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14) In chapter 23 we shall apply the periodic orbit theory to the quantization of helium. In particular, we will study collinear helium, a doubly charged nucleus with two electrons arranged on a line, an electron on each side of the nucleus. 3. 4: A typical colinear helium trajectory in the r1 – r2 plane; the trajectory enters here along the r1 axis and then, like almost every other trajectory, after a few bounces escapes to infinity, in this case along the r2 axis. 2 0 0 2 4 6 8 r1 1 1 2 2 1 H = p21 + p22 − − + .

FLOWS For the solar system, the latitude and longitude in the celestial sphere are enough to completly specify the planet’s motion. All possible values for positions and velocities of the planets form the phase space of the system. More generally, a state of a physical system at a given instant in time can be represented by a single point in an abstract space called state space or phase space M. As the system changes, so does the representative point in phase space. We refer to the evolution of such points as dynamics, and the function f t which specifies where the representative point is at time t as the evolution rule.

2 0 0 2 4 6 8 r1 1 1 2 2 1 H = p21 + p22 − − + . 15) The collinear helium has 2 degrees of freedom, thus a 4-dimensional phase space M, which the energy conservation reduces to 3 dimensions. The dynamics can be visualized as a motion in the (r1 , r2 ), ri ≥ 0 quadrant, fig. 4. It looks messy, and indeed it will turn out to be no less chaotic than a pinball bouncing between three disks. 4, p. 3 Changing coordinates Problems are handed down to us in many shapes and forms, and they are not always expressed in the most convenient way.

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