From 1920 to 1925, Bechert studied physics and mathematics at the University of Munich, in the Institute of Theoretical Physics. He received his doctor of philosophy under Arnold Sommerfeld, in 1925. Under a fellowship of the Rockefeller Foundation, from 1925 to 1926, he accomplished postdoctoral studies and research at the Physics Institute of the University of Madrid. From 1926 to 1933, he was an assistant to Sommerfeld, completing his Habilitation in 1930 and becoming a Privatdozent.
While working for his Ph.D., Bechert helped Sommerfeld extend the Bohr model of the atom and determine the atomic cobalt and vanadium energy terms from their respective spectra. He worked in collaboration with Miguel Ángel Catalán and Ludwig August Sommer. His thesis was on the structure of atomic nickel spectra.
In 1933 he was called as an ordinarius professor and director of the Institute for Theoretical Physics at the University of Giessen. From 1945 to 1946, he was rector at the university. He remained at Giessen until 1946, whereupon he was called as ordinarius professor and director of the Institute for Theoretical Physics at the Johannes Gutenberg University of Mainz, where he remained until his retirement in 1969. He was also a member of the State Commission for School Questions on Lecture Courses at Mainz.
Bechert co-authored a number of books with Christian Gerthsen.
Bechert was raised in the Lutheran Church. He had refused to join the Nazi Party, and in promoting his Jewish colleagues at the Institute for Theoretical Physics of the University of Mainz, he angered the Party. He also protected local citizens from the Nazis by hiding them in the surrounding forest.
In 1945, the Allied Military Government for Occupied Territories appointed Bechert mayor of Donsbach, Westerland and the Oberschulrat in Dillingen, where, on behalf of the Allied Military Government, he built a secondary school/high school. While at the University of Mainz, he was a member of the Senate Commission for Atomic Questions. Since the 1950s, he was a member of the Kuratorium of the Deutsche Friedensgesellschaft (DFG) - the German Peace Society. In 1951, Bechert joined Victor Paschkis in founding the Society for Social Responsibility in Science. In 1955, he became a member the Church and Politics working group of the Evangelist Church in Hesse-Nassau. The next year, he joined the Social Democratic Party of Germany – the SPD. From 1957 to 1972, he was an SPD delegate in the German Bundestag. From 1962 to 1965, he was also chairmen of the Committee for Atomic Energy and Water Management. Bechert was also part of the Pugwash movement and member of the World Union for Protection of Life.