Prayer probably works, even if you are an atheist.
Prayer almost certainly helps the person who prays. Studies on the external effects of prayer are mixed. This is just what one would expect if Christianity were true.
First, God is love and answers the prayers of all His children, even the heart cry of an atheist, when this is best. However, the cosmos is very complicated and interconnected, so one action often impacts billions of future events. As a result, Christians know many requests cannot be answered as we wish.
I was once asked why one sees no healings of amputations. Of course, the question is why no recent reports of healings of amputations. There are many well documented miracles, or at least plausible miracles. It is important to recall that even when there is no natural explanation for a miracle, as with some at Lourdes, a person who is very skeptical can always wait, assuming there will be a natural explanation, even if none comes for centuries. A committed anti-theist can (rationally) live in hope of a solution if his anti-theism has other grounds for denying the supernatural than the existence of miracles.
The notion that a healed “amputation” would change minds is implausible. A good anti-theist would postulate an unknown (and very rare!) natural mechanism for such growth since things such as this happen in other species naturally.
Why would healings of amputations be very rare?
Who knows for sure? Here is a suggestion:
The greater the miracle, the greater the implications. The more visible the miracle, the greater the human implications. God acts for what is best in all cases and the greater the chain of events, especially when impacting free will agents (like humans) the less likely the “miracle will be best.” The creation of matter, or the changing of one form of matter to another, has staggering implications. Imagine the results of changing water to wine! Now contemplate creating wine from nothing and the eternal implications of such an act. Telling a cell to behave as it should is different than creating new cells. Could God heal anything? No. God can only heal what would be best to heal in the light of eternity and all the implications.
I am not arguing the loss of a limb is good: it is bad, horrible. We should pray for healing with faith believing that if a healing is best, then any condition can be healed. Still a miracle of creation would be rare, even more rare perhaps than someone being raised from the dead. In modern times, thanks to natural help available, the likelihood of a miracle being better than the status quo is even less.
Second, Christianity postulates a good God with eternity future to make things as just as they can be in a cosmos with free will. We choose badly and God works carefully, slowly, to make the best possible outcome happen out of those choices. Evil is still evil, but good will come . . . Slowly, carefully, over eternity until beauty brings justice to ugliness and goodness gives meaning to evil actions.
Finally, we do not pray as a transaction, but as a conversation with a person. Conversations are good for our relationship with Eternal Love. If I wish to love my wife, then my conversation with her may have “transactional” elements: please help me with the dishes. However, fundamentally I want Hope to choose and flourish. Our conversations are building a relationship. A “no” from Hope, in that context, is as good as a “yes.”
She chooses. I learn her nature. We grow together.
Thank God for prayers.
Lord Jesus Christ, son of God, have mercy on me a sinner.