Quantum MonodromyQuantum 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 thevibration-rotation transition as the H atom passes around P in HCP.

The aim will be to show how the organisation of the energy level patternsreflects robust consequences of aspects of the classical dynamics, regardlessof 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

22 4

2

1/ 42

1/ 22

2

22 4

2

2

21ˆ

/ 2

2 /

2 /

1 1 1ˆ2 2

mk

H R AR BRR R R R

Scaling

R mA r

E A m

B mA

kh r r r

r r r r

ε

x

y

Matrix elements in degenerate SHO basis

2 4

2

2 2 2

4 2 2

1ˆ ˆ2

ˆ| | | | 1

ˆ2 | | 2 | | / 2

| | | | | |

0 0 0 0

0 0 0

0 0ˆ 0 0

0 0

0 0 0

0 0 0 0

m

h t r r

nk t nk nk r nk n

n k t nk n k r nk n k

n k r nk n k r mk mk r nk

x x x

x x x x

x x x x x

h x x x x x

x x x x x

x x x x

x x x

Bohr-Sommerfeld quantization

2

1

2 21/ 2 2 [ ( ) / 2 ]R

Rv m E V R k mR dR

Corresponds to Johns’s ‘bent state’ label vbent

Alternative linear state label

2 | |linear bentv v k

Both well defined for all states

Classical-Quantum correspondences arising from angle-action transformation (PR,R,Pφ,φ)→(IR,θ,Iφ,φ)

( , , ) ( , )

( 1/ 2)

R

R R

R

Rk R I

v I

H P R P H I I

I v I k

H d

v I dt

H d

k I dt t

Relates energy differences in monodromy plot to radial frequency ωR and ratio(Angle change ΔΦ over radial cycle)/(radial time period Δt)

a

10.0

5.0

0.0

-5.0

-10.0-10.0 -5.0 0.0

10.0

5.0 10.0

(a)

b

a

a

10.0

5.0

0.0

-5.0

-10.0

-10.0 -5.0 0.0 5.0

(b)

10.0

b

a

Classical trajectories

ε< 0 ε > 0

y

xx

(0v20) bending progression of H2O

10000

20000

30000

E/cm-1

ka

-20 0 20-10 10

(0v20) bending progression of H2O

Mathematical origin of monodromy dislocation

2

1

0

1

1

2 4 2 2

2 2 2 2 4 2 2

Arises from confluence between inner turning point

r of the Bohr quantization integral, and singularity at r=0,

as ( ,k) (0,0)

[ ( , ) 1/ 2] 2 2 /

2 / 2 2 /

r

r

r

r

v k r r k r dr

r k r dr r r k r d

2

0

( , ) ( , )

1( , ) Im ( ) ln = multivalued

2 2

( , ) smooth

r

r

a b

a

b

r

f k f k

k if k k i

f k

Quantum correction to Bohr Sommerfeld

1

1/ 2 2 2 4 2 2

21

1

Bohr Sommerfeld arises from standard JWKB wavefunction

( ) sin[ ( ) / 4], ( ) 2 /

assuming that ( ) varies linearly with at the turning point .

Invalid for 0 as (

r

JWKB rr q q r dr q r r r k r

q r r r

r

1

1/ 2

2 2

, ) (0,0).

Comparison with Whitakker equation of mathematical physics shows that

( ) sin[ ( ) / 4 ( , )],

| | 1( , ) ln arctan arg

4 4 2 2 2 2

Corrected

r

corrected r

k

r q q r dr k

k k k ik

k

2

1

quantization condition

( 1/ 2) ( )r

rv q r dr

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