You're wrong, but there is admittedly more to it. In addition to forcing the Japanese to surrender, the bombs were used to keep the Soviets out. They were imminently prepared for a ground invasion by August, and the use of the weapons was authorized by the author of our first containment policy President Harry Truman.
I'm not so sure about that. By that point in the war, the USSR had the most powerful army in the world, but its navy sucked. Especially in the Pacific, since quite a few of its ships had been sent to reinforce the Baltic. I was poking around on some alternate history websites a month ago, and it looks like if the bomb hadn't been used, the Russians probably wouldn't have been able to invade Hokkaido until the spring of 1946. And, even then, Hokkaido is basically the Alaska of Japan. The Russians might have been able to help with the invasion of Honshu later in 1946, but, if so, they'd probably would have been doing it in American and Canadian vessels.
If we hadn't used the bomb, there wouldn't be too much "North Japan" to worry about (just Hokkaido, which, present day, only has a population of 5 million), but Korea might have been unified under the Communists.