Problem 1: Single spin in a dc magnetic field. The Hamiltonian for a spin-1/2 electron in a magnetic field, B, is given by 1 H = - B = 39H30 B, where the magnetic moment, = - 29M30, is written in terms of the Land g-factor, g, the Bohr magneton, Mb = e/2me, and the Pauli spin operators, o = (Ox, Oy, 02). a) Write down the matrix and outer product (ket-bra) representations of the 3 Pauli spin operators as well as the identity operator. b) Find the expectation value of the spin projection, (o) = (10x), (Oy), (0z)), for each of the following quantum states: i. 4) = 1T) ii. 14) = 11) iii. 14) = (1T) [))/V2 iv. 10) = (1T) + i||))/V2 Here |T) and (L) are an eigenbasis for Oz, i.e., IT) = (6) and ) = 0. [Note: you can perform these calculations using either the matrix formulation or Dirac bra-ket notation, and you might find it useful to try using both.] c) Evaluate the spin Hamiltonian in matrix form assuming the magnetic field points along the x- direction, B = (Bx,0,0). Find its eigenvalues and the corresponding eigenvectors. The eigenvectors should be written in terms of the z-axis basis vectors from (b). d) Use the Schrdinger equation to evaluate the time dependence for each of the initial quantum states in (b) subject to the Hamiltonian from (c). Then calculate the time-dependent expectation value for the spin projection, (o(t)). Remember that (i are observables and their expectation value must be real show this explicitly. Problem 1: Single spin in a dc magnetic field. The Hamiltonian for a spin-1/2 electron in a magnetic field, B, is given by 1 H = - B = 39H30 B, where the magnetic moment, = - 29M30, is written in terms of the Land g-factor, g, the Bohr magneton, Mb = e/2me, and the Pauli spin operators, o = (Ox, Oy, 02). a) Write down the matrix and outer product (ket-bra) representations of the 3 Pauli spin operators as well as the identity operator. b) Find the expectation value of the spin projection, (o) = (10x), (Oy), (0z)), for each of the following quantum states: i. 4) = 1T) ii. 14) = 11) iii. 14) = (1T) [))/V2 iv. 10) = (1T) + i||))/V2 Here |T) and (L) are an eigenbasis for Oz, i.e., IT) = (6) and ) = 0. [Note: you can perform these calculations using either the matrix formulation or Dirac bra-ket notation, and you might find it useful to try using both.] c) Evaluate the spin Hamiltonian in matrix form assuming the magnetic field points along the x- direction, B = (Bx,0,0). Find its eigenvalues and the corresponding eigenvectors. The eigenvectors should be written in terms of the z-axis basis vectors from (b). d) Use the Schrdinger equation to evaluate the time dependence for each of the initial quantum states in (b) subject to the Hamiltonian from (c). Then calculate the time-dependent expectation value for the spin projection, (o(t)). Remember that (i are observables and their expectation value must be real show this explicitly