After living in France and Italy, the Koopmans immigrated to the United States in 1915. Koopman was a student of George David Birkhoff and his initial work concentrated on dynamical systems and mathematical physics. In 1931/1932, Koopman and John von Neumann proposed a Hilbert space formulation of classical mechanics, known as the Koopman–von Neumann classical mechanics.
During World World War II, he joined the Anti-Submarine Warfare Operations Research Group (ASWORG, later ORG) in Washington, District of Columbia, directed by Philip M. Morse, to work for the United States. Navy.
The work of Koopman and his colleagues at ASWORG concerned the development of techniques for the United States Navy to hunt U-boats. The theoretical work laid the foundations for search theory which subsequently became a field of its own within operations research.
Their results remained classified Confidential for many years after the war. After 1955 Koopman set out to publish three articles on easily declassifiable portions of the work in the Journal of the Operations Research Society of America.
He wrote down the results in detailed form in the book Search and Screening which was declassified in 1958.
A large part of his work is a systematization of the work performed by his group at ASWORG. The portions on optimum allocation of search effort and on probabilistic aspects of search theory were developed by Koopman himself. The Pitman–Koopman–Darmois theorem states that the only families of probability distributions that admit a sufficient statistic whose dimension remains bounded as the sample size increases are exponential families.
Koopman"s mother, née Louise Osgood, was a first cousin of William Fogg Osgood.