This is of course one of the justifications American leaders used, and as always the victor gets to set the perceived historical narrative. Politically it was extremely important for the US to believe the bomb materially shortened the war given the huge amount of resources the Manhattan Project had consumed that otherwise could have been invested elsewhere in the war effort, especially when the military had to justify the incredible expense to Congress (adjusted for inflation the total cost is around 30 billion in 2018 dollars). I've recently been reading the excellent "The Making of The Atomic Bomb" by Richard Rhodes which covers the events of this period in much detail.

The US had already been ridiculously effective using firebombing to level Japanese cities with their B-29s - so much so, they actually had to consider slowing down/changing targets to leave enough behind to use the Atomic Bomb on: there was almost nothing left worth hitting in strategic terms. By the time the bomb was dropped Japan was largely a beaten nation already considering surrender, Tokyo a smoldering rubble pile save for the Imperial Palace.

"The bomb simply had to be used -- so much money had been expended on it. Had it failed, how would we have explained the huge expenditure? Think of the public outcry there would have been... The relief to everyone concerned when the bomb was finished and dropped was enormous." - AJP Taylor.

Of course no one can say with certainty, but I certainly don't consider the answer to this question to be a simple one.