Question

(25 marks) The radial wave function for a hydrogen atom in the \(3 d\) state is given by \(R(r)=A r^{2} e^{-\alpha r}\), where \(A\) and \(\alpha\) are constants. (a) Determine the constant \(\alpha .[\) Hint \(:\) Consider the radial equation given in the lecture note Ch. 9 page 3\(]\) (b) Determine the largest and smallest possible values of the combination \(\sqrt{L_{x}^{2}+L_{y}^{2}}\) for the \(3 d\) state, where \(L_{x}\) and \(L_{y}\) are the \(x\) - and \(y\) -component of the orbital angular momentum, respectively. Express your result in terms of \(\hbar\).

Answer 1

z

Sr NO. EXPERIMENT OBSERVATION Solh (b) in 3d itate n=3 and R(r) = A r² ē ar in 3d state we know 1² = 4 + 4 + 2 L + y Combination (1²+ly where, leond+it 10.) bo ħ ell+1) So, the largest value corresponds to l = 2 ie t 12 (2+1) = 0 =556 1=2, Mez to and the smallest value corresponds to le ħ √ 212+1) - 4 -m m, o z √2 noo (0) d = Z nao we know, radial wave fr, Rls) Ar e a no to gimen no3. for Heaton, z = 1; so, 3. so = Are e ' 'Rl) = Ar 300