For the first atomic bomb: there were multiple motives, and it depends on who you care about (the scientists had different motives than the military who had different motives than the politicians, and the politicians had multiple motives). Certainly there was some consideration that it might end the war swiftly and avoid an invasion. It was also hoped it might end the Pacific war before Soviet entry into it, thus barring the Soviets from some of its spoils. And there were some, like Secretary of State Byrnes, who explicitly stated that one of the benefits would be that it would intimidate the Soviets and help the US in the Cold War. Which of these were dominant? It's not totally clear, because it was so "over-determined": the planners saw a million good reasons to drop the bomb, and almost no representation of the idea that they shouldn't drop the bomb was made to them. So they didn't actually hem and haw over whether they should do it (they were more or less committed to that); they all just felt they ought to, and that "of course, if you make a billion dollar super weapon, you use it."

But the second bomb is more interesting in this respect, because there wasn't much consideration of it at all, and it is unclear that Truman even knew about it. The strike order that Truman saw on July 25th gave the military the power to use more atomic bombs as they became available, without going back for further direction or authority. It is unclear that Truman realized there would be another atomic bomb ready so soon after the first. The military planned the second strike without consulting the White House, and Truman seems to have been surprised by it. Why'd the military want to do it? On the island of Tinian, where the operation was planned, the only considerations that were weighed were those of weather, and bad weather had delayed the Hiroshima bomb and predictions of bad weather had moved the second bombing (originally for Kokura) upward. In other words, they were not going about this in a "strategic" or "political" mindset at all, just as a "tactical" or "operational" one. The strategic thinking had become decoupled from the military activity, is another way to think about it.

There has been a theory that one of the reasons that General Groves, head of the Manhattan Project and architect of the military plans, pushed for using both bombs is because they would justify the huge expenses put out on the project (they used different fuel, and thus each "justified" a different massive facility). I suspect this is still over-thinking it. Groves himself expressed the idea that even two bombs would not likely cause Japan to surrender, and that they might have to use five or more bombs before the war ended. In other words, even thinking of it as "we'll use two bombs" is an after-the-fact rationalization, only necessary when the bombs seem to have "worked" and people are wondering whether or not they were both necessary or not. It does not reflect how the military or scientists saw the situation, in which the atomic bombs were not completely expected to end the war so quickly.

The British had co-signed the US plans under the Quebec Agreement. They supported them. The Soviets said nothing about Hiroshima and Nagasaki at the time; they barely acknowledged them publicly. Privately Stalin accelerated his own atomic bomb program, telling his scientists it was their job to make sure the USSR never ended up vulnerable like Japan had been.

There is a wide literature on the atomic bombings that covers aspects of these questions, but Hasegawa's_ Racing the Enemy_ is the best one-stop-shop for thinking about the various motivations, and Gordin's Five Days in August does a lot of scholarly work in reminding us that many of our approaches to these questions are generated by "after the fact" considerations, rather than capturing the many views of the bomb that were held prior to the sense that they had "worked" in ending the war.

None of these narratives totally support the Zinn/Alperovitz theses that the bombs were just meant to scare the Soviets. But was that part of the motivation? For some of those involved, certainly.

This is definitely a contentious topic and one on which reasonable people can disagree. You can find quite a bit more in our FAQ on the topic, which will hopefully be informative until someone more expert can provide more information.