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

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SUMMARY

The discussion focuses on the application of commutator relations in quantum optics to derive Equation 26 from Equations 25, 22, and 24 as presented in the provided notes. Participants emphasize the importance of maintaining the correct order during multiplication and utilizing the properties of inner products, specifically <1|2> = <2|1> = 0 and <1|1> = <2|2> = 1. The cancellation of certain exponential terms is also highlighted as a crucial step in the derivation process.

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  • Understanding of quantum mechanics, specifically commutator relations
  • Familiarity with inner product notation in quantum states
  • Basic knowledge of exponential functions in quantum equations
  • Ability to manipulate mathematical equations involving operators
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Students and researchers in quantum mechanics, particularly those studying quantum optics and mathematical derivations involving commutators and inner products.

sam_021
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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

 
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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?
 
^^ yes
 

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