The fractional Schrödinger equation is a generalization of the standard Schrödinger equation, incorporating fractional derivatives to describe quantum systems with non-Gaussian Lévy motion. The paper discusses the hermiticity of the fractional Hamiltonian operator and establishes parity conservation in fractional quantum mechanics. It derives the fractional Schrödinger equation from the path integral approach over Lévy paths, showing that the fractional Hamiltonian includes a space derivative of order α instead of the second-order derivative in the standard case. The equation is shown to reduce to the standard Schrödinger equation when α = 2. The paper also presents the time-independent fractional Schrödinger equation and applies it to find the energy spectrum of a hydrogen-like atom (fractional "Bohr atom") and a 1D fractional oscillator in the semiclassical approximation. A new equation for the fractional probability current density is derived and discussed. The fractional Schrödinger equation is shown to include the standard quantum mechanics as a special case when α = 2. The paper concludes that fractional quantum mechanics provides a broader framework for understanding quantum systems with non-standard statistical properties.The fractional Schrödinger equation is a generalization of the standard Schrödinger equation, incorporating fractional derivatives to describe quantum systems with non-Gaussian Lévy motion. The paper discusses the hermiticity of the fractional Hamiltonian operator and establishes parity conservation in fractional quantum mechanics. It derives the fractional Schrödinger equation from the path integral approach over Lévy paths, showing that the fractional Hamiltonian includes a space derivative of order α instead of the second-order derivative in the standard case. The equation is shown to reduce to the standard Schrödinger equation when α = 2. The paper also presents the time-independent fractional Schrödinger equation and applies it to find the energy spectrum of a hydrogen-like atom (fractional "Bohr atom") and a 1D fractional oscillator in the semiclassical approximation. A new equation for the fractional probability current density is derived and discussed. The fractional Schrödinger equation is shown to include the standard quantum mechanics as a special case when α = 2. The paper concludes that fractional quantum mechanics provides a broader framework for understanding quantum systems with non-standard statistical properties.