How Do the Commutator Relations Lead to Equation 26 in Quantum Optics?

[sam_021]

Homework Statement

http://quantum.leeds.ac.uk/~almut/section2.pdf
Please note i am referring to the above notes

I basically don't get how the maths works to get
(eq(25))(eq(22))(eq(24)) = eq(26)

am i missing something interms of the commutator relations ?

Homework Equations

The Attempt at a Solution

 

[e.bar.goum]

Ah, those equations are familiar. Did you try just multiplying through, remembering to keep everything in the right order, and recalling that <1|2> = <2|1> = 0 and <1|1> = <2|2> = 1, and that some of those exponentials will cancel?

 

[sam_021]

^^ yes