Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: QM having difficulty on proofs of operators

  1. Nov 23, 2006 #1

    kel

    User Avatar

    I know this is a simple part of Quantum Mechanics, but I seem to be having trouble with it, I'm not sure if my math is just wrong or if I'm applying it wrong.

    The questions that I have are:

    Prove the following for arbitrary operators A,B and C:
    (hint-no tricks, just write them out in full)

    i- [A,c1B+c2C] = c1[A,B] + c2[A,C]

    So far I've got

    A[c1B+c2C] - [c1B+c2C]A = (Ac1B+Ac2C) - (c1BA + c2CA)

    giving

    Ac1B+Ac2C - c1BA + c2CA = c1[AB] + c2[AC] - c1[BA] + c2[CA]

    but I don't know what to do from here - something should cancel, but I think my workings may be wrong.


    ii-[A,BC] = [AB]C + B[A,C]

    iii-[A,[B,C]] + [B,[C,A]] + [C,[B,A]] = 0

    I may be able to do this one based on number ii above, but I need some help on that one first please.

    Cheers
    Kel
     
  2. jcsd
  3. Nov 23, 2006 #2
    Nothing cancels, just rearrange the terms. Btw, the second 'plus' should actually be a 'minus'.
     
  4. Nov 23, 2006 #3

    kel

    User Avatar

    ok, I've got the first one.


    i- [A,c1B+c2C] = c1[A,B] + c2[A,C]
    = (Ac1B+Ac2C) - (c1BA + c2CA)
    = Ac1B + Ac2C - c1BA - c2CA
    = c1AB - c1BA + c2AC - c2CA
    = c1[AB-BA] + c2[AC-CA]
    = c1[A,B] + c2[A,C]

    but how, do I go about the second one??
    [A,BC] = [AB]C + B[A,C]
    is it?
    [A,BC] = A[BC] - [BC]A

    and if so (or not) where do I go from here??

    Thanks in advance
    Kel
     
  5. Nov 24, 2006 #4

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    [tex] [A,BC]=ABC-BCA=ABC-BAC+BAC-BCA = \ ... \ [/tex]

    Daniel.
     
  6. Nov 24, 2006 #5

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    iii is written incorrectly. The third double commutator should be [C,[A,B]].

    Daniel.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook