# Heisenberg’s Matrix Mechanics

The original foundations of what is now sometimes referred to as ‘*old
quantum mechanics’* was initially forwarded as an extension of Hamiltonian
mechanics with some additional conditions applied, i.e. a classical
orbit must be an integer multiple of Planck‘s quantum of action:

[1]

This definition of action was qualified by the ‘*correspondence
principle’ *as originally formalised by Niels Bohr in 1920, although
he had made use of the idea as early as 1913, when developing his model
of the atom. This principle required the behaviour of a system, being
described in the context of quantum theory, to converge towards
classical physics in the limit of large quantum numbers. Previously,
the classical theory of electrodynamics had also defined such a limit,
which restricted the number of possible transitions between atomic orbits.
However, such ideas were primarily
*heuristic *
schemes, rather than a
consolidated theory, which could be attributed and described in terms
of what is now called quantum mechanics. It was also recognised that
the ‘*old quantum’* schemes failed to explain a growing list of
empirical observations, e.g. helium spectrum, Zeeman effect, fine structure
of atomic spectra etc. So, by 1925, following the start of the wave-particle
debate triggered by *Compton* and
*deBroglie*, a number of physicists at
the forefront of quantum theory were trying to find a new formulation
of quantum mechanics, which would help address the apparently growing
list of issues. It is in this context that Heisenberg would publish a paper entitled ‘*Quantum-mechanical re-interpretation of kinematic
and mechanical relations’*, in July 1925, which many consider to
signal the start of a ‘*new quantum’ *description*. *However, this
milestone paper is generally considered to be very difficult to follow,
primarily because Heisenberg provided so few clues as to how he arrived
at his conclusions. For example,
*Steven Weinberg* would later write:

*“If the reader is mystified at what Heisenberg was doing, he or
she is not alone. I have tried several times to read the paper that
Heisenberg wrote on returning from Heligoland, and, although I think
I understand quantum mechanics, I have never understood Heisenberg’s
motivations for the mathematical steps in his paper. Theoretical physicists
in their most successful work tend to play one of two roles: they are
either sages or magicians....It is usually not difficult to understand
the papers of sage-physicists, but the papers of magician physicists
are often incomprehensible. In that sense, Heisenberg’s 1925 paper was
pure magic.” *

Possibly, for these reasons, Heisenberg paper and the further development of matrix mechanics by Borne and Jordan, would become somewhat over-shadowed by the subsequent publication of Schrodinger’s wave mechanics just a year later, in 1926. Therefore, we shall only be attempting to outline the most salient aspects of matrix mechanics within the chronological timeline being followed. We may start by saying that Heisenberg was guided by the fundamental principle that only observable quantities should be considered within a quantum theory. In classical mechanics, the position [q=x] of a particle and its momentum [p=mv] can be unambiguously determined as a function of time [t] via Newton’s equations of motion. Therefore, the particle can be defined to be moving along a given path by some position function q(t) plus some momentum function p(t). As ordinary variables, [q] and [p] can be multiplied together in any order, i.e.

[2]

However,
in matrix mechanics, the classical idea of position [q] and momentum [p]
have to be modified in terms of a quantum position [**q**] and quantum
momentum [**p**], which are defined as matrices that describe the
intensity and frequency of the emitted or absorbed atomic radiation.

[3]

Within matrix arithmetic, addition, subtraction,
multiplication and division can still be carried out, but the result
of multiplying [**q**] and [**p**]
needs
to be constrained by the rules of matrix arithmetic, such that:

[4]

These matrix variables have to satisfy what is called the ‘*exact
quantum condition’ *through which Heisenberg defined the difference
between the quantum matrix products [**pq**] and [**qp**] in
terms of the following *
non-commutative relationship*:

[5]

Here [i] is a complex
number and
[I]
is a unit matrix. By replacing
the ordinary classical variables
with quantum matrix variables and then applying the assumption defined
in [5], Heisenberg obtained
the correct values of the frequency
and strength of the hydrogen spectra. Finally, a ‘*variational principle’*
was introduced, derived from correspondence considerations, which yielded
the motion equations for a general *Hamiltonian* [H],
which while appearing to follow the classical form, as previously outlined,
the parameters [**q**] and [**p**] now represent the quantum matrices
corresponding to the intensity and frequency of the atomic radiation
observed. Subsequently, the results of Heisenberg’s matrix mechanics
would be shown to align to those obtained
via *Schrodinger’s wave mechanics*. However, in the timeline of developments
being discussed, this was a surprise, as the two approaches appeared
to be very different. Given the compatibility of the results from both
formulations, we shall now focus on only one, i.e. Schrodinger’s wave
mechanics. However, we will return to Heisenberg's matrix
formulation, in *Part-2*, in
order to better explain the the underlying mathematical
relationships between operators, matrices and the Dirac notation.