Question

|jm⟩ are simultaneous eigenstates of J2 and Jz which are normalized and mutually orthogonal. Consider the case j = 1, and use the basis |1, +1⟩, |1, 0⟩, and |1, −1⟩. Write the operators J2, Jz, and Jx in this basis. Use matrix representation.

Answer 1

To write the operators J2, Jz, and Jx in the given basis, we can find their matrix representations. Jz can be represented by a 3x3 matrix using the basis states, J2 can be obtained by squaring Jz's matrix representation, and Jx can be expressed in terms of Jy and Jz and then substituted with their matrix representations.

Step-by-step explanation:

To write the operators J2, Jz, and Jx in the given basis, we need to find their matrix representations. We can start by finding the matrix representation of Jz. Since we are working with j = 1, the dimension of the matrix will be 3x3. Using the basis states |1, +1⟩, |1, 0⟩, and |1, −1⟩, we can represent Jz as:

Jz = ⋅0 0 0 ⋅
⋅0 1 0 ⋅
0 0 -1 0

We can find the matrix representation of J2 by squaring Jz. It turns out that J2 = ½(Jz² + Jz + Jz²). Therefore, the matrix representation of J2 can be obtained by squaring the matrix representation of Jz. Finally, to find the matrix representation of Jx, we can use the commutation relation [Jx, Jz] = i⋅Jy. From this relation, we can express Jx in terms of Jy and Jz, and then substitute their matrix representations.