1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Eigenkets of the creation operator

  1. Oct 28, 2008 #1
    1. The problem statement, all variables and given/known data

    The problem is to find the eigenkets for the creation operator ,[tex]a^{\dagger}[/tex] if they exist

    2. Relevant equations

    [tex]a^{\dagger}|\Psi>=\lambda|\Psi>[/tex]
    [tex]a^{\dagger}=\frac{1}{\sqrt{2*\hbar*m*\omega}}*(-\hbar*\frac{d}{dx}+m*\omega*x)[/tex]

    3. The attempt at a solution
    I use the expression for the creation operator above and set up the differential equation.
    The boundary conditions for the wavefunction should be that it is zero when x goes to +- infinity.
    When I solve it I get this result.
    [tex]0=\lambda*x\right|_{-\infty}^{+\infty} - \frac{1}{2}\sqrt{\frac{m*\omega}{2*\hbar}}*x^{2} \right|_{-\infty}^{+\infty}[/tex]

    I know the creation operator is not hermitian and it is not an observable but could this answer be correct?
    Could someone please comment on this
     
  2. jcsd
  3. Oct 28, 2008 #2

    olgranpappy

    User Avatar
    Homework Helper

    If eigenkets of [itex]a^\dagger[/itex] exist. What is their overlap with the vacuum?
     
  4. Oct 28, 2008 #3
    What is their overlap with vaccum?
    I don't understand!!
     
  5. Oct 28, 2008 #4

    olgranpappy

    User Avatar
    Homework Helper

    By "vacuum" I just mean the state [itex]|0\rangle[/itex] that satisfies
    [tex]
    a|0\rangle=0
    [/tex]

    By "overlap with the vacuum" I mean, what is
    [tex]
    \langle 0 |\lambda\rangle
    [/tex]
    equal to?
     
  6. Oct 28, 2008 #5

    olgranpappy

    User Avatar
    Homework Helper

    ...where [itex]|\lambda\rangle[/itex] is the eigenket of [itex]a^\dagger[/itex] with eigenvalue [itex]\lambda[/itex]
     
  7. Oct 29, 2008 #6
    The overlap with the vacuum is zero
    but how does this help me?:confused:
     
  8. Oct 29, 2008 #7

    olgranpappy

    User Avatar
    Homework Helper

    So, the overlap with the vacuum is zero. Good.

    Now, what is the overlap with the "one-particle" state
    [tex]
    |1\rangle = a^\dagger |0\rangle\;
    [/tex]
    ?
     
  9. Oct 29, 2008 #8
    I believe the overlap of [tex]|\lambda>[/tex] with the one particle state to be zero
     
  10. Oct 29, 2008 #9
    So, given that, what's the overlap of [tex]|\lambda>[/tex] with the 2-particle state |2>? I hope you realize what the nth question is going to be.
     
  11. Oct 29, 2008 #10
    I think I understand this.
    since the overlap of [tex]\lambda[/tex] and the vaccum is zero and also the overlap with the one, two and so one particle functions is zero, [tex]\lambda[/tex] must be zero.
    Am I right?
     
  12. Oct 29, 2008 #11

    olgranpappy

    User Avatar
    Homework Helper

    The notation you are using above confuses the state with it's eigenvalue.

    What we have just seen is that if the *eigenvalue* [itex]\lambda[/itex] is non-zero then the *state* [itex]|\lambda>[/itex] must be zero--i.e., there are no eigenstate of [itex]a^\dagger[/itex] with non-zero eigenvalues.
     
  13. Oct 30, 2008 #12
    Thank you all for your help:smile:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?