I struggle between two assumptions - either Jack Napier* just "went crazy", but has still somewhere human emotions in him, or the chemicals somehow damaged the parts of his brain which are there for emotions, like the psychopaths' are, so he isn't capable of feeling any (genuine) and became, in fact, a psychopath.

And no - "being crazy" and "being a psychopath" is not the same thing.

Before I forget: for all the people who wanted to pick a "Heath Ledger's Joker isn't a psychopath!" - you're wrong. He is, and it is so obvious that there is no choice for you to pick that mentioned statement. Sorry.

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* I mean by Jack Napier a certain origin story of the Joker. Google it, if you don't know who "Jack Napier" is supposed to be... I highly recommend it for that pick. Also, I act on the assumption that the Heath Ledger's Joker isn't the Jack Napier as we know him, or rather not even him.
We do not know his origin, but I'm sure he didn't fall into a vat of chemicals, which made his face look like that (obviously, it is make up), or his mouth scarred.

^ This .

Yes, what do you think - he is (also) the Joker!