The study investigates the electronic structure of the 5$d$ transition-metal oxide Sr$_2$IrO$_4$ using various spectroscopic techniques and first-principles calculations. The system is found to exhibit novel effective total angular momentum ($J_{eff}$) states, where relativistic spin-orbit (SO) coupling plays a crucial role under a large crystal field. Despite the delocalized 5$d$ states, the $J_{eff}$-states form narrow bands that lead to a $J_{eff} = 1/2$ Mott ground state with unique electronic and magnetic properties. This suggests a new class of $J_{eff}$ quantum spin-driven correlated-electron phenomena. The formation of these $J_{eff}$ bands is well-supported by first-principles calculations, which predict a half-filled $J_{eff} = 1/2$ single band system with a Mott gap. The ARPES, optical conductivity, and X-ray absorption spectroscopy (XAS) results confirm the presence of a $J_{eff} = 1/2$ Mott state, characterized by a sharp peak at around 0.5 eV and a broad peak at around 1 eV in the optical conductivity, and an orbital ratio of $xy : yz : zx = 1 : 1 : 1$ in the O 1s XAS spectra. The magnetic properties of the system are also discussed, showing weak ferromagnetism with a canted antiferromagnetic order. This work highlights the importance of strong SO coupling in driving the Mott instability in 5$d$ transition-metal oxides, leading to a new class of materials with unique electronic and magnetic behaviors.The study investigates the electronic structure of the 5$d$ transition-metal oxide Sr$_2$IrO$_4$ using various spectroscopic techniques and first-principles calculations. The system is found to exhibit novel effective total angular momentum ($J_{eff}$) states, where relativistic spin-orbit (SO) coupling plays a crucial role under a large crystal field. Despite the delocalized 5$d$ states, the $J_{eff}$-states form narrow bands that lead to a $J_{eff} = 1/2$ Mott ground state with unique electronic and magnetic properties. This suggests a new class of $J_{eff}$ quantum spin-driven correlated-electron phenomena. The formation of these $J_{eff}$ bands is well-supported by first-principles calculations, which predict a half-filled $J_{eff} = 1/2$ single band system with a Mott gap. The ARPES, optical conductivity, and X-ray absorption spectroscopy (XAS) results confirm the presence of a $J_{eff} = 1/2$ Mott state, characterized by a sharp peak at around 0.5 eV and a broad peak at around 1 eV in the optical conductivity, and an orbital ratio of $xy : yz : zx = 1 : 1 : 1$ in the O 1s XAS spectra. The magnetic properties of the system are also discussed, showing weak ferromagnetism with a canted antiferromagnetic order. This work highlights the importance of strong SO coupling in driving the Mott instability in 5$d$ transition-metal oxides, leading to a new class of materials with unique electronic and magnetic behaviors.