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I Calculating Hamiltonian matrix elements in a chaotic system

  1. Mar 14, 2017 #1
    The system in which I tried to calculate the Hamiltonian matrix was a particle in a stadium (Billiard stadium). And I used the principle where we take a rectangle around the stadium in which the parts outside the stadium have a very high potential V0.
    We know the wave function of a rectangular potential well and through this I found the following formula for the Hamiltonian matrix elements:
    $$H_{nm} = [(\frac{m_{1}}{a_{1}})^{2} +(\frac{m_{2}}{a_{2}})^{2}]\delta_{nm} + V_{0}v_{nm}$$
    where vnm is given by:
    $$v_{nm} = \int_{\Gamma^{'}}\phi_{n}\phi_{m}$$
    where ##\Gamma^{'}## denotes the area in the rectangle which isn't part of the stadium. And the wave functions are given by:
    $$\phi_{m_{1},m_{2}}(x_{1},x_{2}) = \int sin(\frac{\pi m_{1}x_{1}}{a_{1}})sin(\frac{\pi m_{2}x_{2}}{a_{2}})$$
    but here's my problem, the system we work in is two dimensional and thus we get energy levels corresponding to the doublets: ##n = (n_{1},n_{2})## and ##m = (m_{1},m_{2})##. How am I supposed to interpret the matrix with this because I can't imagine a matrix from that can correspond to ##H_{(2,1)x(1,1)}##

    Please note that I didn't bother to write several constants in front of the formula of the hamiltonian and the wave function because they don't matter to the answer.
     
  2. jcsd
  3. Mar 15, 2017 #2
    It is difficult to say, what is your final objective with those matrix elements but they will be just a version of your first formula
    $$
    H_{\mu, \nu} = \epsilon_{\mu} \delta_{\mu, \nu} + V_0 v_{\mu, \nu},
    $$
    where ##\mu## and ##\nu## are some parametrizations of states in an rectangular box, ##\epsilon_\mu## are respective energies and so on. For instance, ##|\mu \rangle = |m_1, m_2 \rangle ## and
    $$
    \langle \mu | H | \nu \rangle =: H_{\mu, \nu} = H_{m_1, m_2; n_1, n_2} : = \langle m_1, m_2 | H | n_1, n_2 \rangle =
    \int dx_1 \ dx_2 \ \phi_{m_1, m_2}^*(x_1, x_2) \widehat{H} \phi_{n_1, n_2}(x_1, x_2).
    $$
    You deal with such "direct product" matrices in a way very similar to usual matrices. For example,
    $$
    \langle \mu | \widehat{A} \widehat{B} | \nu \rangle = \sum_\lambda A_{\mu, \lambda} B_{\lambda, \nu} =
    \sum_{l_1, l_2} \langle \mu | \widehat{A} | l_1, l_2 \rangle \langle l_1, l_2 | \widehat{B} | \nu \rangle
    $$
    and so on.
     
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