B.1.1. Free Particle

We shall denote the eigenstate corresponding to the eigenvalue a of an operator a ˆ  by the ket |a  so that .  For the position and momentum operators, we write

                      

Setting , we have .  Taking the hermitian adjoint  of  gives

           

where * denotes the complex conjugate operation.  On the other hand, the left eigenvalue of  is defined by

         

Thus, for a hermitian operator (  ), we have , i.e., all eigenvalues of a hermitian operator are real.  The orthonormality of the eigenstates is written as

                  for  eigenvalues

The completeness condition for a set of orthonormal states is

                                   for a discrete

                         for a continuous

       in general

The states  are elements of a Hilbert space, which is simply a linear complex vector space with a norm.  A complete set such as  spans the space so that

         

where  is called the wave function of the state  in the r-representation.  The uncertainty principle is embodied in the relation

       

where i, j denote cartesian components of the vector operators.  Taking  of the relation gives

       

Þ           

Thus, for ,

               

Using

                 

which can be proved by multiplying both sides by  or  and integrating by parts, we have

                  

or more generally,

               

                  

Hence,

       

               

               

                 

Similarly, for any analytic function f, it is straightforward to show that

         

Consider now

       

which is a partial differential equation for the momentum eigenstate in the r-representation, , with solution

                    (B.8)

where C is some normalization constant depending on the boundary conditions.

Note that

                          (B.10)

which is the position eigenstate in the p-representation.