Eigenfunctions of Hermitian operators

Theorems about Hermitian Operators. We now prove some important theorems about the eigenvalues and eigenfunctions of Hermitian operators. 

The point of the above example is that all of our proofs about eigenvalues or eigenfunctions of hermitian operators refer to cases where the eigenfunctions satisfy the requirement that /i/t = 0. Square-integrability guarantees this, but some... 

The following properties of Hermitian operators follow from the definition (1.23). The eigenvalues of a Hermitian operator are real. Two eigenfunctions of a Hermitian operator that correspond to different eigenvalues... 

It can be proved that the eigenvalues of hermitian operators are real. Accordingly, observables are represented by hermitian operators, as that guarantees that the outcome of observations is real. Another property of those operators is that their non-degenerate eigenfunctions are orthogonal, that is... 

Hermitian operator - An operator A that satisfies the relation S u Au dx = (I u Au dx), where indicates the complex conjugate. The eigenvalues of Hermitian operators are real, and eigenfunctions belonging to different eigenvalues are orthogonal. 

Extension of the above proofs to the case of more than two operators shows that for a set of Hermitian operators A,B,C,... there exists a common complete set of eigenfunctions if and only if every operator commutes with every other operator. 

Just as before, we shall be interested in the eigenfunctions and eigenvalues of Hermitian operators. These are defined as operators for which... 

A set of eigenfunctions of a Hermitian operator wifh fhe same eigenvalues can be transformed (linearly combined) into orthogonal functions while remaining eigenfunctions of the operator. 

Operators that correspond to physical observables are Hermitian operators. Eigenvalues and expectation values of Hermitian operators are real numbers. Eigenfunctions of a Hermitian operator are orthogonal functions. Some pairs of operators commute, and some do not. When a complete set of functions are simultaneously eigenfunctions of a set of operators, then every pair of operators in that set commutes. 

As defined in Section 2.5, any hermitian operator, < , signifies a mathematical operation to be done on a wavefunction, v, which will yield a constant, o, if the wavefunction is an eigenfunction of the operator. 

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

The basis functions constructed in this manner automatically satisfy the necessary boundary conditions for a magnetic cell. They are orthonormal in virtue of being eigenfunctions of the Hermitian operator Ho, therefore the overlapping integrals(6) take on the form... 

However, in order to give an unambiguous answer to the question of how one is to calculate the probability of finding a Klein-Gordon particle at some point x at time t, we must first find a hermitian operator that can properly be called a position operator, and secondly find its eigenfunctions. It is somewhat easier to determine the latter since these should correspond to states wherein the particle is localized at a given point in space at a given time. Now the natural requirements to impose on localized states are ... 

If Ip I and ip2 ( fe eigenfunctions of a hermitian operator A with different eigenvalues a and ai, then ipi and p2 are orthogonal. To prove this theorem, we begin with the integral... 

Consider a set of orthonormal eigenfunctions 0, of a hermitian operator. Any arbitrary function / of the same variables as 0, defined over the same range of these variables may be expanded in terms of the members of set 0,-... 

Suppose the members of a complete set of functions tpi are simultaneously eigenfunctions of two hermitian operators A and B with eigenvalues a,- and j3i, respectively... 

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