PHY4221: Homework 3Due Sep. 25, 20081. Letσy=0-ii0, calculatehσyiandh(Δσy)2i ≡›σ2yfi-›σyfi2for the state being aneigenvector ofσx=0 11 0with the positive eigenvalue.2. The Hamiltonian of a three-level system is described by an operatorH, with the matrixelements:h11=h33=g,h22= 0,h12=h21=g,h23=h32=g, wherehij≡ hei|H|ejiand|eijare the base vectors.(a) A state vector is given by|ψi=c1|e1i+c2|e2i+c3|e3i, find the expectation value of theenergy of the system.(b) Find the eigenvalues and normalized eigenvectors ofH.(c) Verify the completeness and orthogonality relation of the normalized eigenvectors.(d) Rewrite the Hamiltonian in the base vectors defined by eigenvectors ofH, then find theform of the time evolution operatore-iHt/~.(e) What is the general form of an operator
This is the end of the preview.
access the rest of the document.
expectation value, time evolution operator, Hermitian, base vectors, normalized eigenvectors