Hermitian Operators

Important theorems about Hermitian operators
(adapted from Ingvar Lindgren)

  • A hermitian operator has real eigenvalues and the eigenfunctions form a complete basis set.(p17, Sakurai)

  • Eigenfunctions corresponding to different eigenvalues are orthogonal. (p17, Sakurai)

  • Simultaneous eigenfunctions to two commuting Hermitian operators ("compatible observables") form a complete basis set. (p19, p30 Sakurai)

  • Consider two commuting Hermitian operators. Matrix elements of one of the operators vanish if evaluated between eigenfunctions of he other operator corresponding to different eigenvalues. (p30, Sakurai)

  • If a hermitian operator commutes with all components of j, its matrix elements are independent of m (in a jm representation).

The proofs of the last two theorems are outlined below:

[A,B] = 0 A a = a a A a' = a' a'
0 = < a' | AB-BA | a > =(a'-a) < a' | B | a >

[ B, j ] = 0 = > B j+ = j+ B
< j m | B j+ | j m-1 > = < j m | j+ B | j m-1 > = ...

http://fy.chalmers.se/~f3aamp/atom/herm.html, Jan 1996
Ann-Marie.Pendrill@fy.chalmers.se