Question

(a) Given a Hermitian operator A with eigenvalues a, and eigenfunctions un(x) [n = 1, 2, ....., N; 0 ≤ x ≤ L], show that the operator exp(iA) is unitary.

(b) Conversely, given the matrix U_mn of a unitary operator, construct the matrix of a Hermitian operator in terms of U_mn.

(c) Given a second Hermitian operator B with eigenvalues b_m and eigenfunctions w,(x), construct a representation of the unitary operator V that transforms the eigenvectors of B into those of A.

Answer 1

(a) As A⁺ = A, A being Hermitian,

{exp (iA)}⁺ = exp (-iA⁺) = exp (-iA) = {exp (iA)}^-1

Hence exp(iA) is unitary.

(b) Let

c_mn = U_mn + U*_mn = U_mn + (U⁺)_mn,

i.e., C = U + U⁺.

As U⁺⁺ = U, C⁺ = C. Therefore C_mn = U_mn + U*_nm is the matrix representation of a Hermitian operator.

(c) The eigenkets of a Hermitian operator form a complete and orthonormal set. Thus any 1 urn) can be expanded in the complete set \(|v_n\rangle :\)

showing that V is unitary. Thus V is a unitary operator transforming the eigenvectors of B into those of A.