The result of a measurement of the observable A must yield one of the eigenvalues of A. Thus, we see why A is required to be a hermitian operator: Hermitian operators have real eigenvalues. If we denote the set of eigenvalues of A by , then each of the eigenvalues satisfies an eigenvalue equation
where is the corresponding eigenvector. Since the operator A is hermitian and is therefore real, we have also the left eigenvalue equation
The probability amplitude that a measurement of A will yield the eigenvalue is obtained by taking the inner product of the corresponding eigenvector with the state vector , . Thus, the probability that the value is obtained is given by
Another useful and important property of hermitian operators is that their eigenvectors form a complete orthonormal basis of the Hilbert space, when the eigenvalue spectrum is non-degenerate. That is, they are linearly independent, span the space, satisfy the orthonormality condition
and thus any arbitrary vector can be expanded as a linear combination of these vectors:
By multiplying both sides of this equation by and using the orthonormality condition, it can be seen that the expansion coefficients are
The eigenvectors also satisfy a closure relation:
where I is the identity operator.
Averaging over many individual measurements of A gives rise to an average value or expectation value for the observable A, which we denote and is given by
That this is true can be seen by expanding the state vector in the eigenvectors of A:
where are the amplitudes for obtaining the eigenvalue upon measuring A, i.e., . Introducing this expansion into the expectation value expression gives
The interpretation of the above result is that the expectation value of A is the sum over possible outcomes of a measurement of A weighted by the probability that each result is obtained. Since is this probability, the equivalence of the expressions can be seen.
Two observables are said to be compatible if AB=BA. If this is true, then the observables can be diagonalized simultaneously to yield the same set of eigenvectors. To see this, consider the action of BA on an eigenvector of A. . But if this must equal , then the only way this can be true is if yields a vector proportional to which means it must also be an eigenvector of B. The condition AB=BA can be expressed as
where, in the second line, the quantity is know as the commutator between A and B. If [A,B]=0, then A and B are said to commute with each other. That they can be simultaneously diagonalized implies that one can simultaneously predict the observables A and B with the same measurement.
As we have seen, classical observables are functions of position x and momentum p (for a one-particle system). Quantum analogs of classical observables are, therefore, functions of the operators X and P corresponding to position and momentum. Like other observables X and P are linear hermitian operators. The corresponding eigenvalues x and p and eigenvectors and satisfy the equations
which, in general, could constitute a continuous spectrum of eigenvalues and eigenvectors. The operators X and P are not compatible. In accordance with the Heisenberg uncertainty principle (to be discussed below), the commutator between X and P is given by
and that the inner product between eigenvectors of X and P is
Since, in general, the eigenvalues and eigenvectors of X and P form a continuous spectrum, we write the orthonormality and closure relations for the eigenvectors as:
The probability that a measurement of the operator X will yield an eigenvalue x in a region dx about some point is
The object is best represented by a continuous function often referred to as the wave function. It is a representation of the inner product between eigenvectors of X with the state vector. To determine the action of the operator X on the state vector in the basis set of the operator X, we compute
The action of P on the state vector in the basis of the X operator is consequential of the incompatibility of X and P and is given by
Thus, in general, for any observable A(X,P), its action on the state vector represented in the basis of X is