FOUNDATIONS OF QUANTUM MECHANICS

38 CHAPTER II. THE FORMALISM
To introduce the adjoint of an operator we first delimit the domain. Let Dom A † be the set of all
vectors |ϕ⟩ such that a vector |η⟩ exists for which
⟨ϕ | A | ψ⟩ = ⟨η | ψ⟩ ∀ |ψ⟩ ∈ Dom A. (II. 121)
Using the assumption that Dom A is dense in H it is possible to show that if such a vector |η⟩ exists
it is also unique. The adjoint A † of operator A is now, by definition, the mapping
A † : |ϕ⟩ ∈ Dom A † ↦→ |η⟩ := A † |ϕ⟩, (II. 122)
and the operator is called self - adjoint if
A = A † and Dom A = Dom A † . (II. 123)
This requirement is stronger than Hermiticity; it can be shown that in general it holds for Hermitian
operators that Dom A ⊂ Dom A † , instead of (II. 123).
EXERCISE 15. Verify that the domain of P † , with P as in the example above, is indeed larger
than the domain of P .
II. 6. 2. 2
CONTINUOUS SPECTRA
Another aspect in which infinite - dimensional Hilbert spaces deviate from finite - dimensional
ones is the possibility for an operator to have a continuous spectrum, a mathematical impossibility
in the finite - dimensional case since the term ‘spectrum’ was defined as the set of eigenvalues of
operators. Examples of operators with continuous spectra are, again, the position operator and the
momentum operator, whose spectra consist of the entire line of real numbers R. Therefore, the
term ‘spectrum’ needs to be redefined. The spectrum of operator A is now defined as the set of all
values λ ∈ C for which the operator A − λ11 has no inverse operator. To illustrate the deviations from
the finite - dimensional case we give two examples, the angle operator and the angular momentum
operator.
EXAMPLE
Consider the Hilbert space L 2( [0, 2π] ) and the angle operator
Q : ψ(q) ↦→ q ψ(q), 0 q 2 π. (II. 124)
This operator has, analogous to (II. 112), eigenfunctions which are not in H, its spectrum is the
interval [0, 2π], but it is bounded, ∥Q∥ = 2π.