With the limited information you've posted I can only come to the conclusions I have with that. If they're incorrect generalisations, you should post more info.
From what I can deduce is, you get home from work a little after midnight and spend then time on your own until 3am. You've said you do things like relax, therefore I assume you can just choose not do that and go to bed earlier and then wake up earlier because the way I see it is probably how your wife sees it. You could get up earlier and spend time with her and the kid. Instead what you appear to be doing is going to bed later than required, lying in bed until as late as possible and then, conveniently, that absolves you of all responsibility in the home during these long shift times. On top of that, you're dropping the "I work" card to claim you're doing your bit on her. Sorry, that's the worst thing to say and do, she'll resent that comment, I've done it myself.
I'm sorry you're not receiving from me a response you probably hoped. To me you appear to be seeking sympathy and I dare GENERALISE that you want everyone to say "yeah your wife is being an unreasonable cow, tell her to do one". I'm afraid, given your responses, it sounds like she's the one with her work cut out. When does she relax? You say you get 6hrs sleep, I still think that's pretty reasonable! You know it's a difficult job, you clearly need to be putting more effort in, sorry.
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