What I think the OP is trying to achieve is a proof of concept based on living things providing the sustenance for other living things in a balanced setting. Conceptually it sounds like it could work for a short while; with the proper lighting the plants should remove the nitrates, the shrimp and snails are scavengers, the fish are basically cannibalistic (not sure if the shrimp will breed). However, the two things I see going against this is the small volume of the tank (this is not a pond where volume will correct for a lot of things) and the lack of fresh nutrients that come from water changes and additives. Other chemicals besides nitrates will build up and the plants may not necessarily remove them. This may work for short while but things start to go bad, it may go bad quickly and not end well.
This reminds me of the those experiments I used to see in the old science books and encyclopedias. As Old Scales mentioned, we have learned a lot since then.
Passionate responses are natural when the word experiment and fish are used in the same thread on a forum such as this one. Apply the same concept on a dog or cat forum and someone will be contacting PETA. This is an interesting query and you certainly got some responses out of it (when I start a thread I am lucky to get maybe one response).
Now if you were to perform daily testing and set limits (such as the experiment ends when the ammonia, nitrites, nitrates reach certain values) and record your findings, then something might be gained from this venture.