Jensen studied physics, mathematics, physical chemistry, and philosophy at the Albert-Ludwigs-Universität Freiburg and the University of Hamburg from 1926 to 1931, and he received his doctorate at the latter in 1932 under Wilhelm Lenz. Jensen completed his Habilitation in 1936 at the University of Hamburg.

Adolf Hitler took power on 30 January 1933. On 7 April of that year the Law for the Restoration of the Professional Civil Service was enacted; this law, and its subsequent related ordinances, politicized the education system in Germany. Other factors enforcing the politicization of education were Nationalsozialistische Deutsche Arbeiterpartei (NSDAP, National Socialist German Workers Party) organizations in academia and the rise of the Deutsche Physik movement, which was anti-Semitic and had a bias against theoretical physics, especially including quantum mechanics. The Party organizations were the Nationalsozialistischer Deutscher Studentenbund (NSDStB, National Socialist German Student League) founded in 1926, the Nationalsozialistischer Lehrerbund (NSLB, National Socialist Teachers League) founded in 1927, and the Nationalsozialistischer Deutscher Dozentenbund (NSDDB, National Socialist German University Lecturers League) founded in 1933. While membership in the NSDDB was not mandatory, it was tactically advantageous, if not unavoidable, as the district Leaders had a decisive role in the acceptance of an Habilitationsschrift, which was a prerequisite to attaining the rank of Privatdozent necessary to becoming a university lecturer.

While all German universities were politicized, not all were as strict in carrying out this end as was the University of Hamburg, where Jensen received his doctorate and Habilitationsschrift. Upon his 1936 habilitation he had been a member of NSDDB for three years, the NSLB for two years, and a candidate for membership in NSDAP, which he received the next year. The university leader of NSLB had made it clear that active participation was expected from Jensen, and that is what they got.

In 1937 Jensen was Privatdozent (unpaid lecturer) at the University of Hamburg and began working with Paul Harteck, Director of the university's physical chemistry department and advisor to the Heereswaffenamt (HWA, Army Ordnance Office) on explosives. Harteck and his teaching assistant Wilhelm Groth made contact with the Reichskriegsministerium (RKM, Reich Ministry of War) on 24 April 1939 to tell them of potential military applications of nuclear chain reactions. Military control of the German nuclear Energy project, also known as the Uranverein (Uranium Club), began on 1 September 1939, the day that Nazi Germany initiated World War II by invading Poland. Harteck, one of the principals in the Uranverein, brought Jensen into the project. Jensen’s main thrust was on double centrifuges for separation of uranium isotopes (see the section below citing internal reports of the Uranverein). Harteck and Jensen developed a double centrifuge based on a rocking process (Schaukelverfahren) to facilitate the separation effect.

In 1941 Jensen was named extraordinarius professor of theoretical physics at the Technische Hochschule Hannover (today, the University of Hanover), and in 1946 he became an ordinarius professor there. In 1949 he was appointed ordinarius professor at the Ruprecht Karl University of Heidelberg; since 1969 he was emeritus praecox. He was a guest professor at the University of Wisconsin–Madison (1951), the Institute for Advanced Study (1952), University of California, Berkeley (1952), Indiana University (1953), California Institute of Technology (1953), University of Minnesota, Twin Cities (1956), and University of California, San Diego (1961).

In 1963 Jensen shared half of the Nobel Prize in Physics with Maria Goeppert-Mayer for their proposal of the nuclear shell model; the remaining half of the prize was awarded to Eugene Wigner for unrelated work in nuclear and particle physics, especially through the application of fundamental symmetry principles.

The following reports were published in Kernphysikalische Forschungsberichte (Research Reports in Nuclear Physics), an internal publication of the German Uranverein. The reports were Classified Top Secret, they had very limited distribution, and the authors were not allowed to keep copies. The reports were confiscated under the Allied Operation Alsos and sent to the United States Atomic Energy Commission for evaluation. In 1971 the reports were declassified and returned to Germany. The reports are available at the Karlsruhe Nuclear Research Center and the American Institute of Physics.

After World War II the denazification process began. When Jensen faced the proceedings, he turned to Werner Heisenberg, a prominent member of the Uranverein, for a testament to his character – a document known as a Persilschein (whitewash certificate). Heisenberg was a particularly powerful Writer of these documents, as he had never been a member of NSDAP, he had publicly clashed with NSDAP and the Schutzstaffel (SS), and he had been appointed by the British occupation authorities to the chair for theoretical physics and the directorship of the Max-Planck Institut für Physik then in Göttingen. Heisenberg wrote the document and convinced the authorities that Jensen had only joined the Party organizations to avoid unnecessary difficulties in academia.