Not a dr, but a mom who survived postpartum psychosis.
Dads can help by taking night duty with the newborn; getting the mom to take a nap - don't just ask if she wants one, most moms will say no; doing some of the laundry to give her a break; make sure she has time to take a shower/bath alone, without kids in the bathroom.
Also, know the signs of severe illnesses and when to call the dr for help. If the mom stays in bed all day, cries all the time, or tells you she's had thoughts of hurting herself or the baby, get her help immediately!
I'm sure Dr Tim has more ideas....