Quantum Chemistry

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Exercise. This is the solution to exercise 5.1 in the book.

Solution.

We have that:

[Â,B̂] = ÂB̂ B̂Â = (B̂Â ÂB̂) = [B̂,Â]

We have that:

[Â,Ân] = ÂÂn Ân = Ân+1 Ân+1 = 0

We have that:

[kÂ,B̂] = kÂB̂B̂k = kÂB̂kB̂ = k(ÂB̂B̂Â) = k[Â,B̂]

[Â,kB̂] = ÂkB̂kB̂Â = kÂB̂kB̂Â = k(ÂB̂B̂Â) = k[Â,B̂]

We have that:

[Â,B̂+Ĉ] = Â(B̂+Ĉ)(B̂+Ĉ)Â = ÂB̂+ÂĈB̂ÂĈÂ = [Â,B̂]+[Â,Ĉ]

Similarly,

[Â+B̂,Ĉ] = (Â+B̂)ĈĈ(Â+B̂) = ÂĈ+B̂ĈĈÂĈB̂ = [Â,Ĉ]+[B̂,Ĉ]

We have that:

[Â,B̂Ĉ] = ÂB̂ĈB̂ĈÂ = ÂB̂ĈB̂ÂĈ+B̂ÂĈB̂ĈÂ = [Â,B̂]Ĉ+B̂[Â,Ĉ]

Similarly,

[ÂB̂,Ĉ] = ÂB̂ĈĈÂB̂ = ÂĈB̂ĈÂB̂+ÂB̂ĈÂĈB̂ = [Â,Ĉ]B̂+Â[B̂,Ĉ]