Question

An electron in a Hydrogen atom is in a state with orbital angular momentum 2 (a) Using the general raising and lowering operator formalism e.g Construct the linear combinations of mi ms states which have 2) j 5/2,my 3/2 3) j-3/2, m,-3/2 (b) An external magnetic field B is applied in the z-direction. The interaction between the external field and the magnetic moment of the electron is given by Hmag_ 2mc Find the energy splitting induced between the states (1) and (2) of part (a) above. Assume a weak external B-field. Ignore the fine-structure correction

Answer 1

Given, l = 2: m_l = -2, -1, 0, 1, 2 for hydrogen atom, S = 1/2: M_s = + 1/2, -1/2 Total angular momentum, j = 2 exclus 1/2 = 5/2, 3/2: M_j(j = 5/2) = -5/2, -3/2, -1/2, 1/2, 3/2, 5/2 M_j(j = 3/2) = -3/2, -1/2, 1/2, 3/2Vertical-bar j, M_j > = sigma_M_l, M_s(_m_l, m_s Vertical-bar M_l, M_s > Objective is to find,Vertical-bar 5/2, 5/2 > = sigma_M_l, M_s() Vertical-bar M_l, M_s > Vertical-bar 5/2, 3/2 > = sigma () Vertical-bar M_l, M_s > Vertical-bar 3/2, 3/2 > = sigma () Vertical-bar M_l, M_s > Using ladder operator method Note: While adding two angular momentum, out of all possible Combinations of M_l and M_s Only M_l + M_s = M_j alone will be non-zero Components Rest of them will be zero \r\n There are 5 M_l 's and 2 M_s 's are available, So unique Combination among them are, ^5C_2 = 5!/3! 2! = 5 times 4 times 3!/3! 2! = 10 ways (-3/2) (-5/2) (-1/2) (-3/2)