Energy levels of a particle confined in a super-circular box
The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics - Tập 46 - Trang 41-50 - 2007
Tóm tắt
We find the energy levels of a free particle confined in a two
dimensional infinite potential well having super-circular boundary
(|x|n+|y|n=an where n is a rational number and a is a
positive real number) by perturbing about the equivalent circle
(n=2). The ground state energies are very accurate over a wide
range of n and can be improved further by introducing a
phenomenological constant
determined from the knowledge of exact
results available for diamond (n=1). For excited states, we find
that the shape effect can cause parametric resonance which can lead
to singlet-triplet crossing.
Tài liệu tham khảo
H.R. Krishnamurthy, H.S. Mani, H.C. Verma, J. Phys. A: Math. Gen. 15, 2131 (1982)
J.K. Bhattacharjee, K. Banerjee, J. Phys. A: Math. Gen. 20, L759 (1987)
K. Lis, S. Bednarek, B. Szafran, J. Adamowski, Physica E 17, 494 (2003)
P.S. Drouvelis, P. Schmelcher, F.K. Diakonos, Phys. Rev. B 69, 155312 (2004)
I. Magnúsdóttir, V. Gudmundsson, Phys. Rev. B 60, 16590 (1999)
H. Ichikawa, K. Sakata, Intern. J. Quant. Chem. 87, 135 (2002) and referrences therein
S. Sakai, J. Phys. Chem. A 110, 6339 (2006)
M. Gardner, “Piet Hein's Superellipse”, in Mathematical Carnival: A new Round-Up of Tantalizers and Puzzles from Scientific American (Vintage, New York, 1977), Ch. 18, pp. 240–254
G. Lamé, Examen des différentes méthodes employées pour résoudre les problémes de geometrie (Oxford University, 1818)
N.T. Gridgeman, Math. Gaz. 54, 31 (1970)
J.A. Gielis, Am. J. Bot. 90, 333 (2003)
I. Peterson, Science News 163, 18 (2003)
J. Whitfield, Nature Science Update (April 2, 2003)
Y. Shimizu, Chaos, Solitons Fractals 5, 1337 (1995)
M.V. Berry, Eur. J. Phys. 2, 91 (1981)
I.M. Erham, H. Taseli, J. Comput. Appl. Math. 194, 227 (2006)
Handbook of Mathematical functions with Formulas, Graphs and Mathematical Tables, edited by M. Abramowitz, I. Stegun, National Bureau of Standards, Washington, D.C. (1964)
K.F. Riley, M.P. Hobson, S.C. Bence, Mathematical methods for physics and engeneering (Cambridge University Press, 2002), Ch. 19
M.R. Spiegel, Vector Analysis (Schaum's Outline Series, New York, 1959)
G. Arfken, Mathematical Methods for Physicists (Academy Press, 1995)
D.J. Griffiths, Introduction to Quantum Mechanics, 2nd edn. (Prentice Hall, 2004)
F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, 2001)
W.D. Carl, Chemistry Education Material, University of Connecticut (2006)
S. Agmon, Lectures on Elliptic Boundary Value Problems (Van Nostrand, Princeton, N.J., 1965)
I. Babuŝka, U. Banerjee, J.E. Osborn, Survey of meshless and generalized finite element methods: A unified approach (Acta Numerica, 2003), pp. 1–125