#### 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.