FOUNDATIONS OF QUANTUM MECHANICS

36 CHAPTER II. THE FORMALISM
The proof that these vector spaces are complete is not simple, however, the proof that the remaining
requirements for a Hilbert space have been met, is.
These two spaces correspond to two versions of quantum mechanics, where L 2 (R) corresponds
to Schrödingers wave mechanics (1926) and l 2 (N) to the matrix mechanics of Heisenberg, Born, and
Jordan (1925), that is, if we take matrix mechanics in the enriched version of Von Neumann, since
the original version did not contain a ‘state space’. These two versions of quantum mechanics are
mathematically equivalent, see F.A. Muller (1997a, 1997b and 1999) for historical details.
II. 6. 2
OPERATORS
More serious complications occur when introducing of operators on infinite - dimensional Hilbert
spaces. First, we will see that such operators are in general ‘unbounded’, which entails that
they cannot be defined on the entire Hilbert space. Consequently, the definition of sum and product
of operators, as well as their adjoints, becomes more cumbersome, and the terms ‘self - adjoint’
and ‘Hermitian’ no longer coincide. Second, these operators do not always have eigenvectors in H.
Therefore it is more difficult to give a useful version of the spectral theorem.
The second problem is independent of the first, i.e., it can also appear for bounded self - adjoint
operators. ▹
For position and momentum both complications occur together which is shown by an example.
EXAMPLE
Consider the position operator
Q : ψ(q) ↦→ q ψ(q), (II. 112)
and the momentum operator
P : ψ(q) ↦→ − i d ψ(q), (II. 113)
dq
both acting on L 2 (R).
The first problem is that these operators do not map every vector in L 2 (R) to another vector
in L 2 (R). For instance, every non - differentiable function in L 2 (R) is outside the domain of P .
Vice versa, taking for Q, for example, ψ (q) = (a + q) − 3 2 with a ∈ R, we have ψ ∈ L 2 (R),
but Qψ ∉ L 2 (R).
The second problem is that the eigenvalue equation for momentum, −i d dq
ψ (q) = pψ (q), has
solutions ψ (q) ∝ e i pq for p ∈ R, but these functions are not square integrable and therefore
they are not in L 2 (R). Something similar applies to the eigenvalue equation Qψ(q) = q 0 ψ(q)
and its solutions ψ(q) = δ(q − q 0 ).