What's the Normal Order of A + B?

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SUMMARY

The discussion centers on the normal order of operators A and B in quantum mechanics, specifically addressing the notation N[A + B]. The user presents equations involving the normal ordering of the product of operators, such as N[aa^{\dagger}] and N[aa^{\dagger}b^{\dagger}b], and highlights a contradiction in the results derived from the linearity of the normal ordering operation. The key conclusion is that the normal ordering of the product of creation and annihilation operators requires careful consideration of their commutation relations.

PREREQUISITES
  • Understanding of quantum mechanics and operator algebra
  • Familiarity with creation and annihilation operators
  • Knowledge of normal ordering in quantum field theory
  • Basic grasp of linearity in mathematical operations
NEXT STEPS
  • Study the properties of creation and annihilation operators in quantum mechanics
  • Learn about the normal ordering process in quantum field theory
  • Explore the implications of commutation relations on operator products
  • Review examples of normal ordering in various quantum mechanical systems
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Students and researchers in quantum mechanics, particularly those studying quantum field theory and operator algebra, will benefit from this discussion.

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Homework Statement



Hi, guys, I just wondering what's the normal order of A +B

Homework Equations



if N[A+B]=N[A]+N
then I have
[itex]N[aa^{\dagger}]=N[1-a^{\dagger}a]=1-a^{\dagger}a[/itex]
however we know that
[itex]N[aa^{\dagger}]=-a^{\dagger}a][/itex]
what's wrong?

The Attempt at a Solution



another example,
[itex]N[aa^{\dagger}b^{\dagger}b]=N[(1-a^{\dagger}a)b^{\dagger}b]=b^{\dagger}b+a^{\dagger}b^{\dagger}ab[/itex]

thank you!
 
Physics news on Phys.org
[itex]N \left[ 1 \right] = N \left[ aa^\dagger + a^\dagger a \right][/itex] and linearity gives?
 

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