A quantum Hamiltonian is a self-adjoint operator 
on a Hilbert
space that generates unitary time evolution. In units
with 
,
the evolution operators are 
. For a time-independent Hamiltonian, the Schrödinger
equation is
 |
Self-adjointness, including the choice of domain, ensures that the spectral calculus defines 
for every real 
(Reed and Simon 1978).
A Schrödinger operator is a differential-operator example, whereas a finite quantum Heisenberg ferromagnet has
a matrix Hamiltonian on a tensor
product of finite-dimensional Hilbert
spaces. Its equilibrium density matrix at inverse
temperature 
is 
.
