Abstract angular momentum operators: In this problem you may assume when required the commutation relations between the general angular momentum operators J, J, J. Also recall that J = JiĴy. Use whenever possible the orthonormality properties of normalised angular momentum eigenstate j, m), namely (j', mj, m) = 8; j6mm, and the facts that: j,m)=2jj+1)j, m), Jaj, m) = lim|j, m) and \j, m) x \j, m±1). == (a) The state of the system is j, m). Write down without calculation the expectation values of J and J, and the uncertainties A. and AJ.. (b) Assuming the system is in the eigenstate lj, m) find the expectation value (+). Hence, find the expectations values of J, and J, in the same state. (c) Show that do not commute.) (J+J+2J2-2J?). (Keep in mind that the angular momentum operators (d) Now, use the result from (c) to find the expectation value (2) if the system is in the state j, m). Using the result from (b) find the uncertainty AJ.

Abstract angular momentum operators: In this problem you may assume when required the commutation relations between the general angular momentum operators J, J, J. Also recall that J = JiĴy. Use whenever possible the orthonormality properties of normalised angular momentum eigenstate j, m), namely (j', mj, m) = 8; j6mm, and the facts that: j,m)=2jj+1)j, m), Jaj, m) = lim|j, m) and \j, m) x \j, m±1). == (a) The state of the system is j, m). Write down without calculation the expectation values of J and J, and the uncertainties A. and AJ.. (b) Assuming the system is in the eigenstate lj, m) find the expectation value (+). Hence, find the expectations values of J, and J, in the same state. (c) Show that do not commute.) (J+J+2J2-2J?). (Keep in mind that the angular momentum operators (d) Now, use the result from (c) to find the expectation value (2) if the system is in the state j, m). Using the result from (b) find the uncertainty AJ.