  We combine the ideas of Dirac's orthonormal representation, Everett's
relative state, and 't Hooft's ontological basis to define the notion of a
world for quantum mechanics. Mathematically, for a quantum system $\mathcal{Q}$
with an associated Hilbert space $\mathbb{H},$ a world of $\mathcal{Q}$ is
defined to be an orthonormal basis of $\mathbb{H}.$ The evolution of the system
is governed by Schr\"{o}dinger's equation for the worlds of it. An observable
in a certain world is a self-adjoint operator diagonal under the corresponding
basis. Moreover, a state is defined in an associated world but can be uniquely
extended to the whole system as proved recently by Marcus, Spielman, and
Srivastava. Although the states described by unit vectors in $\mathbb{H}$ may
be determined in different worlds, there are the so-called topology-compact
states which must be determined by the totality of a world. We can apply the
Copenhagen interpretation to a world for regarding a quantum state as an
external observation, and obtain the Born rule of random outcomes. Therefore,
we present a mathematical formalism of quantum mechanics based on the notion of
a world instead of a quantum state.
