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Fermionic Fock Space

  1. Jan 16, 2016 #1
    1. The problem statement, all variables and given/known data

    Fermionic.png

    2. Relevant equations

    Given in question.

    3. The attempt at a solution

    Hello

    I'm having some difficulty with part e of this question. Not sure how to go about proving that. Would a possibility would be deriving the following equation:

    ##|n⟩= \frac{1}{\sqrt{n!}}(c^†)^n |0⟩ ## ?
     
  2. jcsd
  3. Jan 16, 2016 #2

    strangerep

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    You know that ##c^2 = 0##. So what can you say about ##\left(c^\dagger \right)^2 = \; ?##

    Then think about ##[H, c^\dagger] = \; ?##, etc.
     
  4. Jan 17, 2016 #3
    Thanks for the reply. Then ##\left(c^\dagger \right)^2 = 0##. However I'm not picking up on the reasoning for determining the commutator. If I remember correctly if two operators commute then they share the same eigenvalues, is that correct?
     
  5. Jan 17, 2016 #4

    vela

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    No, it's a bit more subtle. For example, for a free particle, the momentum ##\hat p## and Hamiltonian ##\hat H = \frac{\hat p^2}{2m}## commute, but the eigenvalues of ##\hat p## have units of momentum and the eigenvalues of ##\hat H## are energies. Clearly, they can't be the same values. Moreover, you can have eigenstates of the Hamiltonian which are not momentum eigenstates.
     
  6. Jan 17, 2016 #5

    strangerep

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    Yes.
    You gave an equation for ##|n\rangle##. So what is ##|2\rangle, |3\rangle, \dots## ? From this what can you conclude about the structure of the Fock space?

    Then evaluate ##H |0\rangle## and ##H |1\rangle## ...
     
  7. Jan 18, 2016 #6
    Right, so ##|n\rangle = 0## when ##n=2,3,4...##, in which case the only non vanishing ones are when ##n=0,1## as required. And then evaluating ##H |0\rangle## and ##H |1\rangle## is straight forward using the relations given in part d and before.
     
  8. Jan 18, 2016 #7

    vela

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    Is your expression for ##\lvert n \rangle## justified? Does applying ##c^\dagger## repeatedly generate all possible states?

    I think it would be better if you considered the fact that you can write N in terms of H. Then use what you showed in part (c).
     
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