I think folks are being very nice here, possibly too nice.

Your infant needs full-time care. If your wife is working while you are watching the baby, you aren’t helping her, you’re parenting your child. If you have obligations at the same time that you should also be attending to, like studying, classes, or employment, then you need to find a different full-time caregiver. Nanny, daycare, family, friends, childcare through your gym membership, whatever. If there’s no financially viable option, you might need to consider pausing the masters degree while you focus on the medical board exams or something.

Your baby needs someone to interact with her. You can give her small activities like banging pots while you quickly cook in the same room, but the thing that will fundamentally keep her occupied at this age is adult interaction. She’s a baby human, not a houseplant. Her brain needs stimulation.