1 Origin of Quantum Mechanics—Inadequacy of Classical Laws of Mechanics and Electromagnetic Radiation
Prof. V.S. Bhasin
TABLE OF CONTENTS
1. Learning Outcomes
2. Introduction
3. Origin of Quantum Mechanics: Prominent Phenomena laying the Foundations of Quantum Mechanics
3.1 Energy Spectrum in Black Body Radiation
3.2 Specific Heat of Solids at Very Low Temperatures
3.3 Photoelectric Effect
3.4 The Compton Effect
4. Summary
1. Learning Outcomes (Times New Roman , size 14)
- After studying this module, you shall be able to
- Learn how Planck proposed his idea of quantization of energy from the study of spectral distribution of black body radiation
- Know Einstein’s extension of Planck’s hypothesis to resolve the paradox of the behavior of specific heat of solids at very low temperatures
- Study Einstein’s generalization of photon picture of light to explain photo-electric effect
- Learn how to deduce the expression for shift in wave length in Compton Scattering of hard X-rays by elements of low atomic number
2. Introduction
Quantum Mechanics is considered as the fundamental theory describing the phenomena of molecules, atoms and subatomic particles of matter. This subject evolved as a result of striking observations from a number of experimental investigations carried out in different areas of physics from the end of 19th century to the beginning of 20th century. These observations could not be adequately explained on the basis of classical laws of physics. Historically, by the end of 19th century, classical mechanics as formulated by Newton and later developed by Hamilton and Lagrange was widely believed and universally accepted as the ultimate theory of dynamics. Similar was the status regarding the wave nature of light as established from the experiments on interference, diffraction and polarization and later developed in the form of electromagnetic theory of light by Clark Maxwell and confirmed by Hertz in 1887 as electromagnetic waves produced by oscillating electric charges.
However, by the end of 19th century these classical theories were found inadequate to explain a growing number of atomic phenomena. Attempts were being made to explore the structure of matter and the nature of radiation and the interaction of radiation with matter. As a result, the first quarter of the 20th century witnessed how the foundations of physics were radically reexamined. In the present module we shall trace the history of some prominent phenomena which point out the limitations of the laws of classical mechanics when used to study microscopic phenomena in physics and show how these were overcome first by Planck, Einstein and Bohr and later by Schrodinger and others.
3 Prominent Phenomena laying the Foundations of Quantum Mechanics
3.1 Energy Spectrum in Black Body Radiation
One of the earliest phenomena for which classical theory was unable to offer any satisfactory explanation till the end of nineteenth century was the nature of the continuous spectrum of black body radiation. We all know that when an object is heated Intensity up sufficiently so that it gets red hot, it emits light at the red end of the spectrum. Further heating it causes the colour of the emitted light to change towards shorter wavelengths or higher frequencies. Recall that a black body, by definition, is one which absorbs all the electromagnetic radiation which is incident on it and reflects none. Such a body is a better radiator at every frequency than any other at the same temperature. A perfect black body does not exist. However, for practical purposes a black body is realized by creating a small hole in the wall of the cavity in which radiation is admitted from outside. The cavity contains radiation which is emitted by the walls of the cavity. The amount of radiant energy in the cavity does not increase indefinitely with time. In fact, the process of emission is opposed by the process of absorption. In the state of thermodynamic equilibrium, the amount of energy, E(ν) dν in the frequency range between ν and ν+dν is determined by the condition at which the rate of emission is balanced by the rate of absorption for a given frequency. It has been demonstrated both experimentally and theoretically that at equilibrium the intensity distribution, I(ν)=?/? E(ν), depends only on the temperature of the walls and is independent of the material of back body. Figure (1.1), given below shows a plot of intensity distribution as a function of wavelength λ plotted at different temperatures.
Fig.(1.1): Plot of Intensity Distribution vs wavelength λ radiated by a black body at different temperatures.
The figure clearly depicts that intensity distribution first rises to a maximum and then falls off as the wavelength is increased,- maxima of the distribution curves slightly shifting towards the lower end of the wavelength as temperature increases.
These observations could not be completely accounted for by the attempts based on classical ideas. Classically, the absorption and emission of radiation was considered to be due to oscillating charges within the black body. It was assumed that in a black body there are a large number of linear harmonic oscillators of all possible frequencies in equilibrium with radiation at a given temperature. Based on these considerations, while Wien suggested a semi-empirical theory which agreed only in the short wavelength limit, Raleigh and Jeans, on the other hand, obtained a law which agreed at long wavelengths limit but was in complete disagreement for short wavelengths. Thus, according to classical electromagnetic theory, the expression for the energy density distribution, E(ν), of a black body radiation can be expressed as
This is known as Rayleigh-Jean’s law, according to which the energy radiated in a given wavelength range dλ increases indefinitely. This is depicted on the right hand side in the plot in (Fig.1.1) for comparison. It also becomes clear from the above expression that the total energy, E, becomes i.e.,
which goes to infinity as ν approaches infinite. This basic draw-back in classical theory is often referred to as ‘ultra-violet catastrophe’.
In 1900, Max Planck, the German physicist, came out with the revolutionary idea that matter absorbs or emits radiation energy not continuously but rather in discrete quanta. Assuming, as in classical theory, that a black body is composed of oscillators in equilibrium with the radiation field, Planck postulated that an oscillator with frequency ν can only take discrete values of energy quanta,∈ = , where n=0,1,2…….. and h is a constant which is now known as Planck’s constant having value6.63 x 10-34 joule sec.
To obtain the Planck’s expression for the intensity distribution, we start by writing the mean energy per oscillator of frequency ν as
where is the number of oscillators with energy ∈ in equilibrium with temperature T, given by the classical Boltzmann expression, viz.,
Substituting this expression in the above equation for u, we have
This expression not only explains the observed spectral distribution of a black body but can also reproduce in the limiting case the Rayleigh-Jean’s law. One can also show that the well-known Stefan’s law can also be obtained from this general result. Thus, for example, in the low frequency or long wavelength limit, one can write
Substituting this in the above expression, Planck’s result reduces to the Rayleigh-Jean’s expression. Also the total radiation density arising from all frequencies can be obtained by integrating Planck’s energy distribution. i.e.,
which shows that energy density is proportional to the fourth power of absolute temperature T. This is the result which was first suggested by Stefan in 1879 and is known as Stefan’s law of radiation.
The success of the Planck’s distribution law based on the quantum hypothesis may be regarded as the first step to look beyond classical laws for the understanding of processes in at least some areas of physics.
It may be remarked here that for quantum effects to show an observable departure of the mean energy u(ν) from its classical value kT ( Note here that from classical considerations Boltzmann’s expression,
reduces to kT) , the frequency should be high enough to get hν/kT comparable to unity. For example at room temperature where T= , the term hν/kT turns out to be of the order of 1/5 for the value of ν= .
3.2 Specific Heats of Solids at Very Low Temperatures
This puzzling problem of classical physics was resolved by Einstein following Planck’s hypothesis. Assuming that a solid can be represented by a collection of harmonic oscillators which can take only discrete energy values (nhν), where n is an imteger and ν is the oscillator frequency, Einstein considered the simplest model where all the oscillators have the same frequency, . Thus the internal energy of the solid is
And the lattice heat capacity at constant value is
This can be expressed as
The above expression may be used to obtain a reasonable fit to the experimental data for a given material by choosing an appropriate value for Einstein temperature, which is found to be of the order of a few hundred degrees absolute. Einstein’s formula is found to be in fair agreement with experiment for ≫ but at low twmperatures, the agreement is poor. For non metals, the heat capacity approaches zero as T3 whereas Einstein’s formula predicts
Debye in 1912 successfully explained the variation at low temperatures by considering the presence of a spectrum of frequencies. Indeed, Einstein’s derivation was by no means the last word, but it was sufficient to demonstrate the inadequacy of classical concepts, namely that material oscillators, like radiation oscillators, can take only discrete energy values in limiting situations.
3.3 Photoelectric Effect
The phenomenon of photoelectric effect was observed for the first time in 1887 by Heinrich Hertz while he was carrying out experimental investigations to establish the existence of electromagnetic waves in order to confirm Maxwell’s theory. He noticed that ultraviolet light falling on metallic electrodes produced high voltage sparks across the detector loop. Light shining on the metal surface facilitated to release free charged particles which were identified y as electrons. Later, this phenomenon was studied in detail by W Hallwachs and Phillip Lenard. The detailed investigations carried out by Lenard revealed the following striking features which can not be explained by classical electromagnetic theory.
- There exists a minimum, called threshold frequency of the radiation characteristic of the surface below which no emission of electrons can take place. It does not matter what the intensity of incident radiation is and for how long it falls on the surface. This observation is in contradiction with the classical wave theory according to which photoelectric effect should occur for any frequency of incident radiation provided the intensity of radiation is large enough for ejecting the electrons.
- The maximum kinetic energy of electrons emitted by the surface is found to depend linearly on the frequency but is independent of the intensity of incident radiation. According to classical electromagnetic theory, the maximum kinetic energy of the electrons should increase with the increase in intensity but should be independent of frequency.
- Experimentally, the photoelectric emission is an instantaneous process without any apparent time lag (less than or of the order of 10−9s ). In wave picture, the absorption of energy takes place continuously. The energy absorbed per electron per unit time turns out to be small. Explicit calculations show that it can take a sufficient long time for a single electron to pick up sufficient energy in order to come out of the metal.
In 1905, Einstein proposed a radically different picture by suggesting that electromagnetic radiation only exists in discrete energy corpuscles or quanta, called photons, each photon having an energy E=hν=hc/λ He thus advanced the idea by generalizing Planck’s postulate of quantization of energy to postulate that electromagnetic radiation by itself exists in the form of light quanta.
According to Einstein, when a photon falls on a metallic surface, its entire energy hν is used to eject an electron from the atom. If hν exceeds the amount of energy, called the work function (W), required to escape the electron from the surface of the metal, the electrons would move out of the surface with energies upto a
maximum value, = ℎ − . This result was verified in a series of experiments carried out using sodium and potassium as photoelectric surfaces. According to the Einstein equation, plot of maximum kinetic energy, , (which is experimentally observed in terms of the stopping potential , 0, just sufficient to stop the fastest electrons) with frequency ν of incident radiation gives a linear fit showing that there exists a threshold frequency below which no electron emission takes place( See Fig.1.3(a)..
Fig. 1.3(a) Plot of Stopping Potential vs Frequency of incident radiation for two metals (A) and (B). Note that slope of the linear plot is same for different metallic surfaces. It is, in fact, proportional to Planck‘s constant and charge of the electron.
Fig.1.3(b) Plot of Photocurrent vs collector Potential for different intensity of radiation but having same frequency
Indeed experimental data obtained by Millikan provide further confirmation of Einstein’s theory as shown in Figures 1.3(a) and 1.3(b) .
3.4 The Compton Effect
About 20 years later, i.e., in 1924, Arthur Compton discovered that when hard X-rays (of shorter wavelength) are scattered by atoms of an element of low atomic number (such as graphite), the scattered radiation contains not only the original wavelength but also softer X-rays of longer wavelength. Compton was able to explain this phenomenon of scattering of X-rays by assuming that X-rays consist of a collection of photons, each characterized by energy, E, and momentum, p. Assuming that X-rays of wavelength consist of a stream of photons of energy E h h c / , Compton argued that when one of these quanta hits a free or loosely bound electron, it would recoil. As a result it would have an energy ’ < after the collision and therefore the corresponding wavelength Based on this picture, quantitative calculations can be made, using the laws of conservation of energy and momentum, to estimate the increase in the wavelength of the scattered photon.
Consider an incident photon of energy 0 and momentum 0, which collides with an electron and is then scattered as shown in the figure ( Fig.1.2 ). The figure shows the scattered photon of energy 1 1 = 1/ making an angle and the recoiling electron of momentum 2 making angle ϕ with the direction of the incident photon.In case of metals having low atomic numbers, the binding energy of the electrons in the atoms may be quite small as compared to the energy of the hard X-rays incident on the metal. We may, therefore, assume the electrons to be at rest having the rest mass
W0 = mc2, where m is the mass of the electron. However, since the recoiling electron from the collision may have velocity comparable to c, it would just be appropriate to use the relativistic relation for the velocity v of the recoiling electron. Using the law of conservation of energy during collision, we have
4. Smmary
In this module we have explored the nature of electromagnetic radiation by studying some typical processes like Black Body Radiation, Specific Heat of Solids, The Photoelectric Effect and Compton Scattering. These processes clearly show that electromagnetic radiation manifests itself in the form of corpuscles or quanta of light. From our earlier studies we also know that the phenomena of interference, diffraction and polarization can only be explained on the basis of wave theory of light. It is therefore clear that electromagnetic radiation plays a dual role: it exhibits both as particles and waves. This ability of electromagnetic radiation to behave both as waves and photons is referred to as wave-particle duality.
| you can view video on Origin of Quantum Mechanics—Inadequacy of Classical Laws of Mechanics and Electromagnetic Radiation | |
References
For More Details ( on this topic and other topics discussed in Text Module) See
1.Quantum Mechanics by Bransden and Joachain, Pearson Education Ltd(2000).
2.Quantum Mechanics ,Vol. I, by C. Cohen- Tanoudji , John Wiley & Sons
3.A Text Book of Quantum Mechanics by P M Mathews and K Venkatesan; Tata McG raw-Hill Publishing Co.
Ltd.
For General Study on Origins of Quantum Theory:
1 Introducing Quantum Theory by J P Mc Evoy, University of Pittsburgh
2. The Origins of the Quantum Theory by Cathryn Carson