way... that the laws of classical electrodynamics would suffice, if one adhered sufficiently to generalities and avoided too special hypotheses..."

"I... first developed in... general terms... the laws of the emission and absorption of a linear ... by a... circuitous route which might have been avoided had I used the electron theory which had just been put forward by H. A. Lorentz."

"The outcome of this long series of investigations... was the establishment of a general relation between the energy of a resonator of a definite free frequency and the energy radiation of the corresponding spectral region in the surrounding field in equilibrium with it."

"The remarkable result was obtained that this relation is independent of the nature of the resonator, and in particular of its coefficient of damping—a result which... introduced the simplification that the energy of the radiation could be replaced by the energy of the resonator, so that a simple system of one degree of freedom could be substituted for a complicated system having many degrees of freedom."

"But this result constituted only a preparatory advance towards the attack on the main problem... [M]y original hope [was] that the radiation emitted by the resonator would differ in some characteristic way from the absorbed radiation... The resonator reacted only to those rays which were emitted by itself, and exhibited no trace of resonance to neighbouring spectral regions."

"[M]y suggestion that the resonator might be able to exert a one-sided, i. e. irreversible, action on the energy of the surrounding radiation field called forth the emphatic protest of Ludwig Boltzmann... showing that according to... classical dynamics... the processes I was considering could take place in... the opposite sense. Thus a spherical wave emitted from a resonator when reversed shrinks... continually decreasing... on to the resonator, is absorbed by it, and so permits the resonator to send out again into space the energy formerly absorbed in the direction from which it came."

"[I]t became more and more evident that... an essential link was missing which should lead to... comprehension... The only way out... was to attack the problem from the opposite side, from... thermodynamics, a domain in which I felt more at home."

"[M]y previous studies on the second law of thermodynamics served me here... in that my first impulse was to bring not the but the of the resonator into relation with its energy, more accurately not the entropy itself but its with respect to the energy... [T]his differential coefficient... has a direct physical significance for the irreversibility of the exchange of energy between the resonator and the radiation."

"But as I was... too much devoted to pure phenomenology to inquire more closely into the relation between entropy and probability, I felt compelled to limit myself to the available experimental results. Now, at that time... 1899, interest was centred on the law of the distribution of energy... proposed by W. Wien... On calculating the relation following from this law between the entropy and energy of a resonator the remarkable result is obtained that the reciprocal value of the above differential coeffcient... R, is proportional to the energy. This extremely simple relation can be regarded as an adequate expression of Wien's law..."

"I believed... that the basis of the law of the distribution of energy could be expressed by the theorem that the value of R is proportional to the energy. But in view of the results of new measurements this conception soon proved untenable."

"[T]wo simple limits were established by direct observation for the function R: for small energies proportionality to the energy, for large energies proportionality to the square of the energy. Nothing... seemed simpler than to put in the general case R equal to the sum of a term proportional to the first power and another proportional to the square of the energy... and thus was found a new radiation formula which... has withstood experimental examination fairly satisfactorily."

"But even if this radiation formula should prove to be absolutely accurate it would after all be only an formula found by happy guesswork..."

"I was... occupied with the task of giving it a real physical meaning, and this... led me, along Boltzmann's line... to the consideration of the relation between and probability... after some weeks of the most intense work of my life clearness began to dawn... and an unexpected view revealed itself..."

", according to Boltzmann, is a measure of a physical probability, and the meaning of the second law of thermodynamics is that the more probable a state is, the more frequently will it occur in nature."

"[W]hat one measures are only the differences of entropy, and never entropy itself, and consequently one cannot speak... of the absolute entropy of a state. But nevertheless the introduction of an appropriately defined absolute magnitude of entropy is... recommended... by its help certain general laws can be formulated with great simplicity."

"[T]he case is... the same as with energy. Energy... cannot itself be measured; only its differences can."

"[T]he concept used by our predecessors was not energy but work, and even Ernst Mach, who devoted much attention to the law of but... avoided all speculations exceeding the limits of observation, always abstained from speaking of energy..."

"[I]n the early days of one was content to deal with heats of reaction, that is to say again with differences of energy, until emphasized that... calculations could be... shortened if energies instead of calorimetric numbers were used."

"The additive constant which... remained undetermined for energy was later finally fixed by the relativistic law of the proportionality between energy and inertia."

"As in the case of energy, it is now possible to define an absolute value of entropy, and thus of physical probability, by fixing the additive constant so that together with the energy (or better still, the temperature) the entropy also should vanish."

"Such... led to a comparatively simple method of calculating the physical probability of a given distribution of energy in a system of resonators, which yielded precisely the same expression for entropy as that corresponding to the radiation law; and it gave me particular satisfaction, in compensation for the many disappointments... to learn from Ludwig Boltzmann of his interest and... acquiescence in my... reasoning."

"To work out these probability considerations the knowledge of two universal constants is required, each of... independent meaning, so... evaluation of these... from the radiation law could serve as an a posteriori test whether the... process is merely a mathematical artifice or has a true physical meaning."

"The first constant... is connected with the definition of temperature. If temperature were defined as the mean of a molecule in a , which is a minute energy indeed, this constant would have the value ⅔. But in the conventional scale of temperature the constant ...[instead] assumes an extremely small value... intimately connected with the energy of a single molecule... [I]ts accurate determination would lead to the calculation of the mass of a molecule and... associated magnitudes. This constant is frequently termed Boltzmann's constant, although to the best of my knowledge Boltzmann... never introduced it (...he, as appears from... his statements, never believed it would be possible to determine this constant accurately)..."

"Nothing can better illustrate the rapid progress of experimental physics within the last twenty years than the fact that... [by] a host of methods ...the mass of a single molecule can be measured with almost the same accuracy as that of a planet."

"While at the time when I carried out this calculation on the basis of the radiation law an exact test of the value... was... impossible... it was not long before E. Rutherford and H. Geiger succeeded, by... a direct count of the α-particles, in determining the value of the electrical as 4.65 10-10, the agreement... with my value 4.69 10-10... a decisive confirmation of my theory. ...[M]ethods ...by E. Eegener, R. A. Millikan, and others... have led to a but slightly higher value."

"Much less simple... was the interpretation of the second universal constant of the radiation law... the product of energy and time (...a first calculation to 6.55 10-27 erg. sec.) I called the elementary quantum of action."

"While this constant was absolutely indispensable to... a correct expression for —for only with its aid could be determined the magnitude of the 'elementary region' or 'range' of probability, necessary for the statistical treatment of the problem—it obstinately withstood all attempts at fitting it... into the frame of the classical theory. So long as it could be regarded as infinitely small, that is to say for large values of energy or long periods of time, all went well; but in the general case a difficulty arose... which became the more pronounced the weaker and... more rapid the oscillations."

"The failure placed one before the dilemma: either the quantum of action was only a fictitious magnitude, and... the entire deduction from the radiation law was illusory and a mere juggling with formulae, or there is at the bottom of this method of deriving the radiation law some true physical concept."

"If the latter were the case, the would have to play a fundamental role in physics, heralding the advent of a new state of things, destined, perhaps, to transform completely our physical concepts which since the introduction of the infinitesimal calculus by Leibniz and Newton have been founded upon the assumption of the continuity of all causal chains of events."

"Experience has decided for the second alternative. But that the decision should come so soon... was due not to the examination of the law of distribution of the energy of heat radiation, still less to my special deduction of this law, but to the steady progress of the work of those investigators who have applied the concept of the quantum of action to their researches."

"The first advance... was made by A. Einstein, who... pointed out that the... quanta of energy associated with the quantum of action seemed capable of explaining... a series of remarkable properties of light action discovered experimentally, such as Stokes's rule, the emission of electrons, and the ionization of gases, and on the other hand, by the identification of the expression for the energy of a system of resonators with the energy of a solid body, derived a formula for the specific heat of solid bodies which on the whole represented it correctly as a function of temperature, more especially exhibiting its decrease with falling temperature."

"With regard to specific heat of solid bodies, Einstein's view, which rests on the assumption of a single free period of the atoms, was extended by M. Born and Th. von Karman to the case which corresponds better to reality, viz. that of several free periods; while P. Debye, by a bold simplification of the assumptions as to the nature of the free periods, succeeded in developing a comparatively simple formula for the specific heat of solid bodies which excellently represents its values, especially those for low temperatures obtained by W. Nernst and his pupils, and... compatible with the elastic and optical properties of such bodies."

"But the influence of the quanta asserts itself also in the case of the specific heat of gases. At the very outset it was pointed out by W. Nernst that to the energy quantum of vibration must correspond an energy quantum of rotation, and it was therefore... expected that the rotational energy of gas molecules would also vanish at low temperatures. ...That 'quantized' rotations of gas molecules... do actually occur in nature can no longer be doubted... although a[n] exhaustive explanation of... rotation spectra is still outstanding."

"The inverse of the process of producing light quanta by the impact of electrons is the emission of electrons on exposure to light-rays, or s, and here... energy quanta following from the action quantum and the vibration period play a characteristic role, as was early recognized from the striking fact that the velocity of the emitted electrons depends not upon the intensity but only on the colour of the impinging light. [T]the relations to the light quantum, pointed out by Einstein, have proved successful in every direction... shown especially by R. A. Millikan, by measurements of the velocities of emission of electrons, while the importance of the light quantum in inducing photo-chemical reactions was disclosed by E. Warburg."

"[T]he results... hitherto quoted... taken in their totality, form an overwhelming proof of the existence of the quantum of action, the quantum hypothesis received its strongest support from the theory of the structure of atoms (Quantum Theory of Spectra) proposed and developed by Niels Bohr... the long-sought key to the gates of the wonderland of spectroscopy which since the discovery of spectrum analysis... stubbornly refused to yield. And... once clear, a stream of new knowledge poured in a sudden flood, not only over this... field but into the adjacent territories of physics and chemistry."

"Its first brilliant success was the derivation of Balmer's formula for the spectrum series of and , together with the reduction of the universal constant of Rydberg to known magnitudes; and even the small differences of the Rydberg constant for these two gases appeared as a necessary consequence of the slight wobbling of the massive atomic nucleus (accompanying the motion of electrons around it). As a sequel came the investigation of other series in the visual and especially the X-ray spectrum aided by Ritz's resourceful combination principle, which only now was recognized in its fundamental significance."

"But whoever may have still felt inclined... in the face of this almost overwhelming agreement... to believe it... a coincidence, must... give up... doubt when A. Sommerfeld deduced, by a logical extension of the laws of the distribution of quanta in systems with several degrees of freedom, and by a consideration of the variability of inert mass required by the principle of relativity, that magic formula before which the spectra of both and revealed the mystery of their ... by the most delicate measurements...of F. Paschen..."

"P. Epstein achieved a complete explanation of the of the electrical splitting of spectral lines, P. Debye obtained a simple interpretation of the K-series of the X-ray spectrum investigated by , and then... a long series of further researches... illuminated... the dark secret of atomic structure."

"[T]he quantum of action, which in every one of the many and most diverse processes has always the same value, namely 6.52 10-27 erg. sec., deserves to be... incorporated into the system of the universal s."

"[A]t just the same time as the idea of general relativity arose... nature revealed, precisely... where ...least ...expected, an absolute and strictly unalterable unit, by means of which the amount of action contained in a space-time element can be expressed by a perfectly definite number, and thus is deprived of its former relative character."

"[T]he mere introduction of the quantum of action does not yet mean that a true Quantum Theory has been established. Nay, the path which research has yet to cover... is perhaps not less long than that from the discovery of the velocity of light by Olaf Römer to the foundation of Maxwell's theory of light."

"The difficulties which the introduction of the quantum of action into the well-established classical theory has encountered from the outset... have gradually increased rather than diminished; and although research... has... passed over some of them, the remaining gaps in the theory are the more distressing..."

"[W]hat in Bohr's theory served as the basis of the laws of action consists of certain hypotheses which a generation ago would doubtless have been flatly rejected by every physicist. That with the atom certain quantized orbits [i.e. picked out on the quantum principle] should play a special role could well be granted; somewhat less easy to accept is the further assumption that the electrons moving on these curvilinear orbits, and therefore accelerated, radiate no energy. But that the sharply defined frequency of an emitted light quantum should be different from the frequency of the emitting electron would be regarded... in the classical school as monstrous and almost inconceivable. But numbers decide... the tables have been turned."

"While originally it was a question of fitting in with as little strain as possible a new and strange element into an existing system... generally regarded as settled, the intruder... having won an assured position, now has assumed the offensive; and... is about to blow up the old system... The only question... is, at what point and to what extent this will happen."

"[O]ut of the classical theory the great principles of thermodynamics will not only maintain intact their central position in the quantum theory, but will perhaps even extend their influence."

"The significant part played in the origin of the classical thermodynamics by mental experiments is now taken over in the quantum theory by P. Ehrenfest's hypothesis of the adiabatic invariance; and just as the principle introduced by R. Clausius, that any two states of a material system are mutually interconvertible on suitable treatment by reversible processes, formed the basis for the measurement of , just so do the new ideas of Bohr show a way into the midst of the wonderland he has discovered."