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Are operators always commutative with respect to operation +?

  1. Jul 5, 2004 #1
    In general AB =/= BA, for example,
    orbital angular momentum operators, L_x, L_y.

    but is A + B = B + A always true?
  2. jcsd
  3. Jul 5, 2004 #2


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    (A+B)f=Af+Bf=Bf+Af=(B+A)f, so A+B = B +A

    You are simply working with functions (Af and Bf), so the addition is commutative.
  4. Jul 5, 2004 #3
    I think so, but this is puzzling to me:

    put hbar = 1
    L_x + L_y = L_y + L_x
    so, exp(-i theta L_x) exp(-i theta L_y) = exp(-i theta L_y) exp(-i theta L_x)

    that means rotation about x-axis and rotation about y-axis are commutative?

    but rotate a book around x-axis 90 deg followed by y-axis 90 deg is not same as the other way round.
  5. Jul 5, 2004 #4


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    Warning: for (non-commuting) operators A and B,
    you cannot write:

    exp(A) exp(B) = exp(A+B)


  6. Jul 5, 2004 #5
    I see,
    things become clear when you expand the exponentials

    Thank you.
  7. Jul 5, 2004 #6
    provided the domains of the operators allow for the sum to be defined, linear operators will respect the addition operation.
  8. Jul 5, 2004 #7


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    You have to be careful with your order of operations (associativity). Exponentiation first and then addition vs. addition first and then exponentiation. The exponentiation makes different operators, say:

    eA = Ω and eB = Λ

    Not only do you have to worry about A + B = B + A, which is almost trivial in a physics context, you must also worry about [Ω,Λ] = 0, which is often enough untrue.
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