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Finding eigenstate for the annhilation operator

  1. Oct 5, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the eigenvector of the annhilation operator a.

    2. Relevant equations
    [itex]a|n\rangle = \sqrt{n}|{n-1}\rangle[/itex]

    3. The attempt at a solution
    Try to show this for an arbitrary wavefunction:
    [itex]|V\rangle = \sum_{n=1}^\infty c_{n}|n\rangle [/itex]

    [itex] a|V\rangle = a\sum_{n=1}^\infty c_{n}|n\rangle = \sum_{n=1}^\infty \sqrt{n} c_{n}|n-1\rangle[/itex]

    Define a new constant, k = n - 1, and put this in:

    - > [itex]\sum_{k=0}^\infty \sqrt{k + 1} c_{k + 1}|k\rangle = \sum_{n=0}^\infty \sqrt{n + 1} c_{n + 1}|n\rangle [/itex]

    This was a trick I found in my notes from a lecture, but I do not quite understand how you can say that k = n in the last part. Is it just because k and n are an arbitrary notation for the basis set? Or is it due to the fact that since the sum goes to infity, the difference of 1 does not matter (since I really defined k to be n - 1)?

    When I continue with this, I get:

    [itex]a( c_{1}|1\rangle + c_{2}|2\rangle + c_{3}|3\rangle + ...) = c( \sqrt{1}c_{1}|0\rangle\ + \sqrt{2}c_{2}|1\rangle + \sqrt{3}c_{3}|2\rangle + \sqrt{4}c_{4}|3\rangle) [/itex]

    I am not really sure where to go next. I tried setting the coeffecient before each basisvector equal, like this:

    [itex] c\sqrt{1}c_{1} = 0 [/itex]
    [itex] c_{1} = \sqrt{2}c_{2}c [/itex]
    [itex] c_{2} = \sqrt{3}c_{3}c [/itex]
    [itex] c_{3} = \sqrt{4}c_{4}c [/itex]

    The problem her is that since the first part does not contain the vector [itex] 0|\rangle[/itex], either c or [itex]c_{1}[/itex] has to be zero in the first equation, and then everything becomes zero, but I know that the annihilation operator has eigenstates.
     
  2. jcsd
  3. Oct 5, 2015 #2

    strangerep

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    Science Advisor

    Is there some reason you didn't start this sum at n=0 ?
     
  4. Oct 6, 2015 #3
    Yes, because when the annihilation operator works on the state zero, we will get zero by definition according to my book so I thought that there was no point in taking it. But I can see that that would get me on the right track again, but I don't know how I can do it?

    But by taking the sum from n = 0, I end up with [itex] c_n = c_{n-1}\frac{c'}{\sqrt{n}} [/itex], where c' = [itex] \frac{1}{c} [/itex]. And now I'm not really sure what to do next. Can I just put this in my expression for [itex] |V\rangle [/itex] and call that an eigenstate with c' as the constant? It doesn't look like the expression I found for the eigenstate, which was [itex] |V\rangle [/itex] = [itex] c_{0} [/itex] [itex] \sum\limits_{n=0}^\infty [/itex] [itex] \frac{c^n}{\sqrt{n!}} [/itex][itex]|n\rangle [/itex] in my notation. I tried writting the expression I found out, but then I got something like:
    [itex] c_0 + \frac{c'c_0}{\sqrt{1}} + \frac{c_0c'^2}{\sqrt{2}} + \frac{c_0c'^3}{\sqrt{6}} [/itex]. It sort of looks like the equations I got in the start, but I'm stuck in trying to get to one place from the other. Any hints?

    And thanks for the help!
     
  5. Oct 6, 2015 #4

    strangerep

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    In your "relevant equations", you didn't write out the action of the creation operator ##a^*##. E.g.,
    $$a^*|0\rangle ~=~ ~?~$$$$(a^*)^2 |0\rangle ~=~ ~?~$$etc. Try to express each term in your sum for ##|V\rangle## in terms of ##a^*## acting on the vacuum.
     
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