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2.R. W. Robinett, Quantum Mechanics, 2nd ed. (Oxford U. P., Oxford, 2006).
3.D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective (University Science Books, Sausalito, CA, 2007).
4.B. Thaller, Visual Quantum Mechanics (Springer, New York, 2000).
5.A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177–186 (1967).
6.M. H. Bramhall and B. M. Casper, “Reflections on a wave packet approach to quantum mechanical barrier penetration,” Am. J. Phys. 38, 1136–1145 (1970).
7.N. Kiriushcheva and S. Kuzmin, “Scattering of a Gaussian wave packet by a reflectionless potential,” Am. J. Phys. 66, 867–872 (1998).
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12.In traditional optics, birefringence occurs when light propagates through a material that has an index of refraction for the components of light with a given polarization, and a different index of refraction for the perpendicular component. As a consequence, the wave is split into two components.
14.D. Bailer-Jones, Scientific Models in Philosophy of Science (U. Pittsburgh, Pittsburgh, PA, 2009).
15.In optics, the refractive index can be defined equivalently in terms of the ratio of wave numbers or phase velocities: , where is the phase velocity in the medium. Thus vacuum can be seen as the medium with , and typical dielectrics have . In quantum mechanics the dependence on in Eq. (5) indicates that the refractive index is measured relative to the level of zero potential energy. This level should most naturally also correspond to the vacuum, that is, the case of a free particle that can gain no further kinetic energy. We see from Eq. (5) that yields , a situation seemingly unfamiliar from optics. Cases with are encountered in classical optics for X-rays and are also possible for electromagnetic waves propagating in plasmas. See, for example, D. G. Swanson, Plasma Waves (IOP Publishing, Bristol, 2003).
16.M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. P., Cambridge, UK, 1999).
17.As an aside, we point out that the phase and group velocities associated with quantum waves show different trends when crossing between regions of space with different potentials. We define the phase velocity in the usual fashion and find , whereas the group velocity is . Hence, in a case where the particle (group) velocity increases upon transmission, the phase velocity will decrease (and vice versa). This behavior is a consequence of dispersion. More details about quantum phase velocity in situations with a finite external potential are discussed, for example, in L. Bergmann and C. Schaefer, Optics of Waves and Particles (de Gruyter, Berlin, 1999), p. 965,
17.and K. U. Ingard, Fundamentals of Waves and Oscillations (Cambridge U. P., Cambridge, UK, 1988), Sec. 16.5.
18.The interference fringes seen in Fig. 2 are parallel to the time axis, and their only time dependence is a weak modulation due to the propagating envelopes of the incoming and reflected wavepackets. Hence, they can be understood as a purely spatial phenomenon. In contrast, the fringes in Fig. 3 exhibit an oscillation in time for fixed . This effect, which can be attributed to diffraction in time, changes the character of the interference fringes from being spatial in nature to having an additional temporal dimension.
21.For charged particles there are additional, velocity dependent, terms due to the vector potential and the resulting equations depend on the choice of gauge.
22.D. F. Treagust, A. G. Harrison, G. J. Venville, and Z. Dagher, “Using an analogical teaching approach to engender conceptual change,” Int. J. Sci. Educ. 18, 213–229 (1996).
24.The refractive index for classical electromagnetic waves propagating in a waveguide is given by , where is the cut-off frequency for a particular mode. See, for example, J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, Hoboken, NJ, 1999), Sec. 8.3. Specializing the law of refraction in spacetime [Eq. (11)] to this case yields the same expression in terms of refractive indices as is found for quantum waves [Eq. (12b)].
25.J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature (London) 468, 545–548 (2010)
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