Heisenberg matrix mechanics origins

[JohnF]

I picked up a copy of Dirac's public lectures, "Directions in
Physics", Wiley 1978, ISBN 0-471-02997-1, which immediately
(page 5) reminded me of a question I never understood.
Paraphrasing a few page 5 paragraphs:
Heisenberg (in 1925) felt physical theory should concentrate
on quantities closely related to observed quantities. And
observed quantities are only remotely related to Bohr orbits.
Instead, things closely related to observed quantities are all
associated with >>two<< Bohr orbits. The natural way to write
a set of quantities, each associated with two elements, is like
/ x x x . .\
| x x x . . | (note: the book just has x's denoting
| x x x . . | elements, without any further
| . . . . . | elaboration)
\ . . . . ./
where rows are associated with one of the states, and columns
with the other. Heisenberg assumed the whole set of such
quantities together corresponds to a dynamical variable of
Newtonian theory, e.g., coordinates or momenta, etc. Each of
these quantities is replaced by a matrix, the underlying idea
being that theories should be constructed in terms of observable
quantities, and that the observable quantities are these matrix
elements, each associated with two Bohr orbits.

I get Dirac's explanation of Heisenberg's intuition that
observable quantities are each associated with two orbits,
so it's arrays of numbers that should be considered.
But I don't get how Heisenberg sees that it's the dynamical
variables which should be replaced by such arrays.
In 1925 I'd imagine it was "energy jumps" between two orbits
that one would immediately think of as observable quantities,
leading to a (anti-)symmetric matrix of energy jumps, e.g.,
Balmer series. But where can you go from there? I went back
to Goldstein (2nd ed), who points out that the commutator
of two matrices satisfies the same Lie algebra as Poisson
brackets. Was that the "numerical clue/coincidence" which
suggested itself to Heisenberg? But then what could Heisenberg
use for the "x's" in Dirac's illustrated matrix to mathematically
make sense? I don't suppose he could just somehow use energy jumps,
but I can't see how he intuitively arrived at anything else to use.
--
John Forkosh ( mailto: j...@f.com where j=john and f=forkosh )