Even if a battery is new, it could have been sitting on a shelf for a year. Also, a new alternator may test fine on the assembly line but one component may randomly fail when it is put into service.
When I bought the battery the battery gauge read the correct voltage and the car started and ran fine. It took a few hours of driving until the voltage started dropping.
* Check the connectors to the alternator. Maybe one of the pins is not seating properly or a wire is broken.
Those were good looking. I didn't test the wires, as I don't know how but from the looks of it the wires looked fine.
* Check the connections to the battery. Corrosion is often a problem in the ground wire: it looks fine at the terminal connector but there is all sorts of green corroded wire hiding further down in the cable.
Connections are solid. New battery and new terminal ends. Not saying farther down is bad, just everything looks good from what I can see.
* Check the grounds on the motor. If you have a loose connection then this will cause all sorts of issues. When you stab the throttle the motor rotates a little and this can improve a loose connection. Of course, the higher RPM helps the alternator to charge the battery, too. Note that the alternator needs a minimum voltage from the battery to stimulate recharging and the output of the alternator is not constant (it increases with RPM).
I didn't check the grounds yet. When I do hit the throttle the voltage increases back to where they should be.
* Do you have an aftermarket stereo or alarm? If so, the wiring has to be checked because there might be a leak to ground. You did not mention why the battery was replaced but aftermarket electronics/wiring can be a cause of battery problems.
No aftermarket stereo or alarm. PATS is actually turned off. I replaced the battery because the battery light was flashing and voltage was dropping and inconsistent. I bought a new alternator first because the car had sat for winter and part of spring with little to no driving (probably 6-8 months long). I figured the battery needed a good charge and I figured I wasn't getting one from the current alternator (battery voltage was inconsistent with the factory alternator). The new alternator didn't work so I bought a new battery thinking it was done.
The battery is so dead I couldn't even get it jumped. Didn't even want to crank even though it was reading the correct voltage when the jumper cables were hooked up. I had just bought the battery that day and by the next morning it was completely drained.