The Beauty of Physics:
Patterns, principles, and perspectives
Adding a dimension
States and Transformations
Saddles
Coins, classical and quantum
Symmetry
Maps
The Problem of Time
Complexity and Emergence
A. R. P. Rau, Oxford University Press, 2014
Trojan asteroids at Stability at saddles
Lagrange points,
Coriolis forces,
Mechanical analog
Paul Trap for ions,
Charged particles
cannot be trapped
with only static,
electric fields.
Add rf field
Lagrange Points
Points of quasi-stability
Sun – Jupiter
Trojan Asteroids
Artificial Satellites
Earth – Moon
Rotation
Coriolis Forces
(from kinetic energy)
Two-electron potential energy surface
Transformation x, y  Circular
Two-electron Atom
Hyperspherical coordinates
3 Euler angles
6-dimensional coordinates
Two-electron atom’s potential energy
Hyperspherical coordinates
Saddle point in potential surface at
Two electrons at equal and opposite distances from Z
Classes of two-electron states
Valley
Saddle
Independent
particle
Hyperspherical
“pair”
Doubly-excited
and two-electron escape near threshold;
strongly correlated, angular and radial
+
+
He N = 3
3sns
0
1s2s
He N = 1
24.6
1sns
+
He N = 2
65.4
2sns
79
He++
1s 2
2s 2
3s 2
2p 2
3p 2
2pnp
3d 2
S
1 e
H N=2
2sns
H N=3
3sns
+
0.75 H N = 1
0
10.95
14.35
H
1s 2
2s 2
3s 2
2p 2
3p 2
Fig. 5.1
2pnp
3d 2
Doubly-excited states of He and H--
Physics of quasi-stability
Coriolis
Restoring Force
Coupling to another variable, t or R;
comes from kinetic energy “cross terms”
between R and α, crucial to Wannier theory.
More variables, more saddles, varieties, ….