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Homework Statement
The operators J(subscript x)-hat, J(subscript y)-hat and J(subscript z)-hat are Cartesian components of the angular momentum operator obeying the usual commutation relations ([J(subscript x)-hat, J(subscript y)-hat]=i h-bar J(subscript z) etc). Use these commutation relations to show that the operators J(subscript +)-hat=J(subscript x)-hat +i h-bar J(subscript y)-hat and J(subscript -)-hat=J(subscript x)-hat -i h-bar J(subscript y)-hat satisfy the following commutation relations:
[J(subscript z)-hat, J(subscript +)-hat]=h-bar J(subscript +)-hat
[J(subscript z)-hat, J(subscript -)-hat]=h-bar J(subscript -)-hat
The Attempt at a Solution
[J(z),J(+)]
=J(z)(J(x)+(i)J(y)-(J(x)+i(J(y))J(z)
=-i h-bar J(y) +(i)(-i) h-bar J(z)- i h-bar J(y) - (i)(i)h-bar J(z)
=2h-bar J(x) -2i h-bar J(y)
=2h-bar (J9x) -i J(y) =2 h-bar J(-)