We shall denote the eigenstate
corresponding to the eigenvalue *a *of
an operator

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.