American Prometheus revealed the most compelling reason for using both bombs I’ve seen to date:

  Truman had extracted a promise from Stalin that the Soviet Union would declare war on Japan by August 15—an event that he and many of his military planners thought would be decisive. “He’ll [Stalin] be in the Jap war on August 15,” Truman wrote in his diary on July 17. “Fini Japs when that comes about.”

Even more than avoiding a bloody island-hopping invasion of Japan, Truman wanted to avoid having to divide East Asia with the USSR as they had done Europe. The Soviets declared war on Japan and invaded Manchuria on August 7, one day after Hiroshima and two days before Nagasaki. By August 15, the Japanese had surrendered, largely limiting the Soviet sphere of influence to mainland Asia.