Quantum Monodromy
Download
Report
Transcript Quantum Monodromy
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.
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
Model Hamiltonian
2
1
k
2
4
ˆ
H 2m
R
AR
BR
2
R
R
R
R
Scaling
2
R
2
/ 2mA
E 2A
1/ 4
2
B 2mA /
r
/ m
1/ 2
2
2
1 2
1
1
k
4
ˆh
r
r
r
2 r r r r 2 2
ε
y
x
Matrix elements in degenerate
SHO basis
1
hˆ tˆ r 2 r 4
2
nk | tˆ | nk nk | r 2 | nk n 1
n 2k | tˆ | nk n 2k | r 2 | nk n 2 k 2 / 2
nk | r 4 | nk nk | r 2 | mk
m
x
x
x
ˆ
h 0
0
0
0
x
x
0
0
0
x
x
x
x
x
x
0
x
0
0
x
0
x
x
x
x
x
x
x
x
0
0
x
x
x
0
0
0
x
x
0
0
0
0
x
x
x
mk | r 2 | nk
Bohr-Sommerfeld quantization
v 1/ 2 R
R2
2m[ E V ( R) k 2 / 2mR 2 ]dR
1
Corresponds to Johns’s ‘bent state’ label vbent
Alternative linear state label
v
linear
2v
bent
|k |
Both well defined for all states
Classical-Quantum correspondences arising from angleaction transformation (PR,R,Pφ,φ)→(IR,θ,Iφ,φ)
H ( PR , R, P ) H ( I R , I )
I R (v 1/ 2)
I k
H
d
R
v k
dt
I R I
H
k v
I
d
dt
t
IR
Relates energy differences in monodromy plot to radial frequency ωR and ratio
(Angle change ΔΦ over radial cycle)/(radial time period Δt)
Classical trajectories
ε>0
ε< 0
10.0
10.0
5.0
5.0
b
y
0.0
a
a
0.0
-5.0
-5.0
b
10.0
-10.0
(a)
-10.0
-10.0
-5.0
0.0
x
5.0
10.0
(b)
-10.0
-5.0
0.0
5.0
x
10.0
(0v20) bending progression of H2O
30000
E/cm-1
20000
10000
-20
-10
0
10
ka
20
(0v20) bending progression of H2O
Mathematical origin of monodromy dislocation
Arises from confluence between inner turning point
r1 of the Bohr quantization integral, and singularity at r=0,
as ( ,k) (0,0)
[v( , k ) 1/ 2]
r2
2 r 2 2 r 4 k 2 / r 2 dr
r1
r0
r1
2 r k / r dr
2
2
2
r2
r0
2 r 2 2 r 4 k 2 / r 2 dr
f a ( , k ) f b ( , k )
1
k i
f a ( , k ) Im (k i ) ln
2
2
f b ( , k ) smooth
= multivalued
Quantum correction to Bohr Sommerfeld
Bohr Sommerfeld arises from standard JWKB wavefunction
r
JWKB (r ) q 1/ 2 sin[ q (r )dr / 4], q 2 (r ) 2 r 2 r 4 k 2 / r 2
r1
assuming that q 2 ( r ) varies linearly with r at the turning point r1.
Invalid for r1 0 as ( , k ) (0, 0).
Comparison with Whitakker equation of mathematical physics shows that
corrected (r ) q
1/ 2
r
sin[ q (r )dr / 4 ( , k )],
r1
2 k2 k
| k | 1 i
( , k ) ln
arctan
arg
4 4 2 2
2
k
2
Corrected quantization condition
r2
(v 1/ 2) q (r )dr
r1
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 ε
vs k at fixed v as sign of ε changes
• Application to vib assignment for H2O
• Term quantum monodromy explained
• Error in semiclassical theory quantified
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 TW’s PhD