
The Weyl Algebra, Quantum Theory, and Normal Ordering 149
point spectrum, that is, the set of eigenvectors {φ
j
}
j∈N
to the eigenvalues {λ
j
}
j∈N
is an
orthonormal basis for H . Then one has Hφ
j
= λ
j
φ
j
and U(t)φ
j
= e
−iHt
φ
j
= e
−iλ
j
t
φ
j
.
Since {φ
j
}
j∈N
is an orthonormal basis, a given state ψ(0) ∈ H at time t =0canbe
expanded as ψ(0) =
j∈N
c
j
φ
j
for some coefficients c
j
∈ C. At time t,thestateisgivenby
ψ(t)=U(t)ψ(0), that is, by
ψ(t)=
j∈N
c
j
e
iλ
j
t
φ
j
.
The above paragraph sketched in a brief and abstract way the structure of a quantum
mechanical description. In concrete cases the Hilbert space H will be given as a L
2
(M,dμ),
and one has a correspondence between classical and ...