  Non-relativistic quantum mechanics is reformulated here based on the idea
that relational properties among quantum systems, instead of the independent
properties of a quantum system, are the most fundamental elements to construct
quantum mechanics. This idea, combining with the emphasis that measurement of a
quantum system is a bidirectional interaction process, leads to a new framework
to calculate the probability of an outcome when measuring a quantum system. In
this framework, the most basic variable is the relational probability
amplitude. Probability is calculated as summation of weights from the potential
alternative measurement configurations. The properties of quantum systems, such
as superposition and entanglement, are manifested through the rules of counting
the alternatives. Wave function and reduced density matrix are derived from the
relational probability amplitude matrix. They are found to be secondary
mathematical tools that equivalently describe a quantum system without
explicitly calling out the measuring system. Schr\"{o}dinger Equation is
obtained when there is no entanglement in the relational probability amplitude
matrix. Feynman Path Integral is used to calculate the relational probability
amplitude, and is further generalized to formulate the reduced density matrix.
In essence, quantum mechanics is reformulated as a theory that describes
physical systems in terms of relational properties.
