TY - JOUR
T1 - Applying Bogomolny's quantization method to generic classical systems
AU - Kay, Kenneth G.
N1 - Publisher Copyright:
© 2017 Author(s).
PY - 2017/5/28
Y1 - 2017/5/28
N2 - The quantization method of Bogomolny [Nonlinearity 5, 805 (1992)] can potentially provide semiclassical estimates for energy levels of all bound states of arbitrary systems. This approach requires the formation of the transfer matrix TE as a function of energy E. Existing practical methods for calculating this matrix require a recalculation of many classical trajectories for each energy. This has hampered the application of Bogomolny's method to generic systems that do not possess special classical scaling properties. Generalizing earlier work [H. Barak and K. G. Kay, Phys. Rev. E 88, 062926 (2013)], we develop initial value representation formulas for TE that overcome this problem. These expressions are obtained from a generalized Herman-Kluk formula for the propagator that allows one to easily derive a family of semiclassical integral approximations for the Green's function that are, in turn, used to form the transfer matrix. Calculations for two-dimensional systems show that Bogomolny's method with the present expressions for TE produces accurate semiclassical energy levels from small transfer matrices.
AB - The quantization method of Bogomolny [Nonlinearity 5, 805 (1992)] can potentially provide semiclassical estimates for energy levels of all bound states of arbitrary systems. This approach requires the formation of the transfer matrix TE as a function of energy E. Existing practical methods for calculating this matrix require a recalculation of many classical trajectories for each energy. This has hampered the application of Bogomolny's method to generic systems that do not possess special classical scaling properties. Generalizing earlier work [H. Barak and K. G. Kay, Phys. Rev. E 88, 062926 (2013)], we develop initial value representation formulas for TE that overcome this problem. These expressions are obtained from a generalized Herman-Kluk formula for the propagator that allows one to easily derive a family of semiclassical integral approximations for the Green's function that are, in turn, used to form the transfer matrix. Calculations for two-dimensional systems show that Bogomolny's method with the present expressions for TE produces accurate semiclassical energy levels from small transfer matrices.
UR - http://www.scopus.com/inward/record.url?scp=85019977980&partnerID=8YFLogxK
U2 - 10.1063/1.4983748
DO - 10.1063/1.4983748
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C2 - 28571363
AN - SCOPUS:85019977980
SN - 0021-9606
VL - 146
JO - Journal of Chemical Physics
JF - Journal of Chemical Physics
IS - 20
M1 - 204111
ER -