Systematic Convergence in Applying Variational Method to Double-Well Potential
Abstract
In this work, we demonstrate the application of the variational method by computing the ground- and first-excited
state energies of a double-well potential. We start with the proper choice of the trial wave functions using optimized
parameters, and notice that accurate expectation values in excellent agreement with the numerical results can be
aquired by cautious systematic improvement of trial wave functions.
References
Bardeen, J., Cooper, L. N. & Schrieffer, J. R. (1957). Theory of superconductivity. Physical Review, 108, 1175-1204.
Bardeen, J., Cooper, L. N. & Schrieffer, J. R. (1957). Microscopic theory of superconductivity Physical Review, 106, 162-164.
Cohen-Tannoudji, C., Diu, B. & . F. (1977). Quantum mechanics, first edition. Vol. 1 and 2, New York: John Wiley. pp 1148-1155.
DeMille, D., (2015). Diatomic molecules, a window onto fundamental physics. Physics Today, December, 34-40.
Feynman, R. P. (1953). Atomic theory of the λ transition in helium. Physical Review, 91, 1301-1308.
Feynman, R. P. (1954). Atomic theory of the two-fluid model of liquid helium. Physical Review, 94, 262-277.
Feynman, R. P. (1955). Slow electrons in a polar crystal. Physical Review. 97, 660-665.
Feynman, R. P. & Cohen, M. (1956). Energy spectrum of the excitations in liquid helium. Physical Review, 102, 1189-1204.
Griffiths, D. J. (2005). Introduction to quantum mechanics, second edition. Upper Saddle River, NJ: Pearson Preston Hall. Chapter 7 and 8 and the references cited therein.
Gutzwiller, M. C. (1965). Quantized Hall conductivity in two dimensions. Physical Review A, 137, 1726-1735.
Hardy, J. R. & Flocken, J. W. (1988). Possible origins of high-Tc, superconductivity. Physical Review Letters, 60, 2190-2193.
Hedgahl, E. R., Johnson III, T. L., Schnell, S. E. & Ward, A. R. (2008). Systematic convergence in applying the variational method to the anharmonic oscillator potentials. Journal of Undergraduate Research in Physics. 21, http://www.jurp.org/.
Heitler, W. & London, F. (1927). Wechselwirkung neutraler atome und homöopolare bindung nach der quantenmechanik. Zeitschrift für Physik, 44, 455-472.
Hylleraas, E. A. (1929). Neue Berechnung der energie des heliums im grundzustande, sowie des tiefsten terms von ortho-helium. Zeitschrift für Physik. 54, 347-366.
Hylleraas, E. A. (1963). Reminiscences from early quantum mechanics of two-electron atoms. Review of Modern Physics. 35, 421-432.
Hylleraas, E. A. (1970). Mathematical and theoretical physics, Vol. II. New York: John Wiley, pp 412-426, and the references cited therein.
Keung, W. Y., Kovac, E. & Sukhatme, U. (1988). Supersymmetry and double-well potentials. Physical Review Letters, 60, 41-44.
Koch, J., Schuck, C. & Wacker, B. (2008). Excited states of the anharmonic oscillator problems: variational method. ibid, 21, http://www.jurp.org/.
Laughlin, R. B. (1981). Quantized Hall conductivity in two dimensions. Physical Review B. 23, 5632-5633.
Laughlin, R. B. (1983). Quantized motion of three two-dimensional electrons in a strong magnetic field. Physical Review B, 27, 3383-3389.
Laughlin, R. B. (1983). Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Physical Review Letters, 50, 1395-1398.
Manning, M. F. (1935). Energy levels of a symmetrical double minima problem with applications to the NH3 and ND3 molecules. Journal of Chemical Physics, 3, 136-138.
Mei, W. N. (1998). Comments on "Phase space integration method for bound states" by Sharada Nagabhushana, B. A. Kagali, & Sivramkrishna Vijay [Am. J. Phys. 65 (6), 563-564, (1997)]. American Journal of Physics, 66 (6) 541-542.
Mei, W. N. (1996). Demonstration of systematic improvements in the application of the variational method to harmonic oscillator potentials. International Journal of Mathematical Education in Science and Technology, 27, 285-292.
Mei, W. N. (1997). Combined variational-perturbative approach to the anharmonic oscillator problems. International Journal of Mathematical Education in Science and Technology, 28, 495-511.
Mei, W. N. (1998). Combined variational-perturbative approach to the anharmonic oscillator problems. International Journal of Mathematical Education in Science and Technology, 29, 875-893.
Mei, W. N. (1999). Demonstration of systematic improvements in the application of the variational method to weakly bound potentials. International Journal of Mathematical Education in Science and Technology, 30, 513-540.
Merzabacher, E. (1961). Quantum mechanics, first edition. New York: John Wiley. pp 393-396.
Ninemire, B. & Mei, W. N. (2004). Demonstration of systematic improvements in the application of the variational method to strongly bound potentials. International Journal of Mathematical Education in Science and Technology, 35, 565-583.
Razavy, M. (1979). An exactly soluble Schrödinger equation with a bistable potential field. American Journal of Physics, 48, 285-288. Ronveaux, A. (1995). Heun's differential equations. Oxford: Oxford University Press.
Schiff, L. L. (1968). Quantum mechanics, third edition. New York: McGraw-Hill. pp 255-262.
Slavyanov, S. Y. & Lay, W. (2000). Special functions, a unified theory based on singularities. Oxford: Oxford Mathematical Monographs.
Bardeen, J., Cooper, L. N. & Schrieffer, J. R. (1957). Microscopic theory of superconductivity Physical Review, 106, 162-164.
Cohen-Tannoudji, C., Diu, B. & . F. (1977). Quantum mechanics, first edition. Vol. 1 and 2, New York: John Wiley. pp 1148-1155.
DeMille, D., (2015). Diatomic molecules, a window onto fundamental physics. Physics Today, December, 34-40.
Feynman, R. P. (1953). Atomic theory of the λ transition in helium. Physical Review, 91, 1301-1308.
Feynman, R. P. (1954). Atomic theory of the two-fluid model of liquid helium. Physical Review, 94, 262-277.
Feynman, R. P. (1955). Slow electrons in a polar crystal. Physical Review. 97, 660-665.
Feynman, R. P. & Cohen, M. (1956). Energy spectrum of the excitations in liquid helium. Physical Review, 102, 1189-1204.
Griffiths, D. J. (2005). Introduction to quantum mechanics, second edition. Upper Saddle River, NJ: Pearson Preston Hall. Chapter 7 and 8 and the references cited therein.
Gutzwiller, M. C. (1965). Quantized Hall conductivity in two dimensions. Physical Review A, 137, 1726-1735.
Hardy, J. R. & Flocken, J. W. (1988). Possible origins of high-Tc, superconductivity. Physical Review Letters, 60, 2190-2193.
Hedgahl, E. R., Johnson III, T. L., Schnell, S. E. & Ward, A. R. (2008). Systematic convergence in applying the variational method to the anharmonic oscillator potentials. Journal of Undergraduate Research in Physics. 21, http://www.jurp.org/.
Heitler, W. & London, F. (1927). Wechselwirkung neutraler atome und homöopolare bindung nach der quantenmechanik. Zeitschrift für Physik, 44, 455-472.
Hylleraas, E. A. (1929). Neue Berechnung der energie des heliums im grundzustande, sowie des tiefsten terms von ortho-helium. Zeitschrift für Physik. 54, 347-366.
Hylleraas, E. A. (1963). Reminiscences from early quantum mechanics of two-electron atoms. Review of Modern Physics. 35, 421-432.
Hylleraas, E. A. (1970). Mathematical and theoretical physics, Vol. II. New York: John Wiley, pp 412-426, and the references cited therein.
Keung, W. Y., Kovac, E. & Sukhatme, U. (1988). Supersymmetry and double-well potentials. Physical Review Letters, 60, 41-44.
Koch, J., Schuck, C. & Wacker, B. (2008). Excited states of the anharmonic oscillator problems: variational method. ibid, 21, http://www.jurp.org/.
Laughlin, R. B. (1981). Quantized Hall conductivity in two dimensions. Physical Review B. 23, 5632-5633.
Laughlin, R. B. (1983). Quantized motion of three two-dimensional electrons in a strong magnetic field. Physical Review B, 27, 3383-3389.
Laughlin, R. B. (1983). Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Physical Review Letters, 50, 1395-1398.
Manning, M. F. (1935). Energy levels of a symmetrical double minima problem with applications to the NH3 and ND3 molecules. Journal of Chemical Physics, 3, 136-138.
Mei, W. N. (1998). Comments on "Phase space integration method for bound states" by Sharada Nagabhushana, B. A. Kagali, & Sivramkrishna Vijay [Am. J. Phys. 65 (6), 563-564, (1997)]. American Journal of Physics, 66 (6) 541-542.
Mei, W. N. (1996). Demonstration of systematic improvements in the application of the variational method to harmonic oscillator potentials. International Journal of Mathematical Education in Science and Technology, 27, 285-292.
Mei, W. N. (1997). Combined variational-perturbative approach to the anharmonic oscillator problems. International Journal of Mathematical Education in Science and Technology, 28, 495-511.
Mei, W. N. (1998). Combined variational-perturbative approach to the anharmonic oscillator problems. International Journal of Mathematical Education in Science and Technology, 29, 875-893.
Mei, W. N. (1999). Demonstration of systematic improvements in the application of the variational method to weakly bound potentials. International Journal of Mathematical Education in Science and Technology, 30, 513-540.
Merzabacher, E. (1961). Quantum mechanics, first edition. New York: John Wiley. pp 393-396.
Ninemire, B. & Mei, W. N. (2004). Demonstration of systematic improvements in the application of the variational method to strongly bound potentials. International Journal of Mathematical Education in Science and Technology, 35, 565-583.
Razavy, M. (1979). An exactly soluble Schrödinger equation with a bistable potential field. American Journal of Physics, 48, 285-288. Ronveaux, A. (1995). Heun's differential equations. Oxford: Oxford University Press.
Schiff, L. L. (1968). Quantum mechanics, third edition. New York: McGraw-Hill. pp 255-262.
Slavyanov, S. Y. & Lay, W. (2000). Special functions, a unified theory based on singularities. Oxford: Oxford Mathematical Monographs.
Published
2017-02-27
How to Cite
MEI, Wai-Ning.
Systematic Convergence in Applying Variational Method to Double-Well Potential.
European Journal of Physics Education, [S.l.], v. 7, n. 2, p. 28-44, feb. 2017.
ISSN 1309-7202.
Available at: <https://eu-journal.org/index.php/EJPE/article/view/44>. Date accessed: 19 apr. 2024.
Issue
Section
Classroom Physics
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