By Gert Heckman
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As time varies, the position vector r moves along the orbit E, and likewise s moves along the circle C. It is a good question to investigate how the point t moves. 1. The point t equals K/mH and therefore is conserved. Proof. The line N spanned by n = p × L is perpendicular to L. The point t is obtained from s by subtracting twice the orthogonal projection of s − r on the line N , and therefore t = s − 2((s − r) · n)n/n2 . Now s = −kr/rH (s − r) · n = −(H + k/r)r · (p × L)/H = −(H + k/r)L2 /H n2 = p2 L2 = 2m(H + k/r)L2 41 and therefore t = −kr/rH + n/mH = K/mH ˙ = 0 is derived with K = p × L − kmr/r the Lenz vector.
Show that for a reductive Lie algebra g with compact real form g0 and faithful unitary representation g → End(H) on a finite dimensional Hilbert space H the adjoint representation is unitary in the sense that [z, x], y = x, [z , y] for all x, y, z ∈ g. 3. 3 show that hλ = (hλ ) . 4. A Lie subalgebra n of a Lie algebra g is called normal if [x, y] ∈ n for all x ∈ g and y ∈ n. A Lie algebra g is called simple if the only normal subalgebras are the two trivial ones 0 and g itself. Show that a reductive Lie algebra g is a unique direct sum g = z ⊕ g1 ⊕ · · · ⊕ gn with z the Abelian center and the gi all simple Lie algebras.
We conclude that for the hydrogen atom the coherent positive energy E > 0 eigenspaces for H have natural degeneration according to an irreducible representation of the Lorentz algebra. 59 10 Triangular decomposition and Verma representations Suppose we have given a reductive Lie algebra g with compact real form g0 , with defining unitary representation g → End(H) and associated symmetric trace form (·, ·) on g. Suppose we have chosen a Cartan subalgebra h in g with associated root system R = R(g, h) in the Euclidean space a∗ .