Matrix Representation of Operators in a Finite Basis

  1. 1. The problem statement, all variables and given/known data

    I have my quantum mechanics final creeping up on me and I just have a question about something that doesn't appear to be covered in the text.

    Let's say you have a wave function of the following form for a linear harmonic oscillator:

    [itex] \Psi = c_1 | E_1 \rangle + c_2 | E_2 \rangle [/itex]

    The basis is just the first two excited energy states. My question is how the Hamiltonian matrix is represented in this case. Is it

    [itex] H = \hbar \omega \left( \begin{matrix} 0 & 0 & 0 \\ 0 & \frac{3}{2} & 0 \\ 0 & 0 & \frac{5}{2} \end{matrix} \right) [/itex]

    Or do you just use the non-zero elements:

    [itex] H = \hbar \omega \left( \begin{matrix} \frac{3}{2} & 0 \\ 0 & \frac{5}{2} \end{matrix} \right) [/itex]

    Any help would be greatly appreciated. Thanks!
     
    Last edited: Dec 10, 2012
  2. jcsd
  3. Mute

    Mute 1,391
    Homework Helper

    If your hamiltonian is the usual harmonic oscillator hamiltonian, ##\mathcal H = \hbar\omega(a^\dagger a + 1/2)##, with eigenstates ##|n\rangle## for n = 0, 1, 2, etc., then your space is infinite dimensional, and any linear combination of the eigenstates will still leave you with an infinite-dimensional system.

    The 3x3 matrix you wrote down is not correct if the elements are ##\mathcal H_{mn} = \langle m| \hbar \omega(a^\dagger a + 1/2)|n\rangle##, as the ##\mathcal H_{00}## element is ##\hbar\omega/2##, not zero. You would need some sort of additional term in the Hamiltonian to cause this term to vanish, which you presently don't have.

    Your 2x2 matrix, however, is a perfectly legitimate finite-dimensional subset of the full infinite-dimensional space. If your initial state is prepared in a superposition of ##|1\rangle## and ##|2\rangle##, then the system will not leave those states and you can focus on the 2x2 subset.

    Note that the other elements of the matrix ##\mathcal H## are not zero, but since your system was initially prepared in a superposition of the n = 1 and 2 states and there are no off-diagonal terms, meaning no transitions between states of different n, your system will stay in those two states and you can just focus on the 2x2 matrix.
     
    Last edited: Dec 10, 2012
  4. Thanks a lot!
     
  5. I also would like to know how to represent the ladder operators of the harmonic oscillator in a finite basis of [itex] | E_1 \rangle [/itex] and [itex] | E_2 \rangle [/itex] as matrices.

    I get the proper position expectation values using the following representations:

    [itex] a = \left( \begin{matrix} 0 & 0 \\ \sqrt{2} & 0 \end{matrix} \right) \hspace{5 mm} a^{\dagger} = \left( \begin{matrix} 0 & \sqrt{2} \\ 0 & 0 \end{matrix} \right)
    [/itex]

    Using these, however, I do not have the proper relationship for the Hamiltonian matrix and the ladder operator product:

    [itex] H = a a^{\dagger} + \frac{1}{2} = \left( \begin{matrix} \frac{3}{2} & 0 \\ 0 & \frac{5}{2} \end{matrix} \right) [/itex]

    I get a matrix of the following form for [itex]a a^{\dagger}[/itex]:

    [itex] a a^{\dagger} = \left( \begin{matrix} 0 & 0 \\ 0 & 2 \end{matrix} \right) [/itex]

    Am I correct in my initial matrices for the ladder operators and the relationship between H and [itex] a a^{\dagger} [/itex] doesn't hold for finite subsets of [itex] \{ E_n \} [/itex]? Or were my initial ladder operator matrices incorrect?

    Thanks!
     
  6. Mute

    Mute 1,391
    Homework Helper

    First, the Hamiltonian contains ##a^\dagger a##, not ##a a^\dagger##. This doesn't fix the discrepancy, however.

    See what happens if you wrote down 3x3 matrices for ##a## and ##a^\dagger##. I suspect that what you'll find is that there are terms from other (##n \neq 1,2##) states that come into the matrix multiplication that you dropped when writing the creation and annihilation operators as 2x2 matrices.

    The number/energy wavefunctions are not eigenstates of the annihilation and creation operators, so restricting the matrix representation of the operators to a finite subspace of your basis states will not work out very well because the operators will try to take the input state into a state that is outside the restricted 2x2 representation. You don't have this problem with the full hamiltonian because it is diagonal in the basis you are using.

    Also, lastly, use ^\dagger rather than ^\dag to get the daggers in Latex.
     
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