Quantum mechanics is one of two cornerstones of modern physics (the other is general relativity), and has been precisely confirmed in thousands of very exacting experiments.
Many isotopes have been studied, probing a wide range of time scales.
The isotope C and counting the amount of each) allows one to date the death of the once-living things.
Perhaps you have heard of Ice Man, a man living in the Alps who died and was entombed in glacial ice until recently when the ice moved and melted.
The man's body was recovered and pieces of tissue were studied for their C content by accelerator mass spectroscopy.
The best estimate from this dating technique says the man lived between 33 BC. From the ratio, the time since the formation of the rock can be calculated.
The boat of a pharaoh was discovered in a sealed crypt and reassembled in a museum near the pyramids (see Fig. The age of our galaxy and earth also can be estimated using radioactive dating.
Using the decays of uranium and thorium, our galaxy has been found to be between 10 and 20 billion years old and the earth has been found to be 4.6 billion years old. Within experimental error, this estimate agrees with the 15 billion year estimate of the age of the Universe.
A related article on the age of the earth and geologic ages presented the current best known values for these dates: Ages.
The figures shown in that article are based on radiometric dating.
Radiometric dating is rooted in the rates of radioactive decay of various isotopes, which rates have been measured carefully in numerous laboratories beginning in the early 20th century.
Radioactive decay is in turn a very basic physical phenomenon, well understood as a consequence of quantum mechanics.