As in the classical case, we assume a solution of the form
where
and we will assume
Substituting into the Liouville equation and working to first order in small quantities, we find
which is a first order inhomogeneous equation that can be solved by using an integrating factor:
(Note that we have chosen the origin in time to be 
, which is an arbitrary
choice.)
For an observable A, the expectation value is
when the solution for 
is substituted in, this becomes
where the cyclic property of the trace has been used and the Heisenberg evolution for A has been substituted in. Expanding the commutator gives
where the cyclic property of the trace has been used again. Define a function
called the after effect function. It is essentially the antisymmetric quantum time correlation function, which involves the commutator between A(t) and B(0). Then the linear response result can be written as
which is the starting point for the theory of quantum transport coefficients. If we choose to measure the operator B, then we find
