8. The Statistical Interpretation of Quantum-
mechanical Indeterminacy. If the ψ-function charac-
terizes the behavior not of an individual particle but
of a statistical ensemble of particles, as contended in
the "statistical interpretation" of the quantum-
mechanical formalism, then' obviously the uncertainty
relations, at least as far they derive from this formalism,
refer likewise not to individual particles but to statis-
tical ensembles of these. In other words, relation (1)
denotes, in this view, a correlation between the disper-
sion or "spread" of measurements of position, and the
dispersion or "spread" of measurements of momentum,
if carried out on a large ensemble of identically pre-
pared systems. Under these circumstances the idea that
noncommuting variables are not necessarily incompat-
ible but can be measured simultaneously on individual
systems would not violate the statistical interpretation.
Such an interpretation of quantum-mechanical
indeterminacy was suggested relatively early by
Popper (Popper, 1935). His reformulation of the un-
certainty principle reads as follows: given an ensemble
(aggregate of particles or sequence of experiments
performed on one particle which after each experiment
is brought back to its original state) from which, at
a certain moment and with a given precision Δq, those
particles having a certain position q are selected; the
momenta p of the latter will then show a random
scattering with a range of scatter Δp where ΔqΔp≳ℏ
and vice versa. Popper even thought, though errone-
ously as he himself soon realized, to have proved his
contention by the construction of a thought-experiment
for the determination of the sharp values of position
and momentum (Popper, 1934).
The ensemble interpretation of indeterminacy found
an eloquent advocate in Henry Margenau. Distin-
guishing sharply between subjective or a priori and
empirical or a posteriori probability, Margenau pointed
out that the indeterminacy associated with a single
measurement such as referred to in Heisenberg's
gamma-ray experiment is nothing more than a qualita-
tive subjective estimate, incapable of scientific verifi-
cation; every other interpretation would at once revert
to envisaging the single measurement as the constituent
of a statistical ensemble; but as soon as the empirical
view on probability is adopted which, grounded in
frequencies, is the only one that is scientifically sound,
the uncertainty principle, now asserting a relation
between the dispersions of measurement results, be-
comes amenable to empirical verification. To vindicate
this interpretation Margenau pointed out that, contrary
to conventional ideas, canonical conjugates may well
be measured with arbitrary accuracy at one and the
same time; thus two microscopes, one using gamma
rays and the other infra-red rays for a Doppler-experi-
ment, may simultaneously locate the electron and de-
termine its momentum and no law of quantum me-
chanics prohibits such a double measurement from
succeeding (Margenau, 1937; 1950). This view does not
abnegate the principle, for on repeating such measure-
ments many times with identically prepared systems
the product of the standard deviations of the values
obtained will have a definite lower limit.
Although Margenau and R. N. Hill (Margenau and
Hill, 1961) found that the usual Hilbert space formalism
of quantum mechanics does not admit probability
distributions for simultaneous measurements of non-
commuting variables, E. Prugovečki has suggested that
by introducing complex probability distributions the
existing formalism of mathematical statistics can be
generalized so as to overcome this difficulty. For other
approaches to the same purpose we refer the reader
to an important paper by Margenau and Leon Cohen,
and the bibliography listed therein (Margenau and
Cohen, 1967), and also to the analyses of simultaneous
measurements of conjugate variables carried out by E.
Arthurs and J. L. Kelly (Arthurs and Kelly, 1965),
C. Y. She and H. Heffner (She and Heffner, 1966),
James L. Park and Margenau (J. L. Park and Margenau,
1968). William T. Scott (Scott, 1968), and Dick H.
Holze and William T. Scott (Holze and Scott, 1968).
These investigations suggest the result that neither
single quantum-mechanical measurements nor even
combined simultaneous measurements of canonically
conjugate variables are, in the terminology of the
introduction, subject to i-indeterminacy, even though
they are subject to u-indeterminacy.
