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Equal commutators

  1. Nov 30, 2009 #1
    Hi.

    Cohen-Tannoudji has this section in his quantum mechanics book where he derives a bunch of relations which are true for operators having the commutation relation [itex][Q,P]=i\hbar[/itex]. Is there any special significance to this value of a commutator? Would things be much different if it had the value 1 ?

    Also, if we have two sets of operators with the same commutator, i.e. [itex][x,p]=[Q,P]=i\hbar[/itex], what does this tell us about the relations between the operators, if anything?
     
  2. jcsd
  3. Nov 30, 2009 #2

    nicksauce

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    Take the hermitian conjugate of each side. If the operators are Hermitian, then you get
    [P,Q] = -ih = -[Q,P], which is exactly what you expect based on the definition of a commutator. If you had [Q,P] = 1, then this process would lead to a contradiction if your operators are Hermitian.
     
  4. Nov 30, 2009 #3
    Ah, so that's why it's important that it's imaginary! Great. What about sets of operators with the same commutator? I know that if we have two Hamiltonians of the same form, i.e.
    [tex]
    H=a^{\dagger} a+\frac{1}{2}
    [/tex]
    [tex]
    H=b^{\dagger} b+\frac{1}{2}
    [/tex]

    and [itex][a^{\dagger},a]=[b^{\dagger},b][/itex] then the Hamiltonians will have the same eigenvalues. Is there more we can say? I've heard that all of quantum mechanics can be based on commutators...
     
  5. Nov 30, 2009 #4

    nicksauce

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    I'm not too sure what we can say about sets of operators with the same commutator... hopefully someone else can help you out.
     
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