C. Cohen-Tannoudji, B. Diu, F. Laloe, Quanten mechanics 1 & 2

A is called Hermitian (self-adjoint) if A † = A
Example: A = ∂2
∂x 2 i
= A †
(AB) † = B † A † ,because:
dx 3 ((AB) † φ) ∗ ψ =
dx 3 φ ∗ ABψ =
dx 3 (A † φ) ∗ Bψ =
dx 3 (B † A † φ) ∗ ψ
Identities (Exercises ???):
[AB, C] =A[B, C]+[A, C]B , [A, B] † =[B † , A † ]
Baker - Hausdorff:
e A Be −A = B +[A, B]+ 1 2! [A,[A, B]] + ... where eA =
∞
k=0
= 1 k! Ak
If [[A, B], A] =0=[[A, B], B] then:
e A e B = e B e A e [A,B]
e A+B
= e A e B e −[A,B]/2