What Erwin Schrödinger Thought About Thermodynamics

While the quantum mechanical framework was being developed after Plank’s discovery in 1901, physicists were wrestling with the dual character of light (wave or particle?). Thomas Young’s double slit experiment in 1803, where interference patterns were observed, seemed to show without doubt that light was a wave phenomenon. However, Planck’s interpretation of black body radiation as light quanta, followed by Einstein’s explanation of the photoelectronic effect, both contradicted the light-as-wave theory. Additionally, a shocking discovery was made by Compton in 1925. Compton found that when he let X-rays (a form of light with extremely short wavelengths) collide head-on with a bundle of electrons, the X-rays were scattered as if they were particles. This phenomenon became known as the “Compton scattering experiment.”

At about that time, French physicist Louis de Broglie combined two simple formulas: Plank’s light quanta expression (E = hν, with ν as the frequency) and Einstein’s famous energy‑mass equation (E = mc²). This led to another simple equation: λ = h/mc, with λ as wavelength. This equation really tells us that all matter has wave properties. However, since the mass, m, of most everyday visible objects is so large, their wavelengths are too small for us to notice any wave effect. But when we consider the small masses of atomic particles such as electrons and protons, their wavelengths become relevant and start to play a role in the phenomena we observe.

All this brought Erwin Schrödinger to the conclusion that electrons should be considered waves, and he developed a famous wave equation that very successfully described the behavior of electrons in a hydrogen atom. Schrödinger’s equation used a wave function to describe the probability of finding a rapidly moving electron at a certain time and place. In fact, the equation confirmed many ideas that Bohr used to build his empirical atom model. For instance, the equation correctly predicted that the lowest energy level of an atom could allow only two electrons, while the next level was limited to eight electrons, and so on. In the year 1933 Schrödinger was awarded the Nobel Prize for his wave equation.

Schrödinger had, as did Planck and Einstein, an extensive background in thermodynamics. From 1906 to 1910, he studied at the University of Vienna under Boltzmann’s successor, Fritz Hasenöhrl. Hasenöhrl was a great admirer of Boltzmann and in 1909 he republished 139 of the latter’s scientific articles in three volumes [Hasenöhrl, 1909]. It was through Hasenöhrl that Schödinger became very interested in Boltzmann’s statistical mechanics. He was even led to write of Boltzmann, “His line of thoughts may be called my first love in science. No other has ever thus enraptured me or will ever do so again [Schrödinger 1929].” Later he published books, (Statistical Thermodynamics and What’s Life), and several papers on specific heats of solids and other thermodynamic issues. [1]

 © 2009 Copyright John Schmitz

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[1] Taken from “The Second Law of Life”:http://www.elsevierdirect.com/product.jsp?isbn=9780815515371

 

What Einstein Thought About Thermodynamics

Einstein, like Planck, was very fluent in thermodynamic theory. Before 1905, Einstein published several papers on thermodynamic topics. One of these dealt with the fundamentals of thermodynamic theory [Einstein, 1903]. In this work, he studied whether the thermodynamic laws could be derived from a minimum amount of elementary assumptions. In 1905[1] he published a study in which he explained the photoelectric phenomenon. In that explanation, he not only used the results of Planck’s discrete energy packets for the black body radiation description, but fully acknowledged Boltzmann’s work, calling the expression S = k lnW  “the principle of Boltzmann.”

How highly Einstein regarded thermodynamics can be appreciated in the following quote:

 “A law is more impressive the greater the simplicity of its premises, the more different are the kinds of things it relates, and the more extended its range of applicability. (..) It is the only physical theory of universal content, which I am convinced, that within the framework of applicability of its basic concepts will never be overthrown.”

Einstein is best known for his invention of relativistic theory where time is no longer invariable[2]. Less remembered is that he searched his whole life for a theory that could unify the electromagnetic theory of Faraday and Maxwell on one hand, and the mechanical theory of the material particles of Newton on the other. For instance, Newton unified the observations of falling objects on earth with the fact that the earth and planets orbited the sun. He did this by using a single concept – namely, gravity – to explain both phenomena. Maxwell showed that seemingly quite different magnetic and electric observations could be described by a single theory of electromagnetic waves.

Around 1900 several outstanding physicists were working to explain Planck’s black body radiation. Planck had to introduce quantum theory to explain the experimental observation of the relationship between energy and wavelength. Einstein did not like this explanation, since it introduced yet another theory rather then unifying existing theories. Einstein was convinced that the answers could be found in thermodynamics, since this theory was based on structure-independent assumptions. Indeed, the special theory of relativity can be considered as a theory of principles analogous to the theory of thermodynamics [Klein, 1967].

What brought Einstein to his Special Theory of Relativity was his idea (conceived when he was 16 years old!) that the velocity of light must be the same for all observers, regardless of their respective speeds. He derived this conclusion from Maxwell’s electromagnetic equations and so kept his mind puzzled for a long time. His familiarity with thermodynamic theory also gave him a lot of inspiration. We can appreciate the challenge he found from two questions (taken from the publication of Martin Klein: “Thermodynamics in Einstein’s Thought”; Science, Vol 157, 509 (1967).  In essence what the classical thermodynamic accomplishment was, was to find mathematical expressions to the dilemma:

“What must the laws of nature be like so that it is impossible to construct a perpetual motion machine from either the first or the second kind?”

This question refers to the empirical fact that perpetual machines have never been observed that could violate the first principle that energy cannot be added or destroyed in an isolated system, or contradict the second principle that entropy always increases for spontaneous processes in, again an isolated system. Similarly, while developing the Special Theory of Relativity, Einstein wondered:

“What must the laws of nature be like so that there are no privileged observers?”

This question refers to the fact that the speed of light is the same for all observers, regardless of how fast their platform (a planet, a rocket, or an angel’s wings) is going. Therefore, one must derive expressions that will obey the principle of the constancy of light speed. In the same way that classical thermodynamics does not worry about why energy is conserved or why entropy increases, so Einstein didn’t try to puzzle out why the speed of light was constant, but merely accepted it as fact. Once accepted, the equations that describe this assumption are pretty straightforward!

Thus, the Special Theory of Relativity can be viewed as a theory of principles analogous to thermodynamics, and not as a constructive theory – as, for instance, gravity or the kinetic gas theory[3]. This means that no model is needed (like a model of an atom in the case of quantum mechanics) in either the Special Theory of Relativity or in thermodynamics, in order to arrive at the end results of both theories. The nice thing is that both theories can live on indefinitely with little risk of needing adjustment because of new insights. That is, in fact, what we’ve seen: both thermodynamics and the Special Theory of Relativity have not changed since their conception. [4]

Taken from:

“The Second Law of Life, Energy, Technology and the Future of Earth As We Know It”

[ http://www.elsevier.com/wps/find/bookdescription.cws_home/715243/description#description]

© Copyright 2009 John Schmitz

 

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[1] 1905 was also the year Einstein published his Special Theory of Relativity, along with his articles on the photoelectric effect, the explanation of Brownian movement, and an article where he stated his famous equation, E=mc². Because of Einstein’s overwhelming amount of important material in one year, 1905 is sometimes called Annus Mirabilis (the MiracleYear) [Bushev, Michael, “A Note on Einstein’s Annus Mirabilis”, Annales de la Fondation Louis de Broglie, Vol 25, no 3 (2000)].

[2] Einstein worked for several years at the Swiss patent office in Bern. During that period, because of the ongoing electrification and synchronization of clocks in the cities and across the countries, many patent applications came in that proposed all sort of ingenious ways to implement the synchronization. Because of that Einstein saw of course many proposals dealing with these kinds of problems and that may have very well triggered his interest in time, see also footnote 72, [Galison, Peter, Einstein’s Clocks, Poincaré’s Maps, Empires of Time; W.W. Norton & Company, Inc., New York (2004)].

[3] The kinetic gas theory starts with the existence of gas molecules, their continuous motion, and their finite dimensions. Then, by applying Newton’s mechanical kinetic theory it is possible to derive a relation among the macroscopic gas parameters: pressure, temperature, and volume. In this way a model can be built that has predictive and verifiable power.

[4] I feel that a few more words are needed here. Einstein himself pointed out in an article in 1919 in the Times of London that a theory of principle is based on empirical observations without the need for a particular model whereas a constructive model will first make assumptions about a fundamental structure then will built a mathematical description of that structure that hopefully will give relationships between the empirically observed parameters. In his own words: “Thus the science of thermodynamics seeks by analytical means to deduce necessary conditions, which separate events have to satisfy, from the universally experienced fact that perpetual motion is impossible”. Thus, classical thermodynamics can be regarded as a theory of principles, whereas statistical thermodynamics (i.e., Boltzmann approach) should be categorized as a constructive theory. In 1904 it was Poincaré who made a similar classification in scientific theories in his book The Value of Science.

 

What Max Planck Thought About Thermodynamics

 

Around 1900, Planck was an expert in classical thermodynamics and wrote many articles and books about that theory. The concept of entropy especially held his interest, but he published also in the fields of dilute solutions and thermoelectricity. Of course, being a time-oriented fellow, he was familiar with the results of Boltzmann’s works. However, being a physicist of the “old school,” he was raised without having the concept of atoms in his scientific toolkit. In 1891, for instance, he and Ostwald had a discussion with Boltzmann at a conference where Planck stated that thermodynamic methods without the incorporation of atomistic models were sufficient to explain those days’ physical observations. Also, Planck was not very pleased with the statistical approach of Boltzmann [1]. His main objection was that Boltzmann’s statistical approach allowed that the change of entropy for spontaneous processes could become negative (i.e., an entropy decrease), although at an extremely low probability (see Chapter 3 for more details on this topic).

But Planck was wrestling at the turn of the century with understanding black body radiation behavior. Since 1861, when Kirchhoff[2] first described a black body, the radiation behavior was studied and described by a slew of well-known physicists such as Wien, Stefan, and Boltzmann. However, all these attempts led only to a radiation law that had very limited applicability. The breakthrough in Planck’s understanding came when he started to use Boltzmann’s statistical approach. In fact, it was Planck who wrote the current well-known form of the Boltzmann equation, S = k lnW, in his famous 1901 article[3] . It was in this text that Planck proposed that the radiation might consist of small packets (quanten) of size hv. This was the beginning of quantum mechanical theory. Planck struggled a long time with his own thoughts, since they were so in contrast with the classical belief of continuous energy. For some time he saw his quanten approach merely as a mathematical trick, but slowly became convinced that energy in nature was indeed discrete, rather than a continuum. It also took some time before his ideas were accepted in the scientific community[4]. It was no less than Einstein who used the quanta principle to explain the photoelectronic effect, as we will see shortly.

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[1] Flamm, Dieter, “Einführung zu Ludwig Boltzmanns Entropy und Wahrscheinlichkeit”, this is an introduction of Entropie und Warscheinlichkeit, 1872-1905 von Ludwig Boltzman in Ostwalds Klassiker der Exakten Wissenschaften, Band 286, Verlag Harri Deutsch, Frankfurt am Main (2000). This book is contains a nice compilation of the most important articles from Boltzmann in original version.

[2] It’s likely that Planck got his interest in black body radiation from Kirchhoff, who was his teacher. In 1889, he succeeded Kirchhoff as professor at the University of Berlin.

[3] Planck, Annalen der Physik 1901

[4] Interestingly, Planck once remarked that a new theory gets accepted not because its opponents become convinced, but because they eventually die and new generations of scientists, unhindered by historic baggage, simply assume the theory is true (provided that it is still supported by experimental facts)!

© Copyright 2009 John Schmitz

 

The Law of Conservation of Energy

Hermann von Helmholtz is a famous name in thermodynamics[1] although he also made famous inventions in the fields of ophthalmology, electrochemistry and acoustics. It was Helmholtz who formulated very clearly, using conclusions reached by Kelvin, Joule and Clausius earlier,  the existence of the law of conservation of energy or of “force” (as energy was called at that time) around 1850. In the winter of 1862, he delivered a series of lectures at Carlsruhe on the topic of the “Conservation of Force”. He started with an introduction in which he managed to elaborate on this theme without using any mathematical formula[2]. Below I will give a brief summary of the main points of this introduction (which can be found on numerous web sites[3]).

He starts of by showing that gravity, the most fundamental of all forces,  can be used to do work[4]. For instance a weight can drive a clock by sinking. Although the weight will have lost it capability to perform work when it reaches the floor, it will not loose its weight: gravity remains. The amount of work can then be determined by the weight times the distance travelled.

Heat can also produce work such as occurs in a steam engine. Here he recalls the point that heat must not be considered as a substance but merely as an movement of internal particles (realize that we are still about 50 years before atoms become widely accepted!). For quite a while it was considered that the amount of heat was constant (for instance, the required amount of heat required to melt a piece of ice is the same amount that one needs to extract when the resulting water is converted to ice again). However, he explains that as soon as heat is converted into work, that an equivalent amount of heat is destroyed. The relationship between heat and work was established by the work of Clausius and Joule. Many other examples exist where work is generated at the cost of something:

• a raised weight can do work but while doing that it must sink and no longer do work
• a stretched spring can do work but will become loose
• the velocity of a mass can do work but will eventually come to rest
• chemical forces (=energy) can do work but they will get exhausted
• electrical force can do work but will consume chemical or mechanical forces

Helmholtz concluded that all natural forces (energy) can do work but they are at the same time exhausted to the degree of work performed. He then formulated that the total quantity of all forces capable of doing work in the whole universe remains constant. He compared this with the laws of constant mass or constant chemical elements (both were of course to be found less constant after the theory of relativity and the discovery of radioactivity!!).

Finally he touches briefly on the topic of perpetual motion and states that force cannot be produced from nothing: something must be consumed. I strongly recommend reading Helmholtz’ introduction.

See also: https://secondlawoflife.wordpress.com/2007/12/28/the-origin-and-development-of-the-energy-principle/

Copyright © 2008 John Schmitz

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[1] Hermann von Helmholtz (1821-1888) reported on July 23 in 1847 on the principle of conservation of energy and showed that he had acquired a deep understanding of this principle. He was, together with Rudolf Clausius, the founder of what was called the Berlin School of Thermodynamics where he succeeded Magnus as the director of the Physical Institute. The influence of this school on the development of thermodynamics was crucial. It is almost unbelievable how many famous scientists were connected to this school. To name a few: Walter Nernst, Max Planck, Albert Einstein, Erwin Schrödinger and Leo Szilard.

[2] Although Helmholtz himself had a very good knowledge of mathematics

[3] See for instance: http://www.bartleby.com/30/125.html

[4] Work is simply defined here as lifting a weight.

 

The Origin and Development of the Energy Principle

Today everybody knows that energy can not be created or destroyed. We know this principle as the law of conservation of energy  (or the First Law of thermodynamics). Today we use this law in many situations perhaps without always realizing this. However, there was a time that there was great confusion about what energy, work and heat exactly was and their relation. It was the German physician Mayer who first formulated in 1842 a statement that can be considered as the predecessor of today’s energy law. At the occasion of the 100th anniversary of Mayer’s discovery, a small (91 pages) book was published in 1943 by Prof. Jacob Clay, a physics professor at the university of Amsterdam, entitled: “Onstaan en ontwikkeling van het Energie Beginsel”[1] (the book is in Dutch and the title can be translated by “Origin and Development of the Energy Principle”). The book contains some interesting historical elaborations of how the energy law developed. Because the book is not so easy to get anymore, I will summarize the chapter where Clay describes how Mayer got to his energy law.

Julius Robert Mayer, born in 1814 in Heilbronn (Germany, not too far from the French border) who, after several world voyages, became a medical doctor in 1841 also in Heilbronn. He was probably driven to understand energy issues by the question how chemical energy was transformed in living beings to do things like work[2] and generate heat. Mayer got convinced that perpetual engines did not exist (“…. das mechanische Arbeit sich nicht aus Nichts erzeugen lasse.” translated as “mechanical work can not be created from nothing”). Also, while studying processes where changes happened (such as the decrease of velocity of moving bodies and the simultaneous appearance of heat) Mayer was searching for something that would be constant. His inspiration for the existence of a  constant factor came from his colleagues in chemistry who used the law of conservation of mass (formulated in 1789 by Antoine Lavoisier) quite successfully. Mayer observed that “lebende Kraft” (literally “living force” but nowadays what we call  that kinetic energy) could be transformed into heat[3]. In addition he noted that the amount of heat needed to warm a given quantity of gas was larger at constant volume (no work from the PDV term!) than when heated at constant pressure. This led him to formulate the equivalence of work (or energy) and heat and he made a first estimate of this equivalency factor (which today is known at 1 calorie = 4.184 Joule).

The contributions by Mayer were for quite some years neglected by the scientific community. One reason was that he did not very clearly present his thoughts[4]. In addition he had competition  from an English beer brewer: John Prescot Joule who arrived  -somewhat later though- to similar conclusions but who was better connected to institutes such as the French Scientific Academy[5].

However, from 1862 onwards Mayer’s work became more and more acknowledged and he won several prestigious awards underlining the importance of his contributions. In 1847 Hermann von  Helmholtz published his famous article entitled: “Ueber die Erhaltung der Kraft” (“On the conservation of energy”) and this can be seen as an important milestone of the development of the First Law of thermodynamics. I will come back to that article soon.

See also: https://secondlawoflife.wordpress.com/2008/03/24/the-law-of-conservation-of-energy/

Copyright © 2008  John Schmitz

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[1] Prof. Dr. J. Clay, Onstaan en ontwikkeling van het energy-beginsel; N.V. Servire, The Hague (1943)

[2] With work we mean here the ability of a system to lift weights.

[3] This fact was actually already noted in 1798 by Count Rumford who observed that a lot of heat was generated in boring cannon barrels.

[4] He confused force and work for instance. Therefore, Johann Poggendorf, who was the editor of the Annalen der Physik, refused to place his article in 1841.

[5] The Academy published a letter from Joule on the energy topic in 1847 in their Comptes Rendus.