Whether the atomic bombs were "necessary" depends really on what you define as "necessary." In an absolute sense, nothing is "necessary" — the United States could, for example, have just declared the war over and let Japan keep its occupied lands and that would be that. Obviously that was not what American policymakers (or the general population) thought was in line with American interests, and certainly the nations occupied and attacked by Japan would not have thought that was in their interests as well.

So the question to ask is what aims the US was pursuing, and whether those could have been achieved by other means. The US was explicitly asking for "unconditional surrender," which it knew was a "big ask," and in fact there were people within the US policy structure (and outside of it but important advisors, like Churchill), who thought that was unreasonable if their other goal was a swift conclusion to the war. Even these people, though, thought that the only "condition" acceptable was to be lenient with regards to the postwar disposition of the Emperor and the Imperial House.

But in any event, even with that caveat, there were certainly other options on the table if the US was concerned with avoiding the use of the atomic bombs (for whatever reason, including but not limited to the sparing of civilian life). I have written at some length about these. Whether any of these would have guaranteed that the US would have achieved its aims (either unconditional surrender or almost-unconditional surrender) without the use of the atomic bomb eventually is unknown — we can't "re-run" history as if it were a simulation and see if the Japanese high command would have gone one way or the other under different circumstances. There are some who think that the war would have ended without the atomic bombings, but if you still had the Soviet invasion of Manchuria. There are many who think that Nagasaki was not that important for the Japanese surrender decision, and could have been avoided (this somewhat misses why Nagasaki was bombed, but this is a different question entirely; it was not about some calculated strategic move, but because the weapon was ready to be used, and the assumption was that many atomic bombs would need to be dropped before Japan surrendered — the plan was not "bomb or invade" but "bomb and invade" and their surrender actually caught the US off-guard). I'm personally of the view that we can't really know these things, because the exact circumstances of what actually happened were so very overlapping and feel very "contingent" (could have gone many different directions), so it is pretty impossible to imagine we can "isolate the variables" this way. This does not mean that I think the atomic bombs were "necessary," it just means, I don't think our knowledge of the situation is ever going to be advanced-enough to let us say if they were necessary or not in this specific definition "necessary."

The historical fallacy of this entire line of thinking is that this is not how the US viewed the atomic bombs at all. It was not a big "decision" to use them, they were not in any way seeking to minimize civilian loss of life, they were not in any way seeking to limit the destruction visited upon Japan prior to the use of the atomic bombs. They did not know the end of the war would happen in early August 1945, they did not expect the atomic bomb to just end it, they did not consider it as big of a "moral" question as we do today. The people in charge of the atomic bombing operation saw many good reasons for dropping the bombs and almost no good reasons for not dropping them. It was "overdetermined" in this respect. When we go back and ask whether they thought it was "necessary," we are implying that they cared about that question, which they definitely did not. Maybe they should have — we are allowed to judge them, as people who are living in a world very much shaped by the consequences of their decisions and mindsets. But as a historical question it is sort of moot for this reason, in my mind.

I would not phrase the goal as "psychological damage" (the concept of long-term trauma from war was still in its infancy, and indeed part of the modern idea of PTSD comes out of studies of Japanese atomic bombing survivors done by the psychologist Robert Jay Lifton in the 1960s), but I would phrase it as a "psychological impact," specifically on the Japanese high command and Japanese morale in general. The bombings were not done because they were militarily all that important — what military/industrial targets they destroyed were explicitly chosen to "justify" using such a weapon on an urban target, but the urban target was the real target, because they wanted to "showcase" the power of the new weapon. They did this for multiple reasons, including impressing the Japanese high command and general population (with a hope it would lead to the acceptance of the Allied surrender terms), impressing the Soviet Union with this new weapon (with a hope that it would give the US an advantage in postwar negotiations and power struggles), and impressing the entire world with this new force to be reckoned with (with the hope that it might lead to a better organization of world affairs, possibly treaties to ban the manufacture of such weapons, and so on). Different individuals within the planning process leaned towards different motivations (and any individual could have multiple motivations; people are complex), but these three stand out as major "hopes" on which a proper "understanding" of the weapon — understood by them as a truly horrific display — was necessary. Again, one can judge this through modern eyes (we do not have to accept the terms in which they saw things as being the best or only ones to see them), but it is important to make sense of what their actual goals were at the time, as opposed to imagined or after-the-fact rationales (e.g., a lot of ink is usually spilled explaining why the military base in Hiroshima was a significant target, but it is clear from the targeting documentation that the actual "military value" of Hiroshima was not something that mattered as much to the targeting planners as its size, geography, ability to showcase the power of the bomb, etc.).

Anyway there is obviously much more that can be said about this topic, and you can find a lot of sub-discussions (was Nagasaki necessary? did the Soviet invasion have more to do with surrender than the bombs? why did they choose Hiroshima and Nagasaki specifically? etc.) in the FAQ and /r/AskHistorians archive (most easily searched through a limiting Google query in my experience).