Abstract
Three phenomena greatly puzzled physicists at the turn of the twentieth century: the photoelectric effect, Compton scattering and the ultraviolet catastrophe associated with blackbody radiation. We show in detail how these puzzles led to the startling conclusion that light, previously thought of as a wave, can behave under certain circumstances as a collection of particles. We derive the Compton scattering formula using the relativistic tools acquired in Chap. 7. Next we present a derivation of the Rayleigh-Jeans law that was at the root of ultraviolet catastrophe. We then show how the simple assumption that light is made up of particles leads to the correct formula for blackbody radiation. Since light behaves like both particles and waves depending on the experiment, it stands to reason that electrons (and indeed all particles) can manifest wave-like attributes, including interference and diffraction. In 1924 Louis de Broglie speculated that every particle has associated with it a wavelength that is inversely proportional to its momentum. This led to one of the best known consequences of quantum mechanics, namely the Heisenberg uncertainty principle relating the quantum uncertainty in the position of a particle to the uncertainty in its momentum.
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Notes
- 1.
1905 is called Einstein’s miraculous year because in that year he published three papers that each had a profound effect on a different field of physics. In these papers he announced his special theory of relativity, revealed the cause of Browning motion (the random motion of tiny particles through a fluid) and explained the photoelectric effect.
- 2.
Einstein [1]. It is interesting that Einstein never received a Nobel Prize for either special or general relativity. These theories were thought to be too radical and of little practical use.
- 3.
It is remarkable that Einstein’s intuitive leap along with this simple argument was sufficient to earn Einstein the Nobel prize for physics.
- 4.
We have added a few significant digits to the measured value for good measure.
- 5.
The radiation comes out of the blackbody at all angles. A steradian is a unit solid angle. There are, by definition \(4\pi \) steradians on a unit sphere.
- 6.
The minus sign is simply telling us that as wavelength increases frequency decreases, and vice versa.
- 7.
It is named after Austrian physicist Ludwig Boltzmann who played a major role in the development of statistical mechanics, the microscopic theory that underlies classical thermodynamics.
- 8.
- 9.
Other boundary conditions are possible. For example, if the string is attached via a frictionless hoop to a post so that it is free to move up and down at the ends, then the boundary condition is that the slope of the wave must vanish at the post.
- 10.
Despite the similarities between electromagnetic radiation and waves on a string, there is one big difference. Light propagates in a vacuum, which of course is impossible for so-called mechanical waves such as waves on a string and sound.
- 11.
We use \(\approx \) here and henceforth to denote “approximately equal to”.
- 12.
It must be a positive number.
- 13.
Statistical mechanics is the microscopic theory from which the laws of classical thermodynamics are derived.
- 14.
Temperature is in fact a direct measure of the average kinetic energy of its constituents.
- 15.
see Appendix 15.2 for a general discussion of continuous probabilities.
- 16.
More accurately, the radiation travels equally in all \(4\pi \) directions of the unit sphere, so to get the amount of radiation that hits each wall we must divide by \(4\pi \) instead of 6. This gives the energy per unit time per steradian hitting that wall. This slight error in our calculation comes from assuming cubic symmetry to simplify the calculation of the modes, instead of the correct spherical symmetry.
- 17.
As mentioned above, the error in numerical factor is that we assumed that the radiation in the cavity was moving in one of six directions. In fact it moves in all directions of the unit sphere, i.e. \(4\pi \) steradians, so we should have divided by \(4\pi \) steradians in Eq. (8.47) instead of a factor of 6, which would have given us precisely the right formula for the intensity.
- 18.
As we will see in Sect. 10.2, the energies predicted by quantum mechanics are \(E_m = (m+\frac{1}{2})hf\). The shift of \(\frac{1}{2}\) is called the “zero point energy” and has very interesting consequences. It can be ignored in the present discussion.
- 19.
Recall that \(dN_{tot}\) is in reality 1, but very small compared to \(N_{tot}\), so it can be considered infinitesimal.
- 20.
The existence of this radiation was first predicted in the late 1940s by Ralph A. Alpher, Robert Hermann and George Gamow, but they were largely ignored. Just prior to the Penzias and Wilson discovery, a group of theorists at Princeton University, led by Robert H. Dicke and including Jim Peebles revived the prediction. Jim Peebles, a Canadian by birth and undergraduate education, was awarded a Nobel prize for his contributions to cosmology in 2019.
- 21.
Weinberg [2].
- 22.
Fixsen [3].
- 23.
The CMBR was described in more detail in Sect. 7.8.
- 24.
Crommie et al. [4].
- 25.
This discussion is non-relativistic so all mass is essentially rest mass.
- 26.
We have labeled \(v_w\) with a subscript w to denote that it is a velocity associated with the quantum wave and distinct from the classical velocity v.
- 27.
Tonomura et al. [5].
- 28.
Rauch and Werner [6].
- 29.
Overhauser and Colella [7].
- 30.
The mathematical and physical meaning of these uncertainties will be discussed in Sect. 9.5.2.
References
A. Einstein, Physik 17, 132 (1905)
S. Weinberg, Cosmology (Oxford University Press, 2008), p. 105
D.J. Fixsen, The temperature of the cosmic microwave background. Astrophys. J. 707, 916–920 (2009)
M.F. Crommie, C.P. Lutz, D.M. Eigler, Science 262(5131), 218–220 (1993)
A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, H. Ezawa, Am. J. Phys. 57, 117–120 (1989)
H. Rauch, S.A. Werner, Neutron Interferometry, 2nd edn. (Oxford, 2015)
A.W. Overhauser, R. Colella, Phys. Rev. Lett. 33, 1237 (1974)
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Kunstatter, G., Das, S. (2022). Introduction to the Quantum. In: A First Course on Symmetry, Special Relativity and Quantum Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-92346-4_8
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