The paper explores the fractional Schrödinger equation, a generalization of the standard Schrödinger equation that incorporates fractional derivatives. The authors prove the hermiticity of the fractional Hamiltonian operator and establish the parity conservation law for fractional quantum mechanics. They derive the energy spectrum for a hydrogen-like atom (fractional "Bohr atom") and the energy spectrum of a 1D fractional oscillator in the semiclassical approximation. A new equation for the fractional probability current density is also developed. The paper discusses the relationship between the fractional and standard Schrödinger equations, showing that the fractional quantum mechanics reduces to the standard quantum mechanics when the Lévy index \(\alpha\) is set to 2. The fractional Schrödinger equation provides a broader perspective on the statistical properties of quantum paths and the fundamental equations of quantum mechanics.The paper explores the fractional Schrödinger equation, a generalization of the standard Schrödinger equation that incorporates fractional derivatives. The authors prove the hermiticity of the fractional Hamiltonian operator and establish the parity conservation law for fractional quantum mechanics. They derive the energy spectrum for a hydrogen-like atom (fractional "Bohr atom") and the energy spectrum of a 1D fractional oscillator in the semiclassical approximation. A new equation for the fractional probability current density is also developed. The paper discusses the relationship between the fractional and standard Schrödinger equations, showing that the fractional quantum mechanics reduces to the standard quantum mechanics when the Lévy index \(\alpha\) is set to 2. The fractional Schrödinger equation provides a broader perspective on the statistical properties of quantum paths and the fundamental equations of quantum mechanics.