Answer to Question #191344 in Quantum Mechanics for Sam Pavan

Question

1. The components of the angular momentum operator in terms of the components of position operator and the linear momentum operator are given as:

𝐿̂ 𝑥 = 𝑦̂𝑝̂𝑧 − 𝑧̂𝑝̂𝑦 , 𝐿̂ 𝑦 = 𝑧̂𝑝̂𝑥 − 𝑥̂𝑝̂𝑧 𝑎𝑛𝑑 𝐿̂ 𝑧 = 𝑥̂𝑝̂𝑦 − 𝑦̂𝑝̂𝑥 ,

where the symbols have their usual meanings.

(a) Show that [𝐿̂ 𝑥, 𝐿̂ 𝑦] = 𝑖ℏ𝐿̂ 𝑧 𝑎𝑛𝑑 [𝐿̂2 , 𝐿̂ 𝑧 ] = 0. State other similar commutation relations.

(b) Define angular momentum ladder operators 𝐿̂±

(i) Show that [𝐿̂2 , 𝐿̂±] = 0 and [𝐿̂ 𝑧 , 𝐿̂±] = ±ℏ𝐿̂±.

(ii) Hence, show that |𝑙 𝜄𝑚𝜄 ⟩ = 𝐿̂±|𝑙, 𝑚⟩ is eigenstates of 𝐿̂2 and 𝐿̂ 𝑧 . What are the values of 𝑙 𝜄 and 𝑚𝜄 ?

Answer 1

Dear Sam Pavan, your question requires a lot of work, which neither of our experts is ready to perform for free. We advise you to convert it to a fully qualified order and we will try to help you. Please click the link below to proceed: Submit order