05 November, 2014

Planck, Einstein and the birth of quantum physics

(or the man who created but didn’t believe, and the man who believed but wished it wasn’t so)

One of the disappointing things about the treatment of the birth of quantum physics in most textbooks is that they present an ahistorical story in which Planck and Einstein react to detailed and precise experimental evidence about blackbody radiation and the photoelectric effect. This certainly produces a very clear storyline, but at the expense of making one of the greatest scientific feats of all time appear practically mundane. In reality the experimental evidence Planck and Einstein had to work with was meagre and their brilliance in piecing it together to develop the foundations of quantum physics was astounding.

At the time of Planck’s work it was understood that all bodies radiated energy (electromagnetic radiation) as a function of their temperature. The nature of this radiation was being studied through the use of blackbody cavity radiators. The light coming off an object consists of reflection and emission. Theoretical blackbodies absorb all the incident radiation (ie no reflection) so that the radiation coming off the object consists entirely of emissions due to their temperature. A good approximation of a blackbody can be made by constructing a cavity lined with highly absorbing material such as graphite, and a geometry that makes it very unlikely that any incident radiation will be reflected out the opening before it is absorbed by the walls of the cavity. Therefore, if you measure the radiation coming off a blackbody radiator it will consist entirely of emitted radiation

The techniques used to study blackbody radiation were based on thermodynamics. If you haven't studied thermodynamics it is difficult to convey a good understanding of Planck’s and Einsteins’s achievements but I will try. Thermodynamics attempts to understand a system by looking at the average actions and properties of all the individual components. In the case of emitted radiation, it was understood that all particles oscillate or vibrate and that temperature is a measure of the average energy of these oscillations (the energy of an oscillation is a function of the amplitude and frequency). Furthermore, a body heats up when its particles absorb energy and oscillate more quickly (at a higher frequency) and cool down when its particles emit energy and oscillate more slowly. A body that is ‘hotter’ than its surroundings will cool down by emitting radiation to its environment and the frequency distribution of this radiation will be related to the temperature of the body. It was the study of these frequency distributions that led Planck to his quantum hypothesis.

To appreciate the brilliance of Planck’s contribution we need to realise that when he was working on the ‘blackbody’ problem good data were only available for the visible part of the EMR spectrum. For this part of the spectrum scientist were developing increasingly good theoretical descriptions of the observed distributions based on the Stefan-Boltzman Law (which describes the total amount of energy radiated) and Wein’s Law (which describes how this radiation is distributed across the spectrum). Both these are classical laws which allow particles to oscillate at any energy level (ie the distribution of the energies of oscillation is continuous). In 1900 reliable data started to become available for the infrared part of the spectrum and there was a small discrepancy between the ‘classical’ predictions and the observations. In an effort to make the predictions fit the observations for this part of the spectrum, Planck introduced the restriction that particles could only oscillate with energies which were multiples of hf. More specifically E = nhf where n=1,2,3,4... In other words the energy of oscillation changed in steps rather than continuously. While this produced a good agreement between theory and observation, Planck did not really believe that the energy did come in ‘lumps’ but rather thought that this 'mathematical trick' would lead to further investigation that would eventually produce a classical solution to the problem.

Empirical and theoretical (dotted) blackbody curves
Meanwhile, in Bern, a patent clerk class 3 called Albert was sitting in a dingy office writing papers building up the foundations of statistical mechanics as a way of describing the overall behaviour of physical systems. In 1905 he applied these techniques to the problem of blackbody radiation. Using classical assumptions, he showed that the only possible solution to the blackbody problem was a frequency distribution that increased exponentially in the UV part of the spectrum. This was clearly not possible as it would imply that the total energy of the system (the area under the curve) was infinite. This was later called the ultraviolet catastrophe. However, when Einstein incorporated Planck’s quantum hypothesis into his analyses he found that the theoretical predictions fitted the observed data in the infrared and also solved the problem of the ultraviolet catastrophe (note that UV measurements were not available at the time)

Unless you are familiar with thermodynamics its not easy to understand why introducing energy 'quanta' changes the energy distribution and solves the ultraviolet catastrophe but if all this is making your brain hurt perhaps this MinutePhysics video will help.

The important point here is that Planck’s modification, while brilliant, was purely empirical (a mathematical attempt to make the predictions match the results) while Einstein’s approach gave the quantum hypothesis a strong theoretical basis. Einstein also realised that if particles could only oscillate at discrete energy levels, and if they change between energy levels by emitting or absorbing radiation, then all radiation must be emitted or absorbed in discrete packets or ‘quanta’ with energies equal to hf. We now call these packets photons though the term would not be introduced until 1916. This idea was truly revolutionary because, among other things, it suggested that light (and all other EMR) was behaving as a particle, rather than as a wave as Maxwell’s equations suggested and most experimental evidence demonstrated.

Einstein’s ideas were not well received for many years. Indeed, in 1914 when Einstein was being recommended for membership to the Prussian Academy, those recommending him, including Planck, wrote ‘he may sometimes have missed the target in his speculations, as, for example, in his hypothesis of light-quanta’. Fortunately, Einstein proposed a test of his hypothesis based on the photoelectric effect which was confirmed by Millikan in 1916 and quantum physics was finally born. It is an interesting and ironic footnote that after having provided the basis of quantum physics which would eventually give rise to the uncertainty principle and wave-particle duality, Einstein spent his later years trying to unscramble the egg and find classical, deterministic explanations for these phenomena. He wasn’t successful.

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