I don't know how accurate this information is but they way it was completely explained to me, the Japanese government and population were not that affected by the first bomb in terms of morale. They had already suffered horrible large-scale bombings (e.g. Tokyo) and regardless of the initial effect, the US leadership had already decided to drop two bombs to give the impression that these bombs could keep coming 2-3 times a week.
Emperor Hirohito was nearly overthrown in an (unsuccessful)coup after he announced the desire to capitulate after the second bomb dropped, as his generals wanted the war to continue. So if anything, the two bombs were nearly not enough.
After the second bomb was dropped, and after the Soviet Union declared war and took Manchuria, defeating what on paper was Japan's strongest remaining army. Two bombs, the Soviet Union in, Manchuria defeated, and there was still an attempted coup to keep Japan from surrendering.
The second bomb coming so quickly was key to changing Hirohito’s mind. The military had argued that Hiroshima was a one time thing, and the US couldn’t possibly build another bomb any time soon.
Even after the second bomb, some Japan's military leadership wanted to keep fighting.
> The Kyūjō incident (宮城事件, Kyūjō Jiken) was an attempted military coup d'état in the Empire of Japan at the end of the Second World War. It happened on the night of 14–15 August 1945, just before the announcement of Japan's surrender to the Allies. The coup was attempted by the Staff Office of the Ministry of War of Japan and many from the Imperial Guard to stop the move to surrender.
> The officers murdered Lieutenant General Takeshi Mori of the First Imperial Guards Division and attempted to counterfeit an order to the effect of occupying the Tokyo Imperial Palace (Kyūjō). They attempted to place the Emperor under house arrest, using the 2nd Brigade Imperial Guard Infantry. They failed to persuade the Eastern District Army and the high command of the Imperial Japanese Army to move forward with the action. Due to their failure to convince the remaining army to oust the Imperial House of Japan, they performed ritual suicide. As a result, the communiqué of the intent for a Japanese surrender continued as planned.
“…many U.S. officials and scientists argued that a demonstration would sacrifice the shock value of the atomic attack, and the Japanese could deny the atomic bomb was lethal, making the mission less likely to produce surrender. Allied prisoners of war might be moved to the demonstration site and be killed by the bomb. They also worried that the bomb might be a dud since the Trinity test was of a stationary device, not an air-dropped bomb. In addition, although more bombs were in production, only two would be available at the start of August, and they cost billions of dollars, so using one for a demonstration would be expensive.”
The bombs barely performed their intended effect being dropped on cities. Even after Nagasaki the Japanese military argued they could continue to fight from underground bunkers.
The first response of the Japanese army to Hiroshima was that it was just a big conventional bomb. The second was to deny that the US had more (the Japanese had an atomic project of their own and knew how hard it was). The third (after Nagasaki) was that they were changing their air defense strategy.
The second bomb coming so quickly was key to convincing Hirohito’s to surrender. The military had argued that Hiroshima was a one time thing, and the US couldn’t possibly build another bomb any time soon.
The main argument against this, which I personally believe, is that the first bomb should have been dropped on a remote Japanese island or area. I don't believe this would have lead to an end to the war but I believe a second one in a more populated area would have, saving Nagasaki at least
The worry was that a remote nuclear explosion would not carry the same impact and they only had two bombs at the moment. It would have been easy for the Japanese militarists to downplay its effect.
And it turns out the Japanese military was exactly in such a deep denial. After Hiroshima they said it would take a longtime for the US to build another. After Nagasaki they said so what, we will just retreat to bunkers and force the Allies to invade. After Hirohito decided to surrender, the military attempted a coup.
Conventional wisdom is that you have to prove you can do it on demand rather than it being a one and done. If you have one, it could be the last one. If you have two, you could have infinity.
And the first one was likely to be a dud, thereby completely removing the psychological effect from further atomic bombs, of which there was only one for the next couple months.
An interesting point here, in the Wikipedia article on the atomic bombings, that Japanese nuclear scientists reported to their government that the US could be expected to have between two and four more such bombs ready to go. They were under no illusions that the US had hundreds of these sitting around.
Not to everybody it wasn't. Some Japanese officers wanted to keep fighting even after two cities were wiped out. (Three if you count the firebombing of Tokyo, which you probably should...)
As proven earlier with Operation Ten-Go / The Battle of the East China Sea, many in the Japanese military preferred spiteful suicidal attacks to surrender, even when the hopelessness of their situation was abundantly clear.
