Schmidt Jr. attended St. Bernard's School, Phillips Exeter Academy, and Yale University. Upon completion of his undergraduate studies, Schmidt attended Yale Law School. Following graduation, he became a law clerk for Chief Justice of the United States Earl Warren, and thereafter worked for two years in the U.S. Department of Justice, before joining Columbia Law School as a faculty member. His expertise as a legal scholar includes the First Amendment to the U.S. Constitution, history of race relations in American law, and history of the U.S. Supreme Court.