3. (a) Show that the spherical harmonic Y₁(0, 0) = (³) 1/2 cost is an eigenfunc- tion of the equation: h² a ΟΥ h² ²Y sine + = (1+1)h²Y. sinᎯ ᎧᎾ ae sin²0 ǝo² Give the eigenvalue and associated value for l. მ (b) Given that L+ = hei ( + i cote 2) and Y¹ (0, 0) = =- show that LY (0, 0) is consistent with the expression L+|lm) = [(1+1) - m(m + 1)] 1/2h|l(m + 1)). (c) In this context, 1 and m are related to which operators? (d) What is the significance of the operator L+? Consider a system that is initially in the state: - (3³) 1/2 sinbeid, $(0,0) = ½³×‚¯¹(0, 0) + √7y;º(0, 0) + ½±1;1(0,6). (e) If Ĺ, is measured, what values would be obtained and with what probabil- ities? (f) The equation given in part (a) is an eigenvalue equation for a quantum mechanical operator. What is the physical significance of the finding that this operator commutes with the Ĺ₂ operator?

3. (a) Show that the spherical harmonic Y₁(0, 0) = (³) 1/2 cost is an eigenfunc- tion of the equation: h² a ΟΥ h² ²Y sine + = (1+1)h²Y. sinᎯ ᎧᎾ ae sin²0 ǝo² Give the eigenvalue and associated value for l. მ (b) Given that L+ = hei ( + i cote 2) and Y¹ (0, 0) = =- show that LY (0, 0) is consistent with the expression L+|lm) = [(1+1) - m(m + 1)] 1/2h|l(m + 1)). (c) In this context, 1 and m are related to which operators? (d) What is the significance of the operator L+? Consider a system that is initially in the state: - (3³) 1/2 sinbeid, $(0,0) = ½³×‚¯¹(0, 0) + √7y;º(0, 0) + ½±1;1(0,6). (e) If Ĺ, is measured, what values would be obtained and with what probabil- ities? (f) The equation given in part (a) is an eigenvalue equation for a quantum mechanical operator. What is the physical significance of the finding that this operator commutes with the Ĺ₂ operator?