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The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics. (English) Zbl 1357.81080
Summary: The infinite square well and the attractive Dirac delta function potentials are arguably two of the most widely used models of one-dimensional bound-state systems in quantum mechanics. These models frequently appear in the research literature and are staples in the teaching of quantum theory on all levels. We review the history, mathematical properties, and visualization of these models, their many variations, and their applications to physical systems.

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81-03 History of quantum theory
01A60 History of mathematics in the 20th century
Full Text: DOI
[1] Schrödinger, E., Quantisierung als eigenwertproblem, Ann. Phys., 79, 361-376, (1926) · JFM 52.0965.08
[2] Schrödinger, E., An undulatory theory of the mechanics of atoms and molecules, Phys. Rev., 28, 1049-1070, (1926) · JFM 52.0965.07
[3] de Broglie, L., Recherches sur la théorie des quanta (researches on the quantum theory), Ann de Physique, 3, 22, (1925) · JFM 51.0729.03
[4] Mott, N. F., An outline of wave mechanics, (1930), Cambridge University Press · JFM 56.1289.04
[5] Gamow, G., Thirty years that shook physics: the story of quantum theory, (1966), Doubleday
[6] Fourier, J., Théorie analytique de la chaleur, (1822) · JFM 15.0954.01
[7] Cauchy, A. L., Théorie de la propagation des ondes â la surface dun fluide pesant dune profondeur indéfinie, Mem. Acad. Roy. Sci. Inst. France, Sci. Math. et Phys., 1, 4-318, (1827), written in 1815, published 1827
[8] Lützen, J., The prehistory of the theory of distributions, (1982), Springer
[9] Jammer, M., The conceptual development of quantum mechanics, (1966), McGraw Hill
[10] Kirchhoff, G., Zur theorie der lichtwellen, Berliner Berichte, 641-669, (1882) · JFM 14.0829.02
[11] Heaviside, O., On operators in physical mathematics, Proc. R. Soc. Lond., 52, 504-529, (1893) · JFM 25.1536.01
[12] O. Heaviside, Electromagnetic Theory, Volume II, London, 1899.
[13] Dirac, P. A.M., The physical interpretation of the quantum dynamics, Proc. R. Soc., A, 113, 621-641, (1926) · JFM 53.0846.01
[14] Dirac, P. A.M., The principles of quantum mechanics, (1930), Oxford · JFM 56.0745.05
[15] L. Schwartz, Théorie des Distributions, Vol. I, Paris, 1950; Théorie des Distributions, Vol. II, Paris, 1951.
[16] Griffiths, D. J., Introduction to quantum mechanics, 24-25, (1995), Prentice Hall, Note: the quote is altered slightly in the second edition of the text
[17] Gould, H.; Tobochnick, J.; Christian, W., Introduction to computer simulation methods, (2006), Addison Wesley
[18] Belloni, M.; Christian, W., Time development in quantum mechanics using a reduced Hilbert space approach, Amer. J. Phys., 76, 385-392, (2008)
[19] Dowling, J. P.; Gea-Banacloche, J., Evanescent light-wave atom mirrors, resonators, waveguides, and traps, Adv. At. Mol. Opt. Phys., 37, 1-94, (1996)
[20] Aminoff, C. G.; Steane, A. M.; Bouyer, P.; Desbiolles, P.; Dalibard, J.; Cohen-Tannoudji, C., Cesium atoms bouncing in a stable gravitational cavity, Phys. Rev. Lett., 71, 3083-3086, (1993)
[21] Roach, T. M.; Abele, H.; Boshier, M. G.; Grossman, H. L.; Zetie, K. P.; Hinds, E. A., Realization of a magnetic-mirror for cold atoms, Phys. Rev. Lett., 75, 629-632, (1995)
[22] Landragin, A.; Courtois, J. Y.; Labeyrie, G.; Vansteenkiste, N.; Westbrook, C. I.; Aspect, A., Measurement of the van der Waals force in an atomic mirror, Phys. Rev. Lett., 77, 1464-1467, (1996)
[23] Szriftgiser, P.; Guëry-Odelin, D.; Arndt, M.; Dalibard, J., Atomic wave diffraction and interference using temporal slits, Phys. Rev. Lett., 77, 4-7, (1996)
[24] Dodonov, V. V.; Andreata, M. A.; Andreata, M. A.; Dodonov, V. V., The reflection of narrow slow quantum packets from mirrors, Phys. Lett., Laser Phys., J. Phys. A. Math. Gen., 35, 8373-8392, (2002) · Zbl 1048.81533
[25] Bonvalet, A.; Nagle, J.; Berger, V.; Migus, A.; Martin, J.-L.; Joffre, M., Femtosecond infrared emission resulting from coherent charge oscillations in quantum wells, Phys. Rev. Lett., 76, 4392-4395, (1996)
[26] Nesvizhevsky, V. V., Quantum states of neutrons in the earth’s gravitational field’, Nature, 415, 297-299, (2002)
[27] Nesvizhevsky, V. V., Measurement of quantum states of neutrons in the earth’s gravitational field, Phys. Rev. D, 67, 1-9, (2003)
[28] Hannson, J.; Olevik, D.; Türk, C.; Wiklund, H., Comment on ‘measurement of quantum states of neutrons in the earth’s gravitational field, Phys. Rev. D, 68, 108701, (2003), See also V.V. Nesvizhevsky, et al., Reply to Comment on ‘Measurement of quantum states of neutrons in the Earth’s gravitational field’, Phys. Rev. D 68 (2003) 108702
[29] Nesvizhevsky, V. V., Search for quantum states of the neutron in a gravitational field: gravitational levels, Nucl. Instrum. Methods Phys. Res. A, 440, 754-759, (2000)
[30] Nesvizhevsky, V. V., Study of the neutron quantum states in the gravity field, Eur. Phys. J., C40, 479-491, (2005)
[31] Nesvizhevsky, V. V.; Protasov, K. V., Constraints on non-Newtonian gravity from the experiment on neutron quantum states in the earth’s gravitational field, Classical Quantum Gravity, 21, 4557-4566, (2004) · Zbl 1060.83525
[32] Nesvizhevsky, V. V.; Protasov, K. V., Quantum states of neutrons in the earth’s gravitational field: state of the art, applications, and perspectives, (Moore, D. C., Trends in Quantum Gravity Research, (2006), Nova Science Publishers), 65-107
[33] Voronin, A. Y.; Abele, H.; Baeßler, S.; Nesvizhevsky, V. V.; Petukhov, A. K.; Protasov, K. V.; Westphal, A., Quantum motion of a neutron in a waveguide in the gravitational field, Phys. Rev. D, 73, 044029, (2006)
[34] Miller, D. A.B., Band-edge electroabsorption in quantum well structures: the quantum-confined Stark effect, Phys. Rev. Lett., 53, 2173-2176, (1984)
[35] Frost, A. A., Delta-function model. I. electronic energies of hydrogen-like atoms and diatomic molecules, J. Chem. Phys., 25, 1150-1153, (1956)
[36] Frost, A. A., Delta potential function model for electronic energies in molecules, J. Chem. Phys., 22, 1613, (1954)
[37] Frost, A. A.; Leland, F. E., Delta-potential model. II. aromatic hydrocarbons, J. Chem. Phys., 25, 1154-1160, (1956)
[38] Kronig, R.de L.; Penney, W. G., Quantum mechanics of electrons in crystal lattices, Proc. R. Soc. Lond. Ser. A, 130, 499-513, (1931) · Zbl 0001.10601
[39] Kuhn, H., A quantum-mechanical theory of light absorption of organic dyes and similar compounds, Helv. Chim. Acta, J. Chem. Phys., J. Chem. Phys., 17, 1198-1212, (1949)
[40] Belloni, M.; Robinett, R. W., Less than perfect quantum wavefunctions in momentum-space: how \(\phi(p)\) senses disturbances in the force, Amer. J. Phys., 79, 94-102, (2011)
[41] Ruckle, L. J.; Belloni, M.; Robinett, R. W., The biharmonic oscillator and asymmetric potentials: from classical trajectories to momentum-space probability densities in the extreme quantum limit, Eur. J. Phys., 33, 1505-1525, (2012) · Zbl 1278.81071
[42] Lohmann, B.; Weigold, E.; McCarthy, I. E.; Weigold, E., A real ‘thought’ experiment for the hydrogen atom, Phys. Lett., Amer. J. Phys., 51, 152-155, (1983)
[43] Tan, S., Generalized virial theorem and pressure relation for a strongly correlated Fermi gas, Ann. Phys., Ann. Phys., Ann. Phys., 323, 2987-2990, (2008) · Zbl 1157.82382
[44] Braaten, E.; Platter, L., Exact relations for a strongly interacting Fermi gas from the operator product expansion, Phys. Rev. Lett., 100, 205301, (2008)
[45] Zhang, S.; Leggett, A. J., Universal properties of the ultracold Fermi gas, Phys. Rev. A, 79, 023601, (2009)
[46] Braaten, E., How the tail wags the dog in ultracold atomic gases, Physics, 2, 9, (2009)
[47] Stewart, J. T.; Gaebler, J. P.; Drake, T. E.; Jin, D. S., Verification of universal relations in a strongly interacting Fermi gas, Phys. Rev. Lett., 104, 235301, (2010)
[48] Kuhnle, E. D., Universal behavior of pair correlations in a strongly interacting Fermi gas, Phys. Rev. Lett., 105, 070402, (2010)
[49] Wild, R. J.; Makotyn, P.; Pino, J. M.; Cornell, E. A.; Jin, D. S., Measurements of tan’s contact in an atomic Bose-Einstein condensate, Phys. Rev. Lett., 108, 145305, (2012)
[50] Styer, D. F., The motion of wave packets through their expectation values and uncertainties, Amer. J. Phys., 48, 1035-1037, (1990)
[51] Kennard, E. H., Zur quantenmechanik einfacher bewegungstypen, J. Franklin Inst., Z. Phys., 44, 326-352, (1927), (translated as The quantum mechanics of simple types of motion) · JFM 53.0853.02
[52] Darwin, C. G., Free motion in the wave mechanics, Proc. R. Soc. Lond., A117, 258-293, (1928) · JFM 53.0844.03
[53] Andrews, M., The evolution of free wave packets, Amer. J. Phys., 76, 1102-1107, (2008)
[54] Levy-Leblond, J.-M., Correlation of quantum properties and the generalized Heisenberg inequality, Amer. J. Phys., 54, 135-136, (1986)
[55] Robinett, R. W.; Doncheski, M. A.; Bassett, L. C., Simple forms of position-momentum correlated Gaussian free-particle wave packets in one dimension with the general form of the time-dependent spread in position, Found. Phys. Lett., 18, 455-475, (2005) · Zbl 1085.81019
[56] Hiller, J. R.; Johnston, I. D.; Styer, D. F., Quantum mechanics simulations: the CUPS software, (1995), Wiley New York
[57] Robinett, R. W.; Bassett, L. C., Analytic results for Gaussian wave packets in four model systems: I. visualization of the kinetic energy, Found. Phys. Lett., 17, 607-625, (2004) · Zbl 1063.81045
[58] Hagley, E. W., Measurement of the coherence of a Bose-Einstein condensate, Phys. Rev. Lett., 83, 3112-3115, (1999)
[59] Wigner, E., On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40, 749-759, (1932) · JFM 58.0948.07
[60] Tatarskii˘, V. I., The Wigner representation of quantum mechanics, Sov. Phys. Usp., 26, 311-327, (1983)
[61] Carruthers, P.; Zachariasen, F., Quantum collision theory with phase-space distributions, Rev. Modern Phys., 55, 245-285, (1983)
[62] Bertrand, J.; Bertrand, P., A tomographic approach to wigner’s function, Found. Phys., 17, 397-405, (1987)
[63] Curtright, T.; Fairlie, D.; Zachos, C.; Zachos, C., Deformation quantization: quantum mechanics lives and works in phase-space, Phys. Ref., Internat. J. Modern Phys., A117, 297-316, (2002) · Zbl 1040.81050
[64] Balaczs, N. L.; Jennings, B. K., Wigner’s function and other distribution functions in mock phase space, Phys. Rep., 105, 347-391, (1984)
[65] Hillery, M.; O’Connell, R. F.; Scully, M. O.; Wigner, E. P., Distribution functions in physics: fundamentals, Phys. Rep., 106, 121-167, (1984)
[66] Lee, H.-W., Theory and application of the quantum phase-space distribution functions, Phys. Rep., 259, 147-211, (1995)
[67] Ozorio de Almeida, A. M., The Weyl representation in classical and quantum mechanics, Phys. Rep., 296, 265-342, (1998)
[68] Snygg, J., Wave functions rotated in phase space, Amer. J. Phys., 45, 58-60, (1977)
[69] Stenholm, S., The Wigner function: I. the physical interpretation, Eur. J. Phys., 1, 244-248, (1980)
[70] Kim, Y. S.; Wigner, E. P., Canonical transformations in quantum mechanics, Amer. J. Phys., 58, 439-448, (1990)
[71] Casas, M.; Krivine, H.; Martorell, J., On the Wigner transforms of some simple systems and their semiclassical interpretations, Eur. J. Phys., 12, 105-111, (1991)
[72] Kim, Y. S.; Noz, M. E., (Phase Space Picture of Quantum Mechanics: Group Theoretical Approach, Lecture Notes in Physics Series, vol. 40, (1990), World Scientific Singapore)
[73] Redner, S., Citation statistics from 110 years of physical review, Phys. Today, 58, June, 49-54, (2005)
[74] Hudson, R. L., When is the Wigner quasi-probability density non-negative?, Rep. Math. Phys., 6, 249-252, (1974) · Zbl 0324.60018
[75] Soto, F.; Claverie, P., When is the Wigner function of multi-dimensional systems nonnegative?, J. Math. Phys., 24, 97-100, (1983)
[76] Ford, G. W.; O’Connell, R. F., Wave packet spreading: temperature and squeezing effects with applications to quantum measurement and decoherence, Amer. J. Phys., 70, 319-324, (2002)
[77] Dodonov, V. V.; Andreata, M. A., Shrinking quantum packets in one dimension, Phys. Lett. A, 310, 101-109, (2003) · Zbl 1011.81502
[78] Meekhof, D. M.; Monroe, C.; King, B. E.; Itano, W. M.; Wineland, D. J.; Meekhof, D. M.; Monroe, C.; King, B. E.; Itano, W. M.; Wineland, D. J., Generation of nonclassical motion states of a trapped atom, Phys. Rev. Lett., Phys. Rev. Lett., 77, 2346-1799, (1996), (erratum) · Zbl 1226.81011
[79] Heinzen, D. J.; Wineland, D. J., Quantum-limited cooling and detection of radio-frequency oscillation by laser-cooled atoms, Phys. Rev., A42, 2977-2994, (1990)
[80] Robinett, R. W., Quantum mechanics: classical results, modern systems, and visualized examples, (2006), Oxford University Press Oxford
[81] Averbukh, I. Sh.; Perelman, N. F., The dynamics of wave packets of highly-excited atoms and molecules, Phys. Lett., Acta Phys. Polon., Zh. Eksp. Teor. Fiz., Usp. Fiz. Nauk, 161, 41-81, (1991), [Sov. Phys. Usp. 34 (1991) 572-591]
[82] Robinett, R. W., Quantum wave packet revivals, Phys. Rep., 392, 1-119, (2004)
[83] Ballentine, L. E., Quantum mechanics: A modern development, (1998), World Scientific Singapore · Zbl 0997.81501
[84] Schleich, W. P., Quantum optics in phase space, (2001), Wiley-VCH Berlin · Zbl 0961.81136
[85] Andrews, M. R.; Townsend, C. G.; Miesner, H.-J.; Durfee, D. S.; Kurn, D. M.; Ketterle, W.; Durfee, D. S.; Ketterle, W., Experimental studies of Bose-Einstein condensates, Science, Optics Express, 2, 299-313, (1998)
[86] Wallis, H.; Röhrl, A.; Naraschewski, M.; Schenzle, A., Phase-space dynamics of Bose condensates: interference versus interaction, Phys. Rev. A, 55, 2109-2199, (1997)
[87] Hadzibabic, Z.; Stock, S.; Battelier, B.; Bretin, V.; Dalibard, J., Interference of an array of independent Bose-Einstein condensates, Phys. Rev. Lett., 93, 180403, (2004)
[88] Anderson, P. W., Measurement in quantum theory and the problem of complex systems, (Boer, J. D.; Dal, E.; Ulfbeck, O., The Lesson of Quantum Theory, Proceedings of the Niels Bohr Centenary Symposium, Copenhagen, Denmark, 1985, (1986), Elsevier Amsterdam), 23
[89] Pitaevskii, L.; Stringari, S., Interference of Bose-Einstein condensates in momentum space, Phys. Rev. Lett., 83, 4237-4240, (1999)
[90] Andrews, M., Wave packets bouncing off walls, Amer. J. Phys., 66, 252-254, (1998)
[91] Barton, G.; Bray, A. J.; McKrane, A. J.; Berman, D., Boundary effects in quantum mechanics, Amer. J. Phys., Amer. J. Phys., 59, 937-941, (1991)
[92] Doncheski, M. A.; Robinett, R. W., Anatomy of a quantum ‘bounce’, Eur. J. Phys., 20, 29-37, (1999)
[93] Belloni, M.; Doncheski, M. A.; Robinett, R. W., Exact results for ‘bouncing’ Gaussian wave packets, Phys. Scr., 71, 136-140, (2005) · Zbl 1061.81020
[94] Robinett, R. W., Self-interference of a single Bose-Einstein condensate due to boundary effects, Phys. Scr., 73, 681-684, (2006)
[95] Robinett, R. W., Image method solutions for free-particle wave packets in wedge geometries, Eur. J. Phys., 27, 281-289, (2006)
[96] Kleber, M., Exact solutions for time-dependent phenomena in quantum mechanics, Phys. Rep., 236, 331-393, (1994)
[97] Born, M.; Born, M.; Ludwig, W., Zur quantenmechanik der kräftefreien teilchens, Kgl. Danske Videns. Sels. Mat.-fys. Medd., Z. Phys., 150, 2, 106-117, (1958), Born was addressing concerns made by A. Einstein, Elementare Überlegungen zur Interpretation der Grundlagen der Quanten-Mechanik, in Edinburgh, Oliver and Boyd, Scientific Papers Presented to Max Born, 1953 · Zbl 0107.44006
[98] Mathews, J.; Walker, R. L., Mathematical methods of physics, (1970), W.A. Benjamin Menlo Park
[99] Jung, C., An exactly soluble three-body problem in one-dimension, Can. J. Phys., 58, 719-728, (1980) · Zbl 0982.81522
[100] Richens, P. J.; Berry, M. V., Pseudointegrable systems in classical and quantum mechanics, Physica, 2D, 495-512, (1981) · Zbl 1194.37150
[101] Li, W.-K.; Blinder, S. M., Particle in an equilateral triangle: exact solution of a nonseparable problem, J. Chem. Educ., 64, 130-132, (1987)
[102] Doncheski, M. A.; Robinett, R. W., Quantum mechanical analysis of the equilateral triangle billiard: periodic orbit theory and wave packet revivals, Ann. Phys. (New York), 299, 208-227, (2002) · Zbl 1005.81028
[103] Robinett, R. W., Visualizing the solutions of the circular infinite well in classical and quantum mechanics, Amer. J. Phys., 64, 440-446, (1996)
[104] See the GRE Physics Test (FORM GR9677) offered in 1996 reprinted in: GRE: Practicing to Take the Physics Test, third ed., Princeton: Educational Testing Service, 1997.
[105] Condon, E. U.; Morse, P. M., Quantum mechanics, (1929), McGraw-Hill New York · JFM 55.1172.09
[106] Pauling, L.; Wilson, E. B., Introduction to quantum mechanics: with applications to chemistry, (1935), McGraw-Hill New York
[107] Schleich, W.; Walls, D. F.; Wheeler, J. A., Area of overlap and interference in phase space versus Wigner pseudoprobabilities, Phys. Rev. A, 38, 1177-1186, (1988)
[108] Kenfack, A.; Zyczkowski, K., Negativity of the Wigner function as an indicator of non-classicity, J. Opt. B: Quantum Semiclass. Opt, 6, 396-404, (2004)
[109] Schrödinger, E., Der stetige übergang von der mikro- zur makromechanik, Naturwissenschaften, 14, 664-666, (1926), translated and reprinted as, The continuous transition from micro- to macro mechanics, Collected Papers on Wave Mechanics, Chelsea Publishing, New York, 1982, pp. 41-44 · JFM 52.0967.01
[110] Rojansky, V., Introductory quantum mechanics, (1938), Prentice-Hall New York · JFM 64.1503.02
[111] Styer, D. F., Quantum revivals versus classical periodicity in the infinite square well, Amer. J. Phys., 69, 56-62, (2001)
[112] For a review of revival phenomena, see Ref. [82].
[113] Sukhatme, U. P., WKB energy levels for a class of one-dimensional potentials, Amer. J. Phys., 41, 1015-1016, (1973)
[114] Nieto, M. M.; Simmons, L. M., Limiting spectra from confining potentials, Amer. J. Phys., 47, 634-635, (1979)
[115] Liboff, R. L., On the potential \(x^{2 N}\) and the correspondence principle, Internat. J. Theoret. Phys., 18, 185-191, (1979)
[116] Robinett, R. W., Wave packet revivals and quasirevivals in one-dimensional power law potentials, J. Math. Phys., 41, 1801-1813, (2000) · Zbl 0977.81017
[117] Morrison, M. A., Understanding quantum physics: A user’s manual, (1990), Prentice Hall
[118] Bowen, M.; Coster, J., Infinite square well: a common mistake, Amer. J. Phys., 49, 80-81, (1980)
[119] Sapp, R. C., Ground state of the particle in a box, Amer. J. Phys., 50, 1152-1153, (1982)
[120] Yinji, L.; Xianhuai, H., A particle ground state in the infinite square well, Amer. J. Phys., 54, 738, (1986)
[121] Seki, R., On boundary conditions for an infinite square well potential in quantum mechanics, Amer. J. Phys., 39, 929-931, (1971)
[122] Cummings, E. F., The particle in a box is not simple, Amer. J. Phys., 45, 158-159, (1977)
[123] Branson, D., Continuity conditions on the Schrödinger wave function at discontinuities of the potential, Amer. J. Phys., 47, 1000-1003, (1979)
[124] Andrews, M., Matching conditions on wave functions at discontinuities of the potential, Amer. J. Phys., 49, 281-282, (1981)
[125] Home, D.; Sengupta, S., Discontinuity in the first derivative of the Schrödinger wave function, Amer. J. Phys., 50, 552-554, (1982)
[126] Dirac, P. A.M., The principles of quantum mechanics, 58-61, (1958), Oxford University Press Oxford England, (Sec. 15)
[127] Markley, F. L., Probability distribution of momenta in an infinite square well potential, Amer. J. Phys., 40, 1545-1546, (1972)
[128] Fulling, S. A., Comment on ‘probability distribution of momenta in an infinite square well potential’, Amer. J. Phys., 41, 1374-1375, (1973)
[129] Lee, H.-W., Theory and application of the quantum phase-space distribution functions, Phys. Rep., 259, 147-211, (1995)
[130] Bonneau, G.; Faraut, J.; Valent, G., Self-adjoint extensions of operators and the teaching of quantum mechanics, Amer. J. Phys., 69, 322-331, (2001)
[131] Araujo, V. S.; Coutinho, F. A.B; Perez, J. F., Operator domains of self-adjoint operators, Amer. J. Phys., 72, 203-213, (2004) · Zbl 1219.81123
[132] Garbaczewski, P.; Karwowski, W., Impenetrable barriers and canonical quantization, Amer. J. Phys., 72, 924-933, (2004)
[133] Rokhsar, D. S., Ehrenfest’s theorem and the particle-in-a-box, Amer. J. Phys., 64, 1416-1418, (1996)
[134] Belloni, M.; Doncheski, M. A.; Robinett, R. W., Wigner quasi-probability distribution for the infinite square well: energy eigenstates and time-dependent wave packets, Amer. J. Phys., 72, 1183-1192, (2004)
[135] Ozorio de Almeida, A. M., The Weyl representation in classical and quantum mechanics, Phys. Rep., 296, 265-342, (1998)
[136] Benedict, M. G.; Czirjak, A., Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms, Phys. Rev. A, 60, 4034-4044, (1999)
[137] Dean, C., Simple Schrödinger wave functions which simulate classical radiating systems, Amer. J. Phys., 27, 161-163, (1959)
[138] Waldenstrom, S.; Razi Naqvi, K.; Mork, K. J., The force exerted by the walls of an infinite square well on a wave packet: Ehrenfest theorem, revivals and fractional revivals, Phys. Scr., 68, 45-53, (2003) · Zbl 1055.81528
[139] Tannor, D. J., Introduction to quantum mechanics: A time-dependent perspective, (2007), University Science Books, Chapter 11
[140] Bluhm, R.; Kostelecký, V. A.; Porter, J., The evolution and revival structure of localized quantum wave packets, Am. J. Phys., 64, 944-953, (1996), See for example
[141] Aronstein, D. L.; Stroud, C. R., Fractional wave-function revivals in the infinite square well, Phys. Rev. A, 55, 4526-4537, (1997)
[142] Schrödinger, E., Further studies on solving eigenvalue problems by factorization, Proc. R. Ir. Acad., 46A, 183-206, (1941) · Zbl 0063.06818
[143] Infeld, L.; Hull, T. E., The factorization method, Rev. Modern Phys., 23, 21-68, (1951) · Zbl 0043.38602
[144] Witten, E., Dynamical aspects of supersymmetric quantum mechanics, Nuclear Phys., B188, 513-554, (1981) · Zbl 1258.81046
[145] Cooper, F.; Freedman, B., Aspects of supersymmetric quantum mechanics, Ann. Phys. (NY), 146, 262-288, (1983)
[146] Gendenshtein, L. E.; Krive, I. V., Supersymmetry in quantum mechanics, Sov. Phys. Usp, 28, 645-666, (1985)
[147] Cooper, F.; Khare, A.; Sukhatme, U., Supersymmetry and quantum mechanics, Phys. Rep., Supersymmetry in Quantum Mechanics, 251, 267-385, (2001), World Scientific Singapore
[148] Pöschl, G.; Teller, E., Bemerkungen zur quantenmechanik des anharmonischen oszillators, Z. Phys., 83, 143-151, (1933) · JFM 59.1555.02
[149] French, A. P.; Taylor, E. F., (Introduction to Quantum Physics, The M.I.T. Introductory Physics Series, (1978), W.W. Norton and Company)
[150] Doncheski, M. A.; Robinett, R. W., Comparing classical and quantum probability distributions for an asymmetric well, Eur. J. Phys., 21, 217-228, (2000)
[151] Gilbert, L. P.; Belloni, M.; Doncheski, M. A.; Robinett, R. W., Piecewise zero-curvature energy eigenfunctions in one dimension, Eur. J. Phys., 27, 1331-1339, (2006) · Zbl 1118.81499
[152] Gilbert, L. P.; Belloni, M.; Doncheski, M. A.; Robinett, R. W., More on the asymmetric infinite square well: energy eigenstates with zero-curvature, Eur. J. Phys., 26, 815-825, (2005)
[153] Timberlake, T. K.; Camp, S., Decay of wave packet revivals in the asymmetric infinite square well, Amer. J. Phys., 79, 607-614, (2011)
[154] Tamm, I., Über eine mögliche art der elektronenbindung an kristalloberflächen, Phys. Z. Sowjetunion, 1, 733-746, (1932), (translated as A possible kind of electron binding on crystal surfaces) · JFM 58.0945.05
[155] Ohno, H., Observation of ‘tamm states’ in super lattices, Phys. Rev. Lett., 64, 2555-2558, (1990)
[156] Seitz, F., The modern rheory of solids, (1940), McGraw-Hill New York · Zbl 0025.13403
[157] Rudenberg, K.; Parr, R. G., A mobile electron model for aromatic molecules, J. Chem. Phys., 19, 1268-1278, (1951), The authors use the \(\delta\)-function notation, but note that they use essentially the Kronig-Penney results
[158] Kuhn, H., Elektronengasmodell zur quantitativen deutung der light-absorptoion von organischen farbsteffen II, Helv. Chim. Acta, 34, 2371-2402, (1951), Kuhn uses a model of a finite well inside a larger box, taking the Kronig-Penney limit, effectively discussing some of the results in Section 6.5.1
[159] Fukui, K.; Nagata, C.; Yonezawa, T., A conbribution to the theory of light absorption of symmetrical polymethine dyes, J. Chem. Phys., 21, 186-187, (1953)
[160] Simpson, W. T., Electronic states of organic molecules, J. Chem. Phys., 16, 1124-1136, (1948)
[161] Platt, J. R., Classification of spectra of cata-condensed hydrocarbons, J. Chem. Phys., 17, 484-495, (1949)
[162] Saxon, D. S.; Hutner, R. A, Some electronic properties of a one-dimensional crystal model, Philips Res. Rep., 4, 81-122, (1949)
[163] Morse, P. M.; Feshbach, H., Methods of theoretical physics, 1644-1645, (1953), McGraw-Hill New York
[164] Brennan, J. G., Limit of a one-dimensional square well, Amer. J. Phys., 29, 45-47, (1961)
[165] Lieber, M., Quantum mechanics in momentum space, Amer. J. Phys., 43, 486-491, (1975)
[166] Brownstein, K. R., Calculation of a bound state wavefunction using free state wavefunctions only, Amer. J. Phys., 43, 173-176, (1974)
[167] Damert, W. C., Completeness of the energy eigenstates for a delta function potential, Amer. J. Phys., 43, 531-534, (1974)
[168] Patil, S. H., Completeness of the energy eigenfunctions for the one-dimensional \(\delta\)-function potential, Amer. J. Phys., 68, 712-714, (2000)
[169] Mukunda, N., Completeness of the Coulomb wave functions in quantum mechanics, Amer. J. Phys., 46, 910-913, (1978)
[170] Kiang, D., Contribution of the continuum in perturbation theory, Amer. J. Phys., 45, 308-309, (1977)
[171] Doncheski, M. A.; Robinett, R. W., Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum, Ann. Phys., 308, 578-598, (2003) · Zbl 1037.81027
[172] Condon, E. U., The physical pendulum in quantum mechanics, Phys. Rev., 31, 891-894, (1928) · JFM 54.0978.06
[173] Baker, G. L.; Blackburn, J. A., The pendulum: A case study in physics, (2005), Oxford University Press Oxford, Chapter 8 · Zbl 1088.70001
[174] Hirschfelder, J. O.; Brown, W. B.; Epstein, S. T., Advances in Quantum Chemistry, vol. 1, 256-268, (1964), Academic Press New York, See, e.g.,
[175] Aslangul, C., \(\delta\)-well with a reflecting barrier, Amer. J. Phys., 63, 935-940, (1995)
[176] Lapidus, I. R., One-dimensional hydrogen atom and hydrogen molecule ion in momentum space, Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., 51, 663-665, (1983)
[177] Nogami, Y.; Vallieres, M.; van Dijk, W.; Nielsen, L. D.; Urumov, V.; Moszkowski, S. A.; Stobel, G.; Crandall, R.; Whitnell, R.; Bettega, R.; Martinez, J. C.; Gangopadhyay, G.; Dutta-Roy, B., The Born-Oppenheimer approximation: a toy version, Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., Amer. J. Phys., 72, 389-392, (2004)
[178] Atkinson, D. A.; Crater, H. W., An exact treatment of the Dirac delta function potential in the Schrödinger equation, Amer. J. Phys., 43, 301-304, (1975)
[179] Rellich, F., Storungstheorie der spektralzerlegung. V, Math. Ann., 118, 462-484, (1942), Rellich considers the eigenvalue problem \(u'' + \lambda u\) with the boundary conditions \(u(0) = 0\) and \(\epsilon u^\prime(1) + u(1) = 0\) which is formally equivalent to the more physical problem discussed here. He notes the correct eigenvalue condition in Eq. (362) and focuses on perturbation theory results · JFM 68.0243.02
[180] Vugalter, G. A.; Das, A. K.; Sorokin, V. A., Revivals in an infinite square well in the presence of a \(\delta\) well, Phys. Rev. A, 66, 012104, (2002)
[181] Epstein, S. T., Application of the Rayleigh-Schrödinger perturbation theory to the delta-function potential, Amer. J. Phys., 28, 495-496, (1960) · Zbl 0092.45302
[182] Oseguera, U., Effect of infinite discontinuities on the motion of a particle in one dimension, Eur. J. Phys., 11, 35-38, (1990)
[183] Gilbert, L. P.; Belloni, M.; Doncheski, M. A.; Robinett, R. W., Playing quantum physics jeopardy with zero-energy eigenstates, Amer. J. Phys., 74, 1035-1036, (2006)
[184] Lapidus, I. R., Particle in a square well with a \(\delta\)-function perturbation, Amer. J. Phys., 55, 172-174, (1987)
[185] Gradshteyn, I. S.; Ryzhik, I. M., Table of integrals, series, and products, 36, (1980), Academic Press New York · Zbl 0521.33001
[186] Boya, L. J., Supersymmetric quantum mechanics: two simple examples, Eur. J. Phys., 9, 139-144, (1988)
[187] Goldstein, J.; Lebiedzik, C.; Robinett, R. W., Supersymmetric quantum mechanics: examples with Dirac \(\delta\) functions, Amer. J. Phys., 62, 612-618, (1994)
[188] de Broglie, L., Einführung in die wellenmechanik, (1929), Akad. Verlag Leipzig · JFM 55.1172.10
[189] Breit, G., The propagation of schroedinger waves in a uniform field of force, Phys. Rev., 32, 273-276, (1928) · JFM 54.0967.03
[190] Mott, N. F.; Sneedon, I. N., Wave mechanics and its applications, (1948), Claredon Oxford · Zbl 0032.32601
[191] Delbourgo, R., On the linear potential Hill, Amer. J. Phys., 45, 1110-1112, (1977)
[192] Branson, D., Correspondence principle and scattering from potential steps, Amer. J. Phys., 47, 1101-1102, (1979)
[193] Landau, L. D.; Lifschitz, E. M., Quantum mechanics: non-relativistic theory, (1977), Pergamon Press Oxford
[194] Winter, R. G., Quantum physics, (1986), Faculty Publishing Davis
[195] Sakurai, J. J., (Tuan, S. F., Modern Quantum Mechanics: Revised Edition, (1994), Addison-Wesley Reading)
[196] Goswami, A., Quantum mechanics, (1992), Wm.C. Brown Dubuque
[197] Mavromatis, H. A., Exercise in quantum mechanics: A collection of illustrative problems and their solutions, (1987), Reidel Dordrecht · Zbl 0606.22012
[198] Langhoff, P. W., Schrödinger particle in a gravitational well, Amer. J. Phys., 39, 954-957, (1971)
[199] Gibbs, R. L., The quantum bouncer, Amer. J. Phys., 43, 25-28, (1975)
[200] Desko, R. D.; Bord, D. J., The quantum bouncer revisited, Amer. J. Phys., 51, 82-84, (1983)
[201] Goodings, D. A.; Szeredi, T., The quantum bouncer by path integral method, Amer. J. Phys., 59, 924-930, (1991)
[202] Whineray, S., An energy representation approach to the quantum bouncer, Amer. J. Phys., 60, 948-950, (1992)
[203] Gea-Banacloche, J., A quantum bouncing ball, Amer. J. Phys., 67, 776-782, (1999)
[204] Vallée, O., Comment on ‘A quantum bouncing ball’, Amer. J. Phys., 68, 672-673, (2000)
[205] Goodmanson, D. M., A recursion relation for matrix elements of the quantum bouncer, Amer. J. Phys., 68, 866-868, (2000)
[206] Doncheski, M. A.; Robinett, R. W., Expectation value analysis of wave packet solutions for the quantum bouncer: short-term classical and long-term revival behaviors, Amer. J. Phys., 69, 1084-1090, (2001)
[207] Vallèe, O.; Soares, M., LES fonctions d’airy pour la physique, (1998)
[208] Albright, J. R., Integrals of products of Airy functions, J. Phys. A., 10, 485-490, (1977) · Zbl 0355.33013
[209] Migdal, A. B.; Krainov, V., Approximation methods in quantum mechanics, 111-144, (1969), Benjamin New York
[210] (Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions, (1965), National Bureau of Standards Washington), 446-452
[211] Dymski, T. C.; Johnson, N.; Churchill, J., Agreement between classical and quantum mechanical solutions for a linear potential inside a one-dimensional infinite potential well, Amer. J. Phys., Amer. J. Phys., 37, 1287-1288, (1969), and comments therein by
[212] Churchill, J. N.; Arntz, F. O.; Johnson, N. M.; Churchill, J. N.; Churchill, J. N.; Holmstrom, F. E., Modeling of a finite tilted quantum well, Amer. J. Phys., Amer. J. Phys., Am. J. Phys., 55, 372-374, (1987), for a variation
[213] Aguilera-Navarro, V. C.; Iwamoto, H.; Ley-Koo, E.; Zimerman, A. H., Quantum bouncer in a closed court, Amer. J. Phys., 49, 648-650, (1981)
[214] Lapidus, I. R., Note on the ‘quantum bouncer in a closed court’, Amer. J. Phys., 51, 84-85, (1983)
[215] Robinett, R. W., Visualizing classical and quantum probability densities for momentum using variations on familiar one-dimensional potentials, Eur. J. Phys., 23, 165-174, (2002)
[216] de la Torre, L.; Gori, F., The bouncing bob: quasi-classical states, Eur. J. Phys., 24, 253-259, (2003) · Zbl 1069.81536
[217] Lekner, J.; Nguyen, H., Quantum bouncer on a spring, Eur. J. Phys., 30, L67-L73, (2009)
[218] Andrews, M., Coherent states for the bouncing pendulum and the paddle ball, Amer. J. Phys., 76, 236-240, (2008)
[219] Ehrenfest, P., Bemerkung uber die angenaherte gultigkeit der klassichen mechanik innerhalb der quantunmechanik, Z. Phys., 45, 455-457, (1927) · JFM 53.0843.01
[220] Tinkham, M.; Ferrell, R. A., Determination of the superconducting skin depth from the energy gap and sum rule, Phys. Rev. Lett., 8, 331-333, (1959)
[221] Bethe, H.; Jackiw, R., Intermediate quantum mechanics, (1968), Benjamin New York, Chapter 11
[222] Jackiw, R., Quantum mechanical sum rules, Phys. Rev., 157, 1220-1225, (1967)
[223] Thomas, W., Über die zahl der dispersionselectronen, die einem starionären zustande zugeordnet sind (vorläufige mitteilung), Naturwissenschaftern, 13, 627, (1925) · JFM 51.0758.14
[224] W. Kuhn, Über die Gesamtstärke der von einem Zustande ausgehenden Absorptionslinien, Z. Phys. 33, 408-412. · JFM 51.0747.05
[225] Reiche, F.; Thomas, W., Über die zahl der dispersionselektronen, die einem stationären zustand zugeordnet sind, Z. Phys., 34, 510-525, (1925) · JFM 51.0748.06
[226] Belloni, M.; Robinett, R. W., Quantum mechanical sum rules for two model systems, Amer. J. Phys., 76, 798-806, (2008)
[227] Bethe, H., Zur theorie des durchgangs schneller korpuskularstrahlen durch materie, Ann. Phys. (Leipzig), 5, 325-400, (1930), Translated as Theory of the passage of fast corpuscular rays through matter, in Selected Works of Hans A. Bethe with commentary, H. Bethe, World Scientific Series in 20th Century Physics, vol. 18, World Scientific, Singapore, 1997, pp. 77-154 · JFM 56.1326.03
[228] Wang, S., Generalization of the Thomas-reiche-Kuhn and the Bethe sum rules, Phys. Rev. A, 60, 262-266, (1999)
[229] Condon, E. U.; Shortley, G. H., The theory of atomic spectra, 397, (1953), Cambridge University Press Cambridge
[230] Borowitz, S., Fundamentals of quantum mechanics, (1967), W.A. Benjamin New York · Zbl 0161.45705
[231] Merzbacher, E., Quantum mechanics, (1970), Wiley New York
[232] Epstein, P. S., The Stark effect from the point of view of schroedinger’s quantum theory, Phys. Rev., 28, 695-710, (1926) · JFM 52.0980.06
[233] Bethe, H. A.; Salpeter, E. E., Quantum mechanics of one- and two-electron atoms, 228-235, (1957), Springer-Verlag Berlin · Zbl 0089.21006
[234] Ruffa, A. R., Continuum wave functions in the calculation of sums involving off-diagonal matrix elements, Amer. J. Phys., 41, 234-241, (1973)
[235] Dalgarno, A.; Lewis, J. T., The exact calculation of long range forces between atoms by perturbation theory, Proc. R. Soc. Lond. Ser. A, 70, 70-74, (1955) · Zbl 0065.44905
[236] Maize, M. A.; Burkholder, C. A., Electric polarizability and the solution of an inhomogeneous differential equation, Amer. J. Phys., 63, 244-247, (1995)
[237] Popov, V. S.; Eletskii, V. L.; Turbiner, A. V., Higher orders of perturbation theory and summation of series in quantum mechanics and field theory, Zh. Eksp. Teor. Fiz., 74, 445-465, (1978), [Sov. Phys. JETP 47 (1978) 232-242]
[238] Fernandez, F. M.; Castro, E. A, Stark effect in a one-dimensional model atom, Amer. J. Phys., 53, 757-760, (1985)
[239] Herbst, I. W.; Simon, B., Stark effect revisited, Phys. Rev. Lett., 41, 67-69, (1978)
[240] Fowler, R. H.; Nordheim, L., Electron emission in intense electric fields, Proc. R. Soc. Lond. Ser. A, 119, 173-181, (1928) · JFM 54.0988.03
[241] Postma, B. J., Polarizability of the one-dimensional hydrogen atom with a \(\delta\)-function interaction, Amer. J. Phys., 52, 725-730, (1984)
[242] Maize, M. A.; Williams, M., The nonrelativistic frequency-dependent electric polarizability of a bound particle, Amer. J. Phys., 72, 691-694, (2004)
[243] Postma, B. J., Photoelectric effect for the one-dimensional hydrogen atom with a \(\delta\)-function interaction, Amer. J. Phys., 53, 357-360, (1985)
[244] Mavromatis, H. A., The dalgarno-Lewis summation technique: some comments and examples, Amer. J. Phys., 59, 738-744, (1991)
[245] Mavromatis, H. A., New summation expressions obtained by combining perturbation theory formalisms, Int. J. Comput. Math., 50, 119-123, (1993) · Zbl 0822.47017
[246] Alexander, M. H., Exact treatment of the Stark effect in atomic hydrogen, Phys. Rev., 178, 3440, (1969)
[247] Jannussis, A. D.; Leodaris, A. D.; Brodimas, G. N., Study of the Stark and Zeeman effects in parabolic coordinates, Phys. Lett. A, 71, 205207, (1979)
[248] Robinett, R. W., The polarizability of a particle in power-law potentials: a WKB analysis, Eur. J. Phys., 19, 31-39, (1998) · Zbl 0905.34053
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