Quantum mechanics is not based on special relativity at all. In fact, you can show that you can write Hamilton-Jacobi's equation (purely Classical Mechanics) as

Quantum mechanics is not based on special relativity at all. In fact, you can show that you can write Hamilton-Jacobi's equation (purely Classical Mechanics) as a limiting case of Schrodinger's equation. It is true that QM came about initially as a problem with the quantization of the electromagnetic field, but that was a bit of an accident.