FOUNDATIONS OF QUANTUM MECHANICS

The angular momentum operator
II. 6. ADDENDUM: INFINITE - DIMENSIONAL HILBERT SPACES 39
L : ψ(q) ↦→ − i d ψ(q), (II. 125)
dq
with domain
Dom L =
{
}
ψ : ∥L ψ∥ < ∞, ψ(0) = ψ(2π) , (II. 126)
does have normalized eigenfunctions,
ψ(q) = 1 √
2 π
e i l q , (II. 127)
and a discrete spectrum l ∈ Z. But, since l can be arbitrarily large, it is unbounded.
II. 6. 2. 3
SPECTRAL THEOREM
Von Neumann succeeded in proving the spectral theorem, in the version of II. 3. 1, for infinite -
dimensional Hilbert spaces for which we can formulate the theorem now.
SPECTRAL THEOREM:
To every normal operator A, bounded or unbounded, corresponds a unique mapping of
subsets of Spec A to the set P (H) of projectors on H, ∆ ↦→ P A (∆), having the following
properties:
(i) P ∅ = 0
(ii) P C = 11
(iii) P ∪i ∆ i
= ∑ i
P ∆i for all ∆ i mutually disjoint. (II. 128)
For the position operator Q we have an explicit expression for the spectral family of eigenprojectors
of Q,
P Q (∆) ψ(q) =
{ q ψ(q) if q ∈ ∆
0 otherwise
, (II. 129)
hence, P Q (∆) is in fact a multiplication with the characteristic function of ∆. The spectral family of
the momentum operator is obtained by applying a Fourier transform to the aforementioned expression.
The probability of finding upon measurement for the physical quantity A, which corresponds to
the normal operator A if the physical system is in the state ψ ∈ H, a value a ∈ ∆ ⊂ R, is
Prob ψ (A : ∆) = ⟨ψ | P A (∆) | ψ⟩, (II. 130)