The brilliant leap made by Heisenberg's quantum mechanics
was made possible by a number of precursors. The BKS (Bohr, Kramers, Slater)
theory provided the model that had to be disproved. An American
scientist named John Slater had proposed the idea of a virtual
radiation field that carried no energy but that was continually
emitted and absorbed by virtual oscillators in an atom. Bohr and
his assistant, Kramers, took this idea to suggest that a quantum jump
could be induced by a virtual field without any real energy transference.
Such a theory required the abandonment of the sacred laws of energy
and momentum conservation, and violated a basic idea of causality,
as absorption and emission of energy were no longer necessarily
correlated.

Heisenberg himself was skeptical at first, but soon found
the theory quite captivating and hoped to connect it with his own
work. Back in Göttingen, Born was using a formula, derived by Kramers in
connection with the BKS theory, to progress in quantum mechanics.
However, not long after, when Heisenberg met Einstein for the first
time, he was discouraged by the master's objections to the BKS theory.
Einstein refused to accept the abandonment of certain essential
principles, and he wrote to Bohr that if they had to be given up,
"then I would rather be a shoemaker or an employee in a gambling
casino than a physicist."

As Born was to be away in 1924, Bohr arranged for a longer Copenhagen
retreat for Heisenberg. Soon, tension grew between Heisenberg and
Kramers, mostly due to competition for Bohr's attention and approval.
Nevertheless, their research coincided on one point, and Bohr pushed
them to write a joint paper on the topic. The paper advanced a
quantum theory of dispersion, which treated light as a wave rather
than as quanta. The theory made particular use of the virtual oscillators;
the point was to show how they correspond to harmonics in classical
theory.

Soon afterward, in late 1924 and early 1925, Pauli would
once again take on Heisenberg's core model, this time using a relativistic approach.
The mass of an object increases with speed according to relativity,
and the electrons in Heisenberg's model would be traveling fast
enough to require consideration of this factor. But Pauli found
no evidence of any mass change, and argued convincingly against
Heisenberg's core model. Rather than feeling frustrated, Heisenberg
praised Pauli's insight and looked for ways to build from it. Soon
after Pauli's discovery, the BKS theory too was being disproved
by experimental evidence.

Heisenberg's achievement came at a time when all of his
colleagues were working on different tangents, and with no systematic approach
to the answers they all desired. Nevertheless, their topics of
research often complemented each other in unexpected ways. While
Born and his new assistant Pascual Jordan were working on the quantum
theory of aperiodic systems, Heisenberg returned to the problem
of virtual oscillators in the atom. The amplitude of the oscillations
could be broken down to a Fourier series, but what Heisenberg recognized
was that this function had continued to use classical relationships.
Assuming that the basic Fourier function held true on the quantum
level, he then set about reinterpreting the frequencies and amplitude
with quantum principles in mind, as he had in formulating the Zeeman
principle.

Since the amplitudes of classical motions could be squared
to find the intensity of the emitted radiation, Heisenberg determined
to find a corresponding multiplication rule for the amplitude of
virtual oscillators, which would yield the intensity of the spectral
lines that had long given physicists trouble. The multiplication
rule that Heisenberg devised looked familiar to Born as he critiqued
the paper. What Born recognized was that the rule involved the same
principle used in multiplying matrices. Before long Born, Jordan, and
Heisenberg wrote the groundbreaking paper that expressed quantum
physics in matrices. Their equations satisfied the prior principles
of quantum physics while accomplishing the long-sought goal of
quantifying the discrete energy states of an atom.