The following are some details of linear operator theory that may be useful to the students of PHY 571: Quantum Physics. Many of the rigorous mathematical details have been omitted in the interest of brevity, although some details are indicated.

The connection between the operator methods and the matrix methods is outlined in this page. Specific matrix methods are outlined in another page, whose link will be placed here. 

Linear Operator Theory Overview

All wave or state functions will be considered to be square integrable, unless otherwise indicated. They can also be differentiable, even if very sharp (e.g., the unit step is considered to have the delta function as its derivative.) 

An operator A is linear if for every complex number c and functions u and v, A(cu+v) =cA(u)+A(v). 

A linear space or subspace is a collection of functions such that for all complex numbers c and d and all functions u and v in the linear space, cu+dv is also in the space. 

Two functions are called orthogonal if their scalar product (also called the "inner product") is zero; i.e.,  . 

The magnitude (or "size") of a wave function is the scalar product of the function with itself; i.e.,  . As is the case with the magnitude of real and complex numbers, the magnitude is always non-negative, and is only zero when u=0. The magnitude is more formally called the "norm". This is why a function is called "normalized" when its magnitude is 1. 
 

Eigenvalues

A complex number E is an eigenvalue of A if there is a nonzero function u such that Au=Eu. The function u is called an eigenvector (or eigenstate) of A corresponding to E. The set of all eigenvectors corresponding to E is a subspace, called the eigenspace. Eigenspaces corresponding to different eigenvectors are orthogonal. Eigenstates from orthogonal eigenspaces are orthogonal. 
 

Dimension

A space may be said to consist of "points", "functions", or "vectors". For the purpose of generality, the space will be said to consist of "vectors" in the discussion that follows. The dimension of a space is the smallest number of vectors of the space that can be used to represent any vector in the space. For example, the plane is two-dimensional since every vector can be expressed as a linear combination of the vectors (0,1) and (1,0), but not as a linear combination of fewer vectors. These vectors that are used to represent the other vectors are called a basis. If the basis vectors are pairwise orthogonal, we have an orthogonal basis. If the magnitude of each vector in the basis is 1, then we have an orthonormal basis. 

The scalar product of a vector in the space with a basis vector is the projection of that vector onto the basis vector. Thus, for a vector v and orthonormal basis vectors un, . 

Most of the operators that we will be working with are diagonalizable. What this means is the following: every wave function in the space is in at least one of the eigenspaces. In this case, it is possible to choose representative functions from each eigenspace such that each representative has magnitude one, and such that the representatives are orthogonal to each other. It will then be the case that these representatives form an orthonormal basis for the space of all functions. This allows any wave function to be given as the linear combination of eigenstates. We are used to seeing this in the form . Note that we sum over infinitely many eigenstates un. Since most of our spaces will have infinitely many eigenstates in this representation, our function spaces are infinitely dimensional. 

A linear operator can be represented by a matrix (where the matrix may be infinite). The corresponding matrix is obtained by setting the n,m entry to . Thus, the matrix and the operator representations for linear operators are essentially equivalent. 
 

Adjoint

The adjoint of A, denoted  , is the linear operator such that for all u and v in the space,  . This operator is also sometimes called the conjugate operator. For most of the operators with which we will be dealing in class,  . 

If , then A is called self adjoint. Self-adjoint operators have several nice properties: 

1. All of the eigenvalues of A are real. This can be seen by noting that if Au=ku for nonzero u, then 

2. If the operator is on a finite-dimensional space, its matrix will be Hermitian (M* = M). 

3. i and -i are not eigenvalues for A. 

4. Any function v in the space can be given by Au+iu for some u in the space. 
 

References

ASU offers a course in Linear Algebra (MAT 442) that offers an excellent introduction to linear operator theory. The following text is often used for the course: 

Friedberg, Insel, and Spence, Linear Algebra, 3rd ed., Prentice Hall, 1997.

Another course, Linear Operator Theory (MAT 551), will meet the needs of more advanced students with an understanding of Partial Differential Equations. One text sometimes used for this course is 

Stakgold, Green's Functions and Boundary Value Problems, John Wiley & Sons, 1979.

More information on these courses can be found here. 
Latest version of this document: 10 March 2001, originally produced by Jennifer Trelewicz and Hu Zhan, April 1998.