We roughly knew when the bombs would be ready. It had already been decided to drop them on Japan, on the simple grounds that Japan was NOT as advanced as Germany and if the bomb did not WORK, and fell into enemy hands, the Japanese had less technical ability to figure out what we did wrong and have an atomic bomb themselves.
Thus by the time of FDR's death the decision to drop the bomb had already been made. The targets had also been made (The B-29s bombing Japan from 1944 onward had strict order NOT to bomb those cities, no one told them why, but the reason was we wanted to see HOW destructive were these weapons and not have to worry about prior damages done by other bombs).
This was reinforced by Truman's fear of the Soviet Union. In the Potsdam conference, Truman had Stalin agree to attack Japan within 3 months of the end of the war in Europe (War in Europe ended officially in May 8th 1945, three months later was August 8th, the Day the Russian Attack Manchuria). The thinking in the US was the Russian intervention would force Japan's hand (and it appears to have been the main reason for the Surrender NOT the atomic bombing), but Truman saw the Atomic Bomb as a way to force Japan to surrender without the Russians.
Analysis of Japanese leadership and their papers indicate that the First Atomic Bombing was a Shock to the Japanese leadership, but a Shock they recovered within a few days. The Second Atomic Bombing seems to have had no affect on the Japanese leadership.
On the other hand in between the two bombings the Russians had attacked Manchuria, and the attack was massive and within a week it was clear Russia would have all of Manchuria by September 1st, and All of Korea by October 1st and they was NOTHING the Japanese could do about it (The Soviet Union also had plans to attack the northern most main island on November 1st, the same day the US was planning to invade the Southernmost Main Island, neither invasions took place).
Now, since the Surrender of Germany, Japan was waiting for Russia to give it an Ultimatum, at which time Japan was planning to agree to the demands of the Soviet Union and ask the Soviet Union to act as middle man for talks with the US. Stalin's decision just to attack not give an Ultimatum appears to have been an option no one in the Japanese leadership thought was possible. When Stalin did just attack, the Japanese Government were shock even more then by the First Atomic Bombing.
The Speed of the Soviet Attack also caught the Japanese by Surprise, now the Japanese had been pulling troops out of Manchuria for the defense of Japan, so the Japanese Army in Manchuria in 1945 was NOT the Japanese Army of 1940, but it still was the largest single Army Japan possessed and in August 1945 suffered the most causalities of any Japanese force during WWII (Some of the actions in the islands were more intense on a per soldier basis, but in total number Manchuria was the most severe loss).
Thus Truman had an option, leave the Russians attack and see what the Japanese did at that time.
On the other hand, the US had the bomb and Truman was under pressure to use it, more as a show of Force against the Soviets then anything to do with the war with Japan. The pressure was from the Military (most of the Generals and Admirals were far right wing even then) and leading members not only of the GOP but his own party (and many of the leading advisers of FDR who had stayed on after FDR's death).
In my mind, Truman had not choice. The GOP was already running of "Who lost Eastern Europe" and the American People did not want to hear that the US Army was at the end of its supply line, its system of forming up military units was showing why every other Army refused to do it that way (Almost all if not All Us Army units needed extensive retraining so they would work as a team, not as individuals, the replacement on line based on the concept every soldier is interchangeable was making the army come apart at the seems. The US used the same system in Korea with Similar results, and again in Vietnam, but in Vietnam the war dragged on so long that the Army could no longer hide the affect of its policy of replacement on the line and had to adopt what other armies do, move units in and out of combat as units NOT as individuals).
In simple terms the Army was a mess.
In terms of Supply, the US was at its limit, in fact even with peace in Europe from May 1945 onward, the US ended up starving 3000 German POWS to death in the winter of 1945-1946 do to a lack of food, a lack of food do to the massive need for food by everyone in Europe in 1945. US supply lines were stretched ot the breaking point and the US plan was to move the US Army in Europe, back to the USA and then to Japan.
On the other hand the Soviet Union had managed to get itself in order after the disasters of 1941 and was slowly improving its Military. The Soviet Red Army was still depended on US and British aid, but it amounted to no more then 20% of their supplies (and the percentage was dropping even as aid increased in 1945). The Soviet Army was in fact growing in strength. Some question if the Soviet Army could drive out the US Army (the Soviet Army was weaker in equipment, its Tanks and Artillery was more fit for the Steeps then the rolling hills of Germany, while the US Tanks and Artillery was more fit for the rolling hills of Germany. On the other hand it did not have a food problem and it had potential allies in form of the most effective Resistance forces in Western Europe, the French and Italian Communist Parties.
If you ignore the need for US retraining do to the problems with replacements on the line, the US Army was more then strong enough to stop a Soviet Advance into Western Germany. On the other hand, so was the Soviet Army to any Advance into Eastern Europe. The US was stronger in air power, but the Soviet Red Air Force of 1945 was much better then the Nazi Air Force had been in 1944-1945.
A further complication is once past Denmark, who had Naval Superiority became a factor, and that was the US and Britain, Stalin made the Comment when asked why he told the Communists or Yugoslavia and Bulgaria to stop supporting the Communists in Greece, after observing that to hold Greece you need a fleet "and we have no Fleet". In terms of Germany, in any attack past Denmark his forces could be subject to naval attack to their rear (Thus Stalin tried even as late as 1952 to create a united but neutral Germany, with such a Germany he had all his other potential invasion points covered and his hold on the Soviet Union would have been truly total).
A further complication was the near east, Iran and Iraq. Three British independent Brigades (i.e. the same number as in a British Division BUT without Divisional Artillery, engineerings and other Support troops) faced off three Soviet DIVISIONS.
Thus in 1945, the world wide situation was a lot more complicated then if you just look at Japan. Thus the US was facing a world were the Soviets had a lot of Cards, but Truman had that Ace, the Atomic Bomb. Truman thus was force to play his Ace, just to show he could. At that time he justified it as a military target (and maintain that position publicly till the day he died), but after second bombing and a letter from a Church Group raising concerns about the number of dead civilians, Truman stopped all further bombing (a third was set for the end of August, when the next bomb would have been ready).
Truman had a policy of NOT looking back at old decisions and second guess them. In simple terms, accept what you had done in the past based on the grounds you used at the time of the decision not what you know today. Prior to the bombings, Truman appears to have accepted that the Targets were all Military Targets, when he found out these were cities full of Civilians he ordered the Atomic Bombing stopped. I suspect what he called a Military Target and what the Pentagon called Military Targets were two different things and he only realized the difference after the second bombing.
Thus, in his mind he made the right decision based on the information he had. That information was wrong is clear, and once corrected he appears to have reversed his thinking (His refusal to use Atomic Bombs in Korea shows this again, through he did say Atomic Bombs were a weapon that Could be used, he just never authorized the use).
Thus Truman was caught in a decision paradigm that had been set in 1943, advised by people who definition of Military Target included Targets with lots of civilians and little military but was an area where items used by the Military were made, and some how trying to stop Soviet Advance, when the US Army was at its worse point since before Pearl Harbor (Again do to HOW the Army replacement system worked, or in better words didn't work) and a rough understanding of the supply problems the US Military was facing in the Summer of 1945, I do not see him or any one else (including Henry Wallace) making any other choice. To many things was forcing his hands, forces he had no control over, but forces he had to deal with. Thus Truman's decision to drop the bomb.
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