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On 15 August, six days after the bombing of Nagasaki, Japan announced its surrender to the United States, signing the instrument of surrender on 2 September, officially ending the war.
The question is: Why didn't the United States drop another atomic bomb between 9 August and 15 August, at the risk of not having the war ended in a short time? How certain are we that the United States didn't have a third atomic bomb to drop over Japan?
Ray Monk in Inside the Center: The Life of J. Robert Oppenheimer tells the story thus:
Truman returned to Washington from Potsdam on the evening of 7 August and was immediately caught up in a whirlwind of activity generated by Groves, who was determined to proceed as quickly as possible with a second bombing of Japan. He and Admiral William Purnell, Groves writes in his autobiography, 'had often discussed the importance of having the second bomb follow the first one quickly', so that the Japanese would not have time to recover their balance'. This second bomb would have to be of the Fat Man type, there being no chance of assembling another uranium bomb at this stage (in fact, the Little Boy remained one of its kind; the Fat Man design, despite its complicated assembly, being easier to manufacture, safer to transport and more powerful). After the success of the Trinity test, the only thing standing in the way of using a Fat Man bomb in Japan was the availability of plutonium. Groves had originally been advised that a plutonium bomb could be ready to use on August 20. At the end of July, this was revised to 11 August. Groves, however, was too impatient to wait that long and, somewhat against the advice he was given by the scientists, saw to it that the bomb was assembled, loaded and ready to use by the evening of 8 August [...]
Immediately after the Nagasaki bombing the Allies did not possess any more atomic bombs. It is true, as Groves puts it, 'our entire organization both at Los Alamos and at Tinian was maintained in a state of complete readiness to prepare additional bombs', but, as he himself reported to General Marshall, the earliest data at which the next bomb could be assembled for use was 17 August, and almost everybody expected the war to be over by then.
BTW, there are also interesting original documents available from the Truman Library. And if you want my personal opinion: two such bombs were more than enough.
A third bomb was in production, though it wouldn't have been ready until after the 15th. Truman specifically rescinded authorization for atomic bomb release prior to the 15th, though.
On August 10, 1945, one day after the second bomb was dropped on Nagasaki, General Groves, the director of the Manhattan project, sent a memo to Truman's chief of staff stating that they were ahead of schedule for the third bomb (another fat man), and it would be ready to drop as early as August 17th. Truman responded, via his chief of staff, that a third, or any further bomb, should not be dropped without his (the president's) express authorization. The text is as follows
The next bomb of the implosion type had been scheduled to be ready for delivery on the target on the first good weather after 24 August 1945. We have gained 4 days in manufacture and expect to ship from New Mexico on 12 or 13 August the final components. Providing there are no unforeseen difficulties in manufacture, in transportation to the theatre or after arrival in the theatre, the bomb should be ready for delivery on the first suitable weather after 17 or 18 August.
The hand written response was:
It is not to be released on Japan without express authority from the president
