Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*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 Δ

*q*, 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 Δ

*p*where Δ

*q*Δ

*p*≳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

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.

Dictionary of the History of Ideas | ||