Postulates of quantum mechanics 371arbitrariness reflects the arbitrariness of the overall normalization of the eigenvectors
ofAˆ. The natural choice for this normalization is〈aj|aj〉= 1, so that the eigenvectors
ofAˆare unit vectors. Therefore, the eigenvectors are orthogonaland have unit length,
hence, they are orthonormal. If some of the eigenvalues ofAˆare degenerate, we can
choose the eigenvectors to be orthogonal by taking appropriatelinear combinations
of the degenerate eigenvectors using a procedure such as Gram-Schmidt orthogonal-
ization to produce an orthogonal set (see Problem 9.1). The last property we need to
prove is completeness of the eigenvectors ofAˆ. Since a rigorous proof is considerably
more involved, we will simply sketch out the main points of the proof. LetGbe the
orthogonal complement space toH. By this, we mean that any vector that lies entirely
inGhas no components along the axes ofH. Let|bj〉be a vector inG. SinceAˆis
defined entirely inH, matrix elements of the form〈bj|Aˆ|ak〉and〈ak|Aˆ|bj〉vanish.
Thus,Aˆ|bj〉has no components along any of the directions|ak〉. As a consequence, the
operatorAˆmaps vectors ofGback intoG. This implies thatAˆmust have at least one
eigenvector inG. However, this conclusion contradicts our original assumption that
Gis the orthogonal complement toH. Consequently,Gmust be a null space, which
means that the eigenvectors ofAˆspanH.^1
The most important consequence of the completeness relation is that an arbitrary
vector|φ〉on the Hilbert space can be expanded in terms of the eigenvectors of any
Hermitian operator. For the operatorAˆ, we have
|φ〉=Iˆ|φ〉=∑
k|ak〉〈ak|φ〉=∑
kCk|ak〉, (9.2.22)where the expansion coefficientCkis given by
Ck=〈ak|φ〉, (9.2.23)and|ak〉〈ak|φ〉is the projection of|φ〉along|ak〉. Thus, to obtain the expansion co-
efficientCk, we simply compute the inner product of the vector to be expandedwith
the eigenvector|ak〉. Finally, we note that any functiong(Aˆ) will have the same eigen-
vectors ofAˆwith eigenvaluesg(ak) satisfying
g(Aˆ)|ak〉=g(ak)|ak〉. (9.2.24)Now that we have derived the properties of Hermitian operators and their eigen-
vector/eigenvalue spectra, we next consider several other aspects of the measurement
process in quantum mechanics. We stated that the result of a measurement of an
observable associated with a Hermitian operatorAˆmust yield one of its eigenvalues.
If the state vector of a system is|Ψ〉, then the probability amplitude that a specific
eigenvalueakwill be obtained in a measurement ofAˆis determined by taking the
inner product of the corresponding eigenvector|ak〉with the state vector,
αk=〈ak|Ψ〉, (9.2.25)(^1) Note that the argument pertains to finite-dimensional discrete vector spaces. In Section 9.2.5,
continuous vector spaces will be introduced, for which suchproofs are considerably more subtle.