Quantum Mechanics for Mathematicians

Kepler’s second law for such motion comes from conservation of angular
momentum, which corresponds to the Poisson bracket relation

{lj,h}= 0

Here we’ll take the Coulomb version of the Hamiltonian that we need for the
hydrogen atom problem

h=

1

2 m

|p|^2 −

e^2

r

The relation{lj,h}= 0 can be read in two ways:

• The Hamiltonianhis invariant under the action of the group (SO(3))
  whose infinitesimal generators arelj.


• The components of the angular momentum (lj) are invariant under the
  action of the group (Rof time translations) whose infinitesimal generator
  ish, so the angular momentum is a conserved quantity.

For this special choice of Hamiltonian, there is a different sort of conserved
quantity. This quantity is, like the angular momentum, a vector, often called
the Lenz (or sometimes Runge-Lenz, or even Laplace-Runge-Lenz) vector:

Definition(Lenz vector).The Lenz vector is the vector-valued function on the
phase spaceR^6 given by

w=

1

m

(l×p) +e^2

q

|q|

Simple manipulations of the cross-product show that one has

l·w= 0

We won’t here explicitly calculate the various Poisson brackets involving the
componentswjofw, since this is a long and unilluminating calculation, but
will just quote the results, which are

• {wj,h}= 0
  This says that, like the angular momentum, the vector with components
  wjis a conserved quantity under time evolution of the system, and its
  components generate symmetries of the classical system.


• 
  

  {lj,wk}=jklwl
  These relations say that the generators of theSO(3) symmetry act onwj
  in the way one would expect for the componentswjof a vector inR^3.