  A time fractional quantum framework has been introduced into quantum
mechanics. A new version of the space-time fractional Schr\"odinger equation
has been launched. The introduced space-time fractional Schr\"odinger equation
has a new scale parameter, which is a fractional generalization of Planck's
constant in quantum physics. It has been shown that the presence of a time
fractional time derivative in the space-time fractional Schr\"odinger equation
significantly impacts quantum mechanical fundamentals. Time fractional quantum
mechanical operators of coordinate, momentum and angular momentum were defined
and their commutation relationships were established. The pseudo-Hamilton
operator was introduced and its Hermicity has been proven. Two new functions
related to the Mittag-Leffler function have been introduced to solve the
space-time fractional Schr\"odinger equation. Energy of a time fractional
quantum system has been defined and calculated in terms of the newly introduced
functions. It has been found that in the framework of time fractional quantum
mechanics there are no stationary states, and the eigenvalues of the
pseudo-Hamilton operator are not the energy levels of the time fractional
quantum system. A free particle solution to the space-time fractional
Schr\"odinger equation was found. A free particle space-time fractional quantum
mechanical kernel has been found and expressed in terms of the H-function.
Renormalization properties of a free particle solution and the space-time
fractional quantum kernel were established. Some particular cases of time
fractional quantum mechanics have been analyzed and discussed.
