Matrices are not explicitly mentioned in Heisenberg’s pa-
per. He did not arrange his quantum-theoretical quantities
into a table or array. In looking back on his discovery,
Heisenberg wrote, “At that time I must confess I did not
know what a matrix was and did not know the rules of ma-
trix multiplication.”
18
In the last sentence of his paper he
wrote “whether this method after all represents far too rough
an approach to the physical program of constructing a theo-
retical quantum mechanics, an obviously very involved prob-
lem at the moment, can be decided only by a more intensive
mathematical investigation of the method which has been
very superFcially employed here.”
27
Born took up Heisenberg’s challenge to pursue “a more
intensive mathematical investigation.” At the time Heisen-
berg wrote his paper, he was Born’s assistant at the Univer-
sity of Göttingen. Born recalls the moment of inspiration
when he realized that position and momentum were
matrices:
28
After having sent Heisenberg’s paper to the
Zeitschrift für Physik for publication, I began to
ponder about his symbolic multiplication, and was
soon so involved in it
…
±or I felt there was some-
thing fundamental behind it
…
And one morning,
about 10 July 1925, I suddenly saw the light:
Heisenberg’s symbolic multiplication was nothing
but the matrix calculus, well known to me since
my student days from the lectures of Rosanes in
Breslau.
I found this by just simplifying the notation a little:
instead of
q
s
n
,
n
+
t
d
, where
n
is the quantum num-
ber of one state and
t
the integer indicating the
transition, I wrote
q
s
n
,
m
d
, and rewriting Heisen-
berg’s form of Bohr’s quantum condition, I recog-
nized at once its formal signiFcance. It meant that
the two matrix products
pq
and
qp
are not identi-
cal. I was familiar with the fact that matrix multi-
plication is not commutative; therefore I was not
too much puzzled by this result. Closer inspection
showed that Heisenberg’s formula gave only the
value of the diagonal elements
s
m
=
n
d
of the ma-
trix
pq
–
qp
; it said they were all equal and had the
value
h
/
2
p
i
where
h
is Planck’s constant and
i
=
Î
²1. But what were the other elements
s
m
Þ
n
d
?
Here my own constructive work began. Repeating
Heisenberg’s calculation in matrix notation, I soon
convinced myself that the only reasonable value of
the nondiagonal elements should be zero, and I
wrote the strange equation
pq
²
qp
=
h
2
p
i
1
,
s
1
d
where
1
is the unit matrix. But this was only a
guess, and all my attempts to prove it failed.
On 19 July 1925, Born invited his former assistant Wolf-
gang Pauli to collaborate on the matrix program. Pauli de-
clined the invitation.
29
The next day, Born asked his student
Pascual Jordan to assist him. Jordan accepted the invitation
and in a few days proved Born’s conjecture that all nondi-
agonal elements of
pq
²
qp
must vanish. The rest of the new
quantum mechanics rapidly solidiFed. The Born and Jordan
paper was received by the Zeitschrift für Physik on 27 Sep-
tember 1925, two months after Heisenberg’s paper was re-
ceived by the same journal. All the essentials of matrix me-
chanics as we know the subject today Fll the pages of this
paper.
In the abstract Born and Jordan wrote “The recently pub-
lished theoretical approach of Heisenberg is here developed
into a systematic theory of quantum mechanics
s
in the Frst
place for systems having one degree of freedom
d
with the aid
of mathematical matrix methods.”
30
In the introduction they
go on to write “The physical reasoning which led Heisenberg
to this development has been so clearly described by him
that any supplementary remarks appear super³uous. But, as
he himself indicates, in its formal, mathematical aspects his
approach is but in its initial stages. His hypotheses have been
applied only to simple examples without being fully carried
through to a generalized theory. Having been in an advanta-
geous position to familiarize ourselves with his ideas
throughout their formative stages, we now strive
s
since his
investigations have been concluded
d
to clarify the math-
ematically formal content of his approach and present some
of our results here. These indicate that it is in fact possible,
starting with the basic premises given by Heisenberg, to
build up a closed mathematical theory of quantum mechanics
which displays strikingly close analogies with classical me-
chanics, but at the same time preserves the characteristic
features of quantum phenomena.”
31
The reader is introduced to the notion of a matrix in the
third paragraph of the introduction: “The mathematical basis
of Heisenberg’s treatment is the
law of multiplication
of
quantum-theoretical quantities, which he derived from an in-
genious consideration of correspondence arguments. The de-
velopment of his formalism, which we give here, is based
upon the fact that this rule of multiplication is none other
than the well-known mathematical rule of
matrix multiplica-
tion
. The inFnite square array which appears at the start of
the next section, termed a
matrix
, is a representation of a
physical quantity which is given in classical theory as a func-
tion of time. The mathematical method of treatment inherent
in the new quantum mechanics is thereby characterized by
the employment of
matrix analysis
in place of the usual
number analysis.”
The Born-Jordan paper
4
is divided into four chapters.
Chapter 1 on “Matrix calculation” introduces the mathemat-
ics
s
algebra and calculus
d
of matrices to physicists. Chapter 2
on “Dynamics” establishes the fundamental postulates of
quantum mechanics, such as the law of commutation, and
derives the important theorems, such as the conservation of
energy. Chapter 3 on “Investigation of the anharmonic oscil-
lator” contains the Frst rigorous
s
correspondence free
d
calcu-
lation of the energy spectrum of a quantum-mechanical har-
monic
oscillator.
Chapter
4
on
“Remarks
on
electrodynamics” contains a procedure—the Frst of its
kind—to quantize the electromagnetic Feld. We focus on the
material in Chap. 2 because it contains the essential physics
of matrix mechanics.
129
129
Am. J. Phys., Vol. 77, No. 2, ±ebruary 2009
William A. ±edak and Jeffrey J. Prentis