Commutation Relationship between Angular Momentum Operators & Their Commutators in Quantum Mechanics

Quantum Chemistry Problem [Q20-05-00]. Commutation Properties of the Angular Momentum Operators: Mx, My, Mz, and M². Calculating commutators: [Mx,My], [My,Mz], [Mz.Mx], and [M²,Mx], [M²,My], [M²,Mz]. The total angular momentum square operator: M². And which angular momentum measurements can be actually made simultaneously on a quantum system?

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Question:
The operators for the components of angular momentum are:
(Mx)=−iℏ(y∂/∂z−z∂/∂y),
(My)=−iℏ(z∂/∂x−x∂/∂z),
(Mz)=−iℏ(x∂/∂y−y∂/∂x).
Show that: (Mx)(My)−(My)(Mx)=iℏMz, and that M²(Mz)=(Mz)M², in which M²≡Mx²+My²+Mz².

Derive the corresponding commutation rules for (My) and (Mz), for (Mz) and (Mx), and for M² with (Mx), and with (My).

[jitkaplucarova]

Great video! It helped me a lot, thanks.