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Properties of Hermitian Operators
If A and B are Hermitian, then
A + B is Hermitian
[A, B] is anti-Hermitian
The Symmeterized Sum ½ (AB + BA) is Hermitian
if additionally [A, B] = 0, AB is Hermitian
If A and B are anti-Hermitian, then
iA is Hermitian
[A, B] is Hermitian
if additionally [A, B] = 0, AB is Hermitian
Theorems Concerning Hermitian Operators
The eigenvalues of a Hermitian Operator are real.
Two eigenfunctions of a Hermitian Operator that correspond to different
eigenvalues are orthogonal.
Eigenfunctions of a Hermitian Operator that belong to a degenerate eigenvalue
can always be chosen to be orthogonal.
Commuting Hermitian Operators have simultaneous eigenfunctions.
The set of eigenfunctions of any Hermitian Operator
(physical observables) form a complete set.
Postulates of Quantum Mechanics
The “State” of a system is described by a function Y of the coordinates and the time.
This function, called the state function or wave function, contains all the information
that can be determined about the system. We further postulate that Y is singlevalued, continuous, and quadratically integrable. For continuum states, the
quadratic integrability requirement is omitted.
To every physical observable there corresponds a linear Hermitian operator. To find this
operator, write-down the classical mechanical expression for the observable in terms of
[cannonical coordinates], and then replace each coordinate x by the operator [multiply
by x] and each momentum component p by the operator –I (h-bar) d/dx.
The only possible values that can result from measurements of the physically observable
property B are the eigenvalues bi in the equation Bgi = bigi, where B is the operator
corresponding to the property B. The eigenfunctions gi are required to be well-behaved.
If B is any linear Hermitian operator that represents a physically observable property,
then the eigenfunctions of gi of B form a complete set.
If Y(q, t) is the normalized state function of a system at time t, then the average value
of a physical observable B at time t is:
<B> = Int (Y* BY dt)
The time development of the state of an undisturbed quantum-mechanical system is
Given by the Schrodinger time-dependent equation:
- (h-bar)/i dY/dt = HY
where H is the Hamiltonian (that is, energy) operator of the system.