The raising (J+) and lowering (J_) operators are defined by J+=JX+ iJy and J_ = JX- iJy.Read the following passage to answer the given questions based on it. Choose the correct options. (Marks: 1)What would be [Jy, J±]OPTIONS ħJx -iħJz -iħJy -iħJ
Question
The raising (J+) and lowering (J_) operators are defined by J+=JX+ iJy and J_ = JX- iJy.Read the following passage to answer the given questions based on it. Choose the correct options. (Marks: 1)What would be [Jy, J±]OPTIONS ħJx -iħJz -iħJy -iħJ
Solution
The commutation relation between Jy and J± can be calculated using the definitions of J± and the commutation relations between the components of the angular momentum operator.
The commutation relations for angular momentum operators are [Jx, Jy] = iħJz, [Jy, Jz] = iħJx, and [Jz, Jx] = iħJy.
First, let's calculate [Jy, J+].
[Jy, J+] = JyJ+ - J+Jy = Jy(Jx + iJy) - (Jx + iJy)Jy = JyJx + iJy^2 - JxJy - iJy^2 = [Jy, Jx] + i[Jy, Jy] = iħJz + 0 = iħJz
Similarly, [Jy, J-] can be calculated as:
[Jy, J-] = JyJ- - J-Jy = Jy(Jx - iJy) - (Jx - iJy)Jy = JyJx - iJy^2 - JxJy + iJy^2 = [Jy, Jx] - i[Jy, Jy] = iħJz - 0 = iħJz
So, [Jy, J±] = iħJz. Therefore, the correct option is -iħJz.
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