In Europe, the conflicts ended in May of 1945. After that, the Japanese rejected the Potsdam Declaration, issued by the Americans calling for their surrender. It seemed like the Japanese weren't willing to surrender and, thus, a possible invasion would be necessary. My questions are: was the use of atomic bombs necessary? Were the atomic bombs used to force a surrender from the Japanese and save the lives of the thousands of soldiers who would be deployed in the invasion? Or was there a different objective, possibly to show the nuclear advantage of the USA (mainly to the Soviet Union) and stop the spread of communist ideology (which had been rising notably since the Bolshevik Revolution in 1917), even prior to the Cold War?

Thank you in advance and correct me if I made any wrong assumptions.