Question

8. Consider one electron in a 1D box of side L. Its wavefunction is given by из where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, H, of a particle in a 1D box, 2m dx2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is Ψ(x) an eigenfunction of A? If it is an eigenfunction, what is the 9. A linear polyene contains 8 -electrons, and absorbs light with412 nm. b) What is the relative population of molecules in the ground state to first eigenvalue? a) What is the length of the polyene? excited state at T 300 K? Would you say this system is likely to exhibit classical or quantum mechanical behavior?

Answer 1

psi(x) = squareroot 3/4 phi_1(x) + squareroot 3/2 squareroot phi_2(x) + 2 + squareroot 3i/4 phi_3(x) To check the normality, we check sigma_i Vertical-bar C_i Vertical-bar ^2 = 1 Now, (squareroot 3/4)^2 + (squareroot 3/2 squareroot 2)^2 + (2 + squareroot 3i/4)^2 = 3/16 + 3/8 + Vertical-bar 4 - 3 + 4 squareroot 3i Vertical-bar /16 = 3/16 + 3/8 + Vertical-bar 1 + 4 squareroot 3i Vertical-bar /16 = 3/16 + 3/8 + 7/16 = 3 + 6 + 7/16 = 16/16 = 1 yes, the wavefunction psi(x) is normalized wave function \r\n Every normalized wave function is eigen function of H. we know, H^^psi = E psi So, If H^^is the eigen function then energy is its eigen value. Energy of this system is given by < E > = Vertical-bar C_i Vertical-bar ^2 lambda_i lambda_i = n^2 pi^2 h^2/2 m L^2 = Vertical-bar (squareroot 3/4)^2 Vertical-bar times pi^2 h^2/2 m L^2 + Vertical-bar (squareroot 3/2 squareroot 2) Vertical-bar ^2 times (2)^2 pi^2 h^2/2 m L^2 + Vertical-bar (2 + squareroot 3i/4)^2 Vertical-bar times (3)^2 pi^2 h^2/2 m L^2 = [3/16 + 3/8 times 4 + 7/16 times 9]