Abstract
By the end of the 19th century it had been established that the electromagnetic (EM) radiation within a cavity at thermal equilibrium (viewed from a hole on one of its walls) is the same for all cavities.
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Notes
- 1.
A detailed exposition of Planck’s theory is given in his book with M. Masius, The Theory of Heat Radiation, McGraw-Hill Book Company, Inc., N.Y., 1914.
- 2.
One can show that, when the linear harmonic oscillator described by Eq. (6.6) is acted upon by a periodic force \( F_{0} cos(\omega t) \), it will execute a periodic motion \( X\left( t \right) = X_{0} cos(\omega t) \) with the frequency ω of the external force. To show this one must add the above periodic force (divided by m) to the right of Eq. (6.6), and then go on to show that \( X\left( t \right) \), with an appropriate value for \( X_{0} \), satisfies the modified equation. In the present case the thing to note is that, the electrically charged one-dimensional harmonic oscillators postulated by Planck, in thermal equilibrium with the EM field in the cavity, will oscillate at the frequencies of the EM eigenmodes of this cavity.
- 3.
I have included a ground-state energy, \( \hbar \omega_{q} /2 \), corresponding to n = 0, to facilitate comparison with results to be presented in later sections. It is of no importance in the present discussion.
- 4.
The reality of the photon as a particle with definite momentum \( \hbar \varvec{q} \) and definite energy \( \hbar cq \), was demonstrated by the American physicist A. Compton, who in 1923 observed the conservation of momentum in the scattering of photons by electrons (see Sect. 15.2.2 ).
- 5.
One can argue as follows: A classical atom, consisting of negatively charged electrons moving about a positive nucleus, is similar to an oscillating electric dipole, and as such it emits continuously radiation (see text about Eq. 10.57). In the present case the process changes to one where the atom emits a quantum of energy \( \hbar \omega \) in an abrupt transition which occurs with some probability per unit time.
- 6.
The first working lasers were built by American and, independently, by Russian scientists in the early 1960s.
- 7.
The paper appeared in volume 26 of the Philosophical Magazine, 1913.
- 8.
Here, for the sake of simplicity, I derive Bohr’s formula assuming that the centre of mass of the atom concides with the centre of the nucleus which is reasonable (the nucleus is about 2,000 times heavier than the electron). The exact formula for the energy levels is obtained by replacing the electronic mass m with the reduced mass \( m_{r} = \, Mm/\left( {M + m} \right) \), where M is the mass of the proton, in the final formula (see Sect. 14.1.2 ). It is also worth remembering that the energy we are interested in here, is the internal energy of the atom: its energy in the centre-of-mass system of coordinates (the origin of coordinates is at the centre of mass of the atom). In any other system of coordinates the (total) energy of a free atom (one not acted upon by external forces) is the sum of its internal energy and its translational (kinetic) energy, \( MV^{2} /2 \), where M is the total mass of the atom and V the velocity of its centre of mass.
- 9.
Z. Physik, 8, 110 (1922); 9,349 (1922).
- 10.
An electron orbiting around the nucleus corresponds to a circular current and as such gives rise to a magnetic moment.
- 11.
Natuwiss, 13, 953 (1925); Nature, 117, 264 (1926).
- 12.
Z. Physik, 43, 601 (1927).
- 13.
Thomson and Davisson were awarded the 1937 Nobel Prize ‘for their experimental discovery of the diffraction of electrons by crystals’.
- 14.
We have assumed that the force acting on the particle can be written as the gradient of a potential energy field \( U(\varvec{r}) \) as described in Sect. 7.3. We note that a Hamiltonian operator can also be defined when the force acting on the particle is the Lorentz force (Eq. 10.58), which involves the magnetic field and can not be written as the gradient of a potential energy field. Schrödinger’s equation remains valid, but the Hamiltonian operator takes a slightly different form.
- 15.
The story appears in Jeremy Bernstein’s book, Einstein, published by Fontana (U.K.), 1973; p. 155.
- 16.
A more systematic (axiomatic) presentation of quantum mechanics (according to the Copenhagen School) will be found in: R. B. Leighton, Principles of Modern Physics, McGraw-Hill Book Company, Inc., N.Y., 1959. Other (tentative) interpretations of quantum mechanics, which accept its mathematical predictions but attempt to propose a physical reality behind the mathematics, nearer to our direct experience of the physical world, have been proposed, but none has been accepted by the scientific community. A most interesting book on this subject has been written by David Bohm: Wholeness and the Implicate Order, Routlege Classics, 2002. N.Y.
- 17.
For the sake of simplicity we give the formulae for motion along the x-direction. The extension to three dimensions is straightforward.
- 18.
- 19.
An alternative way of treating the interaction of an atom with light is presented in Sect. 14.3.3 . In both treatments the interaction of the atom with light is treated by first order perturbation theory.
- 20.
Other examples of seemingly classical trajectories are provided by the tracks of subnuclear particles observed in a Wilson chamber (see, e.g., Fig. 15.2 ). Exact measurements of the positions of the particles would result in points very near but randomly distributed about the shown trajectories.
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Modinos, A. (2014). Quantum Mechanics. In: From Aristotle to Schrödinger. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00750-2_13
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