What is wrong with my 2003 Chevy Impala's Gas Gauge Some Help Please!!?

I have a 2003 Chevorlet Impala and when I start the car the tank is full but then the meter started going to empty about one centimeter every two seconds. So the needle was on full which is where it suppose to be, then it will go back about a centimeter in two second. This repeats until it reaches the half tank then it just stays there i checked around the car for any leaks and i checked around the engine there was nothing wrong can someone please tell me what's going on?
 
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