He earned his B.A. (1953) and Ph.D. (1958) at Columbia University, where Nobel laureate I. I. Rabi was the head of the physics department. Schwartz became an assistant professor at Columbia in 1958. He was promoted to associate professor in 1960 and full professor in 1963. Tsung-Dao Lee, a Columbia colleague who had recently won the Nobel prize at age 30, inspired the experiment for which Schwartz received his Nobel. Schwartz and his colleagues performed the experiments which led to their Nobel Prize in the early 1960s, when all three were on the Columbia faculty. The experiment was carried out at the nearby Brookhaven National Laboratory.
In 1966, after 17 years at Columbia, he moved west to Stanford University, where SLAC, a new accelerator, was just being completed. There, he was involved in research investigating the charge asymmetry in the decay of long-lived neutral kaons and another project which produced and detected relativistic hydrogen-like atoms made up of a pion and a muon.
In the 1970s he founded and became President of Digital Pathways. In 1972 he published a textbook on classical electrodynamics that has become a standard reference for intermediate and advanced students for its particularly clear exposition of the basic physical principles of the theory. In 1991, he became Associate Director of High Energy and Nuclear Physics at Brookhaven National Laboratory. At the same time, he rejoined the Columbia faculty as Professor of Physics. He became I. I. Rabi Professor of Physics in 1994 and retired as Rabi Professor Emeritus in 2000. He spent his retirement years in Ketchum, Idaho, and died August 28, 2006 at a Twin Falls, Idaho, nursing home after struggling with Parkinson's disease and hepatitis C.