The superposition principle for ordinary differential equations states that for a linear homogeneous ordinary differential
equation, if 
and 
are solutions, then so is 
.
More generally, for any linear operator 
, solutions of 
and 
satisfy 
. Thus responses to different sources can be combined
by taking the same linear combination of the
responses as of the sources. Examples include the addition of gravitational fields
in Newton's shell theorem and velocity
fields in Stokes flow.
In quantum mechanics, superposition means forming a linear combination of state vectors in a complex Hilbert
space. For orthonormal vectors 
and 
, a normalized superposition is
 |
where 
and 
are complex numbers satisfying 
. The corresponding pure-state density
matrix 
generally differs from the probabilistic mixture 
. The former includes cross terms
that describe coherence between the two components. A qubit
provides the simplest example. Solutions of the Schrödinger
equation for the same Hamiltonian also obey the superposition principle because
the equation is linear (Nielsen and Chuang 2000).
