mathematics

Paul Sophus Epstein

Paul Sophus Epstein is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Paul Sophus Epstein rather than just read about it. In short: Paul Sophus Epstein (Russian: Павел Зигмундович Эпштейн, romanized: Pavel Zigmundovich Epshteyn; March 20, 1883 – February 8, 1966) was a Russian-American mathematical physicist. He was known for his contributions to fluid dynamics and to the development of quantum mechanics.

Key takeaways

Paul Sophus Epstein belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Paul Sophus Epstein to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Paul Sophus Epstein from memory before moving on to harder problems.

Reference excerpt

Paul Sophus Epstein (Russian: Павел Зигмундович Эпштейн, romanized: Pavel Zigmundovich Epshteyn; March 20, 1883 – February 8, 1966) was a Russian-American mathematical physicist. He was known for his contributions to fluid dynamics and to the development of quantum mechanics.

Early life and studies Paul Epstein's parents, Siegmund Simon Epstein and Sarah Sophia (Lurie) Epstein were of a middle class Jewish family. He said that his mother recognized his potential at the age of four years and predicted that he would be a mathematician. He went to the Hochschule in Minsk, and from 1901 to 1905 studied mathematics and physics at the Imperial University of Moscow under Pyotr Nikolaevich Lebedev. In 1909, he graduated, and became a Privatdozent at the University of Moscow. In 1910, he went to Munich, Germany, to do research under Arnold Sommerfeld, who was his advisor, and Epstein was granted a Ph.D. on a problem in the theory of diffraction of electromagnetic waves. from the Technical University of Munich, in 1914.

Career At the outbreak of World War I he was in Munich, and considered an enemy alien. Thanks to Sommerfeld's intervention he was allowed to stay in Munich as a private citizen, and could continue with his research. By that time Epstein became interested in the quantum theory of atomic structure. In 1916, he published a seminal paper explaining the Stark effect using the Bohr–Sommerfeld quantization from old quantum theory. After the war he went to Zurich, where he remained for two years. He moved to Leiden in 1921, to become Hendrik Lorentz' and Paul Ehrenfest's assistant. Shortly afterwards he was recruited by Robert Millikan to come to the California Institute of Technology (Caltech), where he remained for the rest of his career. In 1930, he was elected a member of the National Academy of Sciences. After World War II, concerned that some young American intellectuals were becoming attracted to communism, he joined the American Committee for Cultural Freedom and in 1951 he served as one of the three US delegates to the seminar the Congress held in Strasbourg. He retired as emeritus professor at Caltech in 1953 and continued to serve as a consultant for a number of industrial companies. The Epstein Memorial Fund was established through donations from more than fifty of his former students and a bronze bust of Epstein stands in the Physics and Mathematics section of the Millikan library at Caltech.

Research

Quantum mechanics In 1922, he published 3 papers, on the explanations of the Stark effect, Zeeman effect and magnetic dipoles using Bohr's quantum theory.

Air and fluid drag As well as quantum theory, Epstein also published papers in other fields. He worked on were the settling of gases in the atmosphere, the theory of vibrations of shells and plates and the absorption of sound in fogs and suspensions. Epstein calculated in 1924 the drag on a sphere moving in a gas in the rarefied (i.e., high, or at least not small, Knudsen number) flow regime. This calculation was an important ingredient in improved measurements of the charge of the electron in the Millikan's oil drop experiment, as cited and described as "masterly" by Millikan himself in his review paper of 1930. Epstein's important result is now known as Epstein drag in honor of his work.

Continental drift Epstein calculated the effect of Polflucht in 1920 as a possible causing force of Alfred Wegener's continental drift. His value was about 10−6 of the gravitational force. Some years later other geophysicists could show that the Polflucht force is far too weak to cause plate tectonics. The toughness of the sublayers of the Earth's crust is much stronger than assumed by Epstein.

Reflection by conducting/absorbing layers In 1930, in two papers in PNAS Epstein addressed reflection of radio waves by conducting layers. Results are also applicable to underwater acoustics, where they gave rise to the notion of an Epstein profile or Epstein layer.

Illustrations

Worked examples

Example 1 — a first encounter with Paul Sophus Epstein

Start with the simplest possible case. Write down what Paul Sophus Epstein claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Paul Sophus Epstein before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Paul Sophus Epstein ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Paul Sophus Epstein

In research: Paul Sophus Epstein appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Paul Sophus Epstein in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Paul Sophus Epstein is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1883 births, 1966 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Paul Sophus Epstein outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Paul Sophus Epstein in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Paul Sophus Epstein means in your own words.
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Frequently asked questions

What is Paul Sophus Epstein in simple terms?

Why does Paul Sophus Epstein matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Paul Sophus Epstein?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Paul Sophus Epstein.