Quantum Monodromy

1
Quantum Monodromy
Quantum monodromy concerns the patterns of
quantum mechanical energy levels close to
potential energy barriers.
Attention will be restricted initially to two
dimensional models in which there is a defined
angular momentum, with particular reference to
the quasi-linear level structures of H2O at the
barrier to linearity and the vibration-rotation
transition as the H atom passes around P in HCP.
The aim will be to show how the organisation of
the energy level patterns reflects robust
consequences of aspects of the classical
dynamics, regardless of the precise potential
energy forms.
The first lecture will relate to assignment of
the extensive computed highly excited
vibrational spectrum of H2O. The second to
modelling spectra close to saddle points on the
potential energy surface.
2
Quantum monodromy in H2O

• Model Hamiltonian and quantum eigenvalues
• Bent and linear state assignments
• Classical motions
• Quantum monodromy defined and illustrated
• Assignment of the Partridge-Schwenke computed
  spectrum
• Relevance to Bohr-Sommerfeld quantization
• Localised quantum corrections

3
Model Hamiltonian
4
e
y
x
5
Matrix elements in degenerate SHO basis
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Bohr-Sommerfeld quantization
Corresponds to Johnss bent state label vbent
Alternative linear state label
Both well defined for all states
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Classical-Quantum correspondences arising from
angle-action transformation (PR,R,Pf,f)?(IR,?,If,f
)
Relates energy differences in monodromy plot to
radial frequency ?R and ratio (Angle change ?F
over radial cycle)/(radial time period ?t)
11
Classical trajectories
elt 0
e gt 0
y
x
x
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(0v20) bending progression of H2O
30000
E/cm-1
20000
10000
0
20
10
-20
-10
ka
15
(0v20) bending progression of H2O
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Mathematical origin of monodromy dislocation
18
Quantum correction to Bohr Sommerfeld
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Summary

• Pattern of quasi-linear eigenvalues analysed by
  semiclassical arguments
• Eigenvalue lattice contains a characteristic
  dislocation, regardless of the precise potential
• Classical trajectories explain sharp change in e
  vs k at fixed v as sign of e changes
• Application to vib assignment for H2O
• Term quantum monodromy explained
• Error in semiclassical theory quantified

22
Acknowledgements

• R Cushman introduced the idea at a workshop for
  mathematicians, physicists and chemists
• J Tennyson extracted and organised the data on
  H2O
• T Weston helped with the semiclassical analysis
• UK EPSRC paid for TWs PhD