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Proof for Problem 1.1.1 of Shankars Prin of QM

  1. Jul 2, 2011 #1
    I think I may be over-thinking this as I have had one formal course in QM and an independent study course in QM, but any help is MORE THAN GREATLY APPRECIATED!!!

    1. The problem statement, all variables and given/known data

    Prove that |-V> = -|V>


    2. Relevant equations

    He instructs us to begin with |V> + (-|V>) = 0|V> = |0>


    3. The attempt at a solution

    Let -|V> = |W>

    Then

    |V> + |W> = 0|V> = |0>

    which implies that

    |W> = |-V> ?

    I'm really confused.
     
  2. jcsd
  3. Jul 4, 2011 #2
    Okay, first I think you mean problem 1.1.2, unless you have a different edition.
    This is the kind of proof where you have to doubt everything. Act as if you are in a mathematical minefield. Only do the things you can completely justify.

    I find myself wanting to say that 1V = V, but that actually isn't an axiom...

    Assuming that for the moment,
    V + (-1)V = 1V + (-1)V = (1 + -1)V = 0V = 0 (by part 1 of the same problem)
    And thus (-1)V is the unique vector -V such that V + -V = 0

    Looking up in other books, 1V = V is listed as an eighth axiom for linear spaces. Shankar described it by interpretation in a paragraph, but didn't actually state it overtly as an axiom.
     
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