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Thread: properties

  1. #1
    Senior Member Sampras's Avatar
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    May 2009
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    301

    properties

    Show the following:

    • $\displaystyle |0 \rangle $ is unique
    • $\displaystyle 0|V \rangle = |0 \rangle $
    • $\displaystyle |-V \rangle = -|V \rangle $
    • $\displaystyle |-V \rangle $ is the unique additive inverse of $\displaystyle |V \rangle $


    So there exists a null vector $\displaystyle |0 \rangle $ obeying $\displaystyle |V \rangle + |0 \rangle = |V \rangle $. Suppose there is another vector $\displaystyle |0' \rangle $ such that $\displaystyle |V \rangle + |0' \rangle = |V \rangle $. Then $\displaystyle |0 \rangle + |0' \rangle = (|V \rangle + |V \rangle)-(|V \rangle + |V \rangle) = |0 \rangle $. Hence $\displaystyle |0 \rangle $ is unique.

    We know that $\displaystyle |0 \rangle = (0+1)|V \rangle + |-V \rangle $. This is equaled to $\displaystyle 0|V \rangle + |V \rangle + |-V \rangle = 0|V \rangle $.

    We know that $\displaystyle |V \rangle + (-|V \rangle) = 0|V \rangle = |0 \rangle $. Then by associativity, $\displaystyle |V \rangle + (|-V \rangle) = 0|V \rangle = |0 \rangle $. Hence $\displaystyle |-V \rangle = -|V \rangle $.

    Suppose there is some $\displaystyle |W \rangle $ such that $\displaystyle |V \rangle + |W \rangle = |0 \rangle $. Then $\displaystyle |V \rangle + |W \rangle = |V \rangle + |-V \rangle $. So $\displaystyle |W \rangle = |-V \rangle $ since $\displaystyle |0 \rangle $ is unique.


    Is this correct?
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  2. #2
    Member
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    Jul 2008
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    Your ideas are mostly correct, but can be simplified to be more elegant.

    If $\displaystyle |0\rangle$ and $\displaystyle |0'\rangle$ are both identity elements, then $\displaystyle |0\rangle=|0\rangle+|0'\rangle=|0'\rangle$, where the left equality comes from $\displaystyle 0'$ being the identity, and the right equality comes from $\displaystyle 0$ being the identity.

    $\displaystyle 0|V\rangle=(1-1)|V\rangle=|V\rangle-|V\rangle=|0\rangle$

    In this next one I assume $\displaystyle |-V\rangle$ is the inverse element of $\displaystyle |V\rangle$. Then $\displaystyle |-V\rangle=|-V\rangle+ 0|V\rangle=|-V\rangle+ (1-1)|V\rangle=|-V\rangle+ |V\rangle-|V\rangle=|0\rangle-|V\rangle=-|V\rangle$

    Suppose $\displaystyle |V\rangle+|W\rangle=|0\rangle$. Then $\displaystyle |W\rangle=|W\rangle+|0\rangle=|W\rangle+|V\rangle+ |-V\rangle=|0\rangle+|-V\rangle=|-V\rangle$
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