# University of Virginia Library

5. Indeterminacies in Quantum Mechanics. The
necessity of introducing indeterminacy considerations
into quantum mechanics became apparent as soon as
the mathematical formalism of the theory was estab-
lished (in the spring of 1927). When Ernst Schrödinger,
in 1926, laid the foundations of wave mechanics he
interpreted atomic phenomena as continuous, causal
undulatory processes, in contrast to Heisenberg's
matrix mechanics in which these processes were inter-
preted as discontinuous and ruled by probability laws.
When in September 1926 Schrödinger visited Niels
Bohr and Heisenberg in Copenhagen, the conflict be-
tween these opposing interpretations reached its climax
and no compromise seemed possible. As a result of this
controversy Heisenberg felt it necessary to examine
more closely the precise meaning of the role of
dynamical variables in quantum mechanics, such as
position, momentum, or energy, and to find out how
far they were operationally warranted.

First he derived from the mathematical formalism
of quantum mechanics (Dirac-Jordan transformation
theory) the following result. If a wave packet with a
Gaussian distribution in the position coordinate q, to
wit ψ(q) = const. exp [-q 2/22(Δq0 2], Δq being the half-
width and consequently proportional to the standard
deviation, is transformed by a Fourier transformation
into a momentum distribution, the latter turns out to
be ϕ(p) = const. exp [-p2/2(ℏ/Δq)2]. Since the corre-
sponding half-width Δp is now given by ℏ/Δq, Heisen-
berg concluded that Δq Δp ≈ ℏ or more generally, if
other distributions are used,

Δq Δp ≳ ℏ

This inequality shows that the uncertainties (or
dispersions) in position and momentum are reciprocal:
if one approaches zero the other approaches infinity.
The meaning of relation (1), which was soon called
the “Heisenberg position-momentum uncertainty rela-
tion,” can also be expressed as follows: it is impossible
to measure simultaneously both the position and the
momentum of a quantum-mechanical system with
arbitrary accuracy; the more precise the measurement
of one of these two variables is, the less precise is that
of the other.

Asking himself whether a close analysis of actual
measuring procedures does not lead to a result in
contradiction to (1), Heisenberg studied what has since
become known as the “gamma-ray microscopic exper-
iment.” Adopting the operational view that a physical
concept is meaningful only if a definite procedure is
indicated for how to measure its value, Heisenberg

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declared that if we speak of the position of an electron
we have to define a method of measuring it. The elec-
tron's position, he continued, may be found by illumi-
nating it and observing the scattered light under a
microscope. The shorter the wavelength of the light,
the more precise, according to the diffraction laws of
optics, will be the determination of the position—but
the more noticeable will also be the Compton effect
and the resulting change in the momentum of the
electron. By calculating the uncertainties resulting
from the Compton effect and the finite aperture of the
microscope, the importance of which for the whole
consideration was pointed out by Bohr, Heisenberg
showed that the obtainable precision does not surpass
the restrictions imposed by the inequality (1). Similarly,
by analyzing closely a Stern-Gerlach experiment of
measuring the magnetic moment of particles, Heisen-
berg showed that the dispersion ΔE in the energy of
these particles is smaller the longer the time Δt spent
by them in crossing the deviating field (or measuring
device):
ΔE Δt ≳ ℏ
It has been claimed that this “energy-time uncertainty
relation” had been implicitly applied by A. Sommer-
feld in 1911, O. Sackur in 1912, and K. Eisenmann
in 1912 (Kudrjawzew, 1965). Bohr, as we know from
documentary evidence (Archive for the History of
Quantum Physics, Interview with Heisenberg, Febru-
ary 25, 1963), accepted the uncertainty relations (1)
and (2), but not their interpretation as proposed by
Heisenberg. For Heisenberg they expressed the limita-
tion of the applicability of classical notions to micro-
physics, whether these notions are those of particle
language or wave language, one language being re-
placeable by the other and equivalent to it. For Bohr,
on the other hand, they were an indication that both
modes of expression, though conjointly necessary for
an exhaustive description of physical phenomena, can-
not be used at the same time. As a result of this debate
Heisenberg added to the paper in which he published
the uncertainty relations (Heisenberg, 1927) a “Post-
script” in which he acknowledged that an as yet un-
published investigation by Bohr would lead to a deeper
understanding of the significance of the uncertainty
relations and “to an important refinement of the results
obtained in the paper.” It was the first allusion to
Bohr's complementarity interpretation, often also
loosely called the “Copenhagen interpretation” of
quantum mechanics (Jammer [1966], pp. 345-61). Bohr
regarded the uncertainty relations whose derivations
(by thought-experiments) are still based on the de
Broglie-Einstein equations E = hv and p = h/λ, that
is, relations between particulate (energy E, momentum
p) and undulatory conceptions (frequency v, wavelength
λ), merely as a confirmation of the wave-particle
duality and hence of the complementarity interpre-
tation (Schiff, 1968).