1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: QM having difficulty on proofs of operators
  1. QM operators (Replies: 3)

  2. Operator proof (Replies: 3)

Loading...