I am a 35-year-old dad of two and my wife is about to have our third kid. I ran D1 and was training extremely hard when my first kid was born (she is now 8). Even if it sounds like trolling, you are actually getting some solid advice on this thread.
Here is the deal: life has changed for you. It will eventually get a little more relaxed, but right now you are not free to do what you want, even if you get up at 5am. Since your child is 6-months old, you can probably take him/her on runs at this point. This saved my marriage.
Buy a BOB SUV stroller with the infant seat attachment. Take the baby on 90-120 minute outings each day. Find a safe path or quiet neighborhood and just give your wife some time. I presume you are working and she is staying home with the baby. So she has absolutely NO time to herself. When you are at work, she sees that as your adult time. In a way, it kind of is your adult time. It is an unspoken secret that men get away from the chaos when they go to work (even though it is often even more stressful and hectic).
So get a good running stroller and it is a win/win. My advice is the BOB SUV. I swear it is 3,000 times better than the BOB Revolution or any other running stroller. I have put serious miles on ours. One week I took my (then) four-year-old son on a 100-mile week. Every mile with the stroller. It has been a solid investment.
To close, I'll just say that your life has changed. Trust me when I say that your life has changed for the better. It may not seem that way right now, but it is true. Also, I get the feeling that your wife feels like you are not pulling your weight. Therefore, figure out ways to pull your weight in a way that you enjoy. It can be running, buying a kid-carrier and going on hikes, buying a bike seat (in a few months). I love spending time with my kids, but I never liked just sitting around and doing nothing. So I made a decision to be a more active dad and on weekends I typically take the kids to go do things, even if it is just hitting up the local University fields to kick soccer balls, throw footballs, run on the track and other active things. You can burn through hours and give your wife some rest/free time. It is a win/win, I swear.
So many of my friends think their kid needs to be doing a baby thing at all hours. They don't understand that you can take your kid to a field and just run around. Kids love that shit. They don't always need to be sitting inside or going to some organized play date. Now that my daughter rides a bike, she goes with me to a local 7-mile trail loop and we do two laps almost every weekend. I run and she rides. It is like her favorite thing. We go get a bagel afterwards and it is like our weekly date.
So here is the deal. Talk to your wife and work it out. Give her two hours a day and she'll give you two hours a day.
The first six months are really hard for a dad (for any girls reading this). The dad has nothing to offer and the kid just wants his/her mother. Once the sixth month rolls around, the dad can start offering more. So the OP knows that there is light at the end of the tunnel, I am a dad with two kids and I am running a lot. I sleep until 7am-8am most days and absolutely love hanging out with my kids. Once #3 is born in a few months, my life will get hard for a while, but I know it'll come around. You are going to be a good dad, just be creative. Congrats on the kid. You have no idea how lucky you are. I promise.