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Operators and eigenstates/values

  1. Oct 7, 2015 #1
    1. The problem statement, all variables and given/known data
    Let the Hermitian operator A^ corresponding to the observable A have two eigenstates |a1> and |a2> with eigenvalues a1 and a2, respectively, where a1 ≠ a2. Show that A^ can be written in the form A^ = a1|a1><a1| + a2|a2><a2|.

    2. Relevant equations


    3. The attempt at a solution
    I reached out the instructor for some guidance but I am still confused.
    To my understanding i should start with A^|ψ>. Where |ψ> is some arbitrary spin state.
    and i don't know where to go from there.
     
  2. jcsd
  3. Oct 8, 2015 #2
    The identity operator can be written as

    $$
    1 = |1\rangle \langle 1| + |2\rangle \langle 2| \\
    $$

    For example suppose ##|\psi\rangle = c_1 |1\rangle + c_2|2\rangle##
    $$ |\psi \rangle = 1|\psi \rangle = |1\rangle \langle 1|\psi\rangle + |2\rangle \langle 2| \psi \rangle \\
    = |1\rangle c1 + |2\rangle c2 \\
    = |\psi \rangle
    $$

    Suppose you tried putting "1" on both sides of your operator?
     
  4. Oct 8, 2015 #3

    blue_leaf77

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    Have you heard about completeness theorem?
     
  5. Oct 8, 2015 #4
    Thank you zhaos, that's actually a lot of help.

    Blue leaf, I have not heard of completeness theorem. But I will give it a Google!
     
  6. Oct 8, 2015 #5

    blue_leaf77

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    Completeness theorem is exactly what zhaos wrote in the first equation in his post.
     
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