FOUNDATIONS OF QUANTUM MECHANICS

24 CHAPTER II. THE FORMALISM
A set of projectors P 1 , . . . , P N is called mutually orthogonal if
P i P j = δ ij P i for i, j = 1, . . . , N, (II. 42)
a set of mutually orthogonal projectors is called complete if
N∑
P i = 11. (II. 43)
i=1
In particular, in accordance with (II. 19), for an orthonormal basis |α i ⟩, . . . , |α N ⟩ it holds that the
associated 1 - dimensional projectors form a complete set,
N∑
|α i ⟩ ⟨α i | = 11. (II. 44)
i=1
II. 3
EIGENVALUE PROBLEM AND SPECTRAL THEOREM
If |β 1 ⟩, . . . , |β N ⟩ is an arbitrary orthonormal basis, an operator A is represented in this basis as
an arbitrary N × N - matrix,
A ij = ⟨β i | A | β j ⟩. (II. 45)
A powerful tool for the study of such matrices is obtained if they can be ‘diagonalized’, i.e., if an
orthonormal basis |α 1 ⟩, . . . , |α N ⟩ can be found where the matrix representation of A is of the form
⎛ ⎞
a 1
A = ⎝ . ..
0
⎠ , (II. 46)
0 a N
or, equivalently,
A ij = a j δ ij . (II. 47)
For such a basis it holds that
A |α i ⟩ = a i |α i ⟩. (II. 48)
Equation (II. 48) is called the eigenvalue equation of the operator A, the values a i are called the
eigenvalues of A, the set of eigenvalues of A the spectrum of A, written as Spec A, the vectors |α i ⟩
are called the eigenvectors, and the system |α 1 ⟩, . . . , |α N ⟩ an eigenbasis of A. For a self - adjoint
operator it holds that the eigenvalues are all real, and the eigenvalues are not negative if the operator
is positive. For a unitary operator U all eigenvalues u i ∈ C are on the complex unit circle, |u i | = 1,
for a projector the eigenvalues are 0 or 1.
The eigenvalue equation does, however, not always have a solution. See as an example operator
(II. 29). The conditions under which the equation can be solved are given by the next important
theorem which we mention without proof.