Calculating Hamiltonian matrix elements in a chaotic system

In summary, the conversation discusses using a rectangular potential well and the wave function to calculate the Hamiltonian matrix elements for a particle in a stadium. The resulting matrix has energy levels corresponding to doublets and can be interpreted in a similar way to traditional matrices.
  • #1
Pyrus96
3
0
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.
 
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  • #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.
 

Related to Calculating Hamiltonian matrix elements in a chaotic system

1. How do you calculate Hamiltonian matrix elements in a chaotic system?

To calculate Hamiltonian matrix elements in a chaotic system, you need to first determine the Hamiltonian of the system. This can be done by analyzing the dynamics of the system and identifying the relevant variables. Once the Hamiltonian is determined, you can use mathematical techniques such as diagonalization or perturbation theory to calculate the matrix elements.

2. Can Hamiltonian matrix elements be calculated analytically?

In most cases, it is not possible to calculate Hamiltonian matrix elements analytically in a chaotic system. This is because the system is highly complex and the equations involved are nonlinear. Therefore, numerical methods are often used to approximate the matrix elements.

3. How do you deal with the nonlinearity in a chaotic system when calculating Hamiltonian matrix elements?

Nonlinearity in a chaotic system can make it difficult to calculate Hamiltonian matrix elements. To deal with this, one approach is to use perturbation theory, which involves approximating the nonlinear equations with simpler linear equations. Another approach is to use numerical methods such as Monte Carlo simulations or variational methods.

4. What is the significance of calculating Hamiltonian matrix elements in a chaotic system?

Calculating Hamiltonian matrix elements in a chaotic system is important for understanding the dynamics of the system. It allows us to study the behavior of the system over time and make predictions about its future evolution. This is particularly useful in fields such as quantum mechanics, where the Hamiltonian matrix elements can provide insights into the behavior of quantum systems.

5. Are there any limitations to calculating Hamiltonian matrix elements in a chaotic system?

One limitation of calculating Hamiltonian matrix elements in a chaotic system is that it can be a computationally intensive task, especially for larger and more complex systems. Additionally, the accuracy of the calculations may be affected by uncertainties or errors in the initial conditions or the mathematical models used. Therefore, it is important to carefully consider the limitations and potential sources of error when interpreting the results of such calculations.

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