Einstein 1938: Continuity—Discontinuity

Reference: Evolution of Physics

This paper presents Chapter IV section 1 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

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Continuity—Discontinuity

A map of New York City and the surrounding country is spread before us. We ask: which points on this map can be reached by train? After looking up these points in a railway timetable, we mark them on the map. We now change our question and ask: which points can be reached by car? If we draw lines on the map representing all the roads starting from New York, every point on these roads can, in fact, be reached by car. In both cases we have sets of points. In the first they are separated from each other and represent the different railway stations, and in the second they are the points along the lines representing the roads. Our next question is about the distance of each of these points from New York, or, to be more rigorous, from a certain spot in that city. In the first case, certain numbers correspond to the points on our map. These numbers change by irregular, but always finite, leaps and bounds. We say: the distances from New York of the places which can be reached by train change only in a discontinuous way. Those of the places which can be reached by car, however, may change by steps as small as we wish, they can vary in a continuous way. The changes in distance can be made arbitrarily small in the case of a car, but not in the case of a train.

The output of a coal mine can change in a continuous way. The amount of coal produced can be decreased or increased by arbitrarily small steps. But the number of miners employed can change only discontinuously. It would be pure nonsense to say: “Since yesterday, the number of employees has increased by 3.783.”

Asked about the amount of money in his pocket, a man can give a number containing only two decimals. A sum of money can change only by jumps, in a discontinuous way. In America the smallest permissible change or, as we shall call it, the “elementary quantum” for American money, is one cent. The elementary quantum for English money is one farthing, worth only half the American elementary quantum. Here we have an example of two elementary quanta whose mutual values can be compared. The ratio of their values has a definite sense since one of them is worth twice as much as the other.

A quantum is a definite, discrete amount.

We can say: some quantities can change continuously and others can change only discontinuously, by steps which cannot be further decreased. These indivisible steps are called the elementary quanta of the particular quantity to which they refer.

We can weigh large quantities of sand and regard its mass as continuous even though its granular structure is evident. But if the sand were to become very precious and the scales used very sensitive, we should have to consider the fact that the mass always changes by a multiple number of one grain. The mass of this one grain would be our elementary quantum. From this example we see how the discontinuous character of a quantity, so far regarded as continuous, can be detected by increasing the precision of our measurements.

If we had to characterize the principal idea of the quantum theory in one sentence, we could say: it must be assumed that some physical quantities so far regarded as continuous are composed of elementary quanta.

The region of facts covered by the quantum theory is tremendously great. These facts have been disclosed by the highly developed technique of modern experiment. As we can neither show nor describe even the basic experiments, we shall frequently have to quote their results dogmatically. Our aim is to explain the principal underlying ideas only.

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FINAL COMMENTS

Some quantities can change only by steps which cannot be further decreased. These indivisible steps are called the elementary quanta of the particular quantity to which they refer.

Some physical quantities so far regarded as continuous are composed of elementary quanta. This is the principle idea of quantum theory.

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