Functions of Operators
Examples
- \(\exp aA = \sum_i \frac{A^i}{i!A}\)
Theorem - Taylor Expansion
If \(F(A)\) is a Linear Operator then you can always Taylor Expand \(F(A)\).
Theorem - Commutations
If \([A,B]=0\) then \([F(A),B] = 0\).
Theorem - Adjoint
\((F(A))^\dagger = F^*(A^\dagger)\).
Corollary
If \(F\) is real and \(A\) is Hermitian, then \(F(A)\) is Hermitian.
Theorem - Eigenfunction
\(F(A)|\psi\rangle=F(a)|\psi\rangle\).
Examples
\(\exp (\frac{1}{i\hbar}\mathcal{H}t) |\varphi_n\rangle = \exp(\frac{1}{i\hbar}E_nt)|\varphi_n\rangle\).