Quantum Mechanics: Concepts and Applications · Problem 5.27

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · angular-momentum-l1-lx-eigenstates

Problem

Consider a particle whose angular momentum is

l = 1

. (a) Find the eigenvalues and eigenvectors,

|1,\ m_x\rangle

, of

\hat{L}_x

. (b) Express the state

|1,\ m_x = 1\rangle

as a linear superposition of the eigenstates of

\hat{L}_z

. Hint: you need first to find the eigenstates of

L_x

and find which of them corresponds to the eigenvalue

m_x = 1

; this eigenvector will be expanded in the

z

basis. (c) What is the probability of measuring

m_z = 1

when the particle is in the eigenstate

|1,\ m_x = 1\rangle

? What about the probability corresponding to measuring

m_z = 0

? (d) Suppose that a measurement of the

z

-component of angular momentum is performed and that the result

m_z = 1

is obtained. Now we measure the

x

-component of angular momentum. What are the possible results and with what probabilities?