Solving Hamiltonian with chain of charge centers?

In summary: This approach can be extended to solve more complex quantum mechanical problems as well. In summary, the problem of solving a potential from an equally spaced chain of static point charges with a single electron moving can be solved using the method of trial wavefunctions in Quantum Mechanics.
  • #1
Gerenuk
1,034
5
I was thinking of how to solve the single particle Hamiltonian
[tex]H=...+\sum_i \frac{1}{\vec{r}-\vec{r}_i}[/tex]
where [itex]\vec{r}_i=i\cdot\vec{a}[/itex]
Transforming it into second quantization k-space I had terms like
[tex]H=...+\sum_G...c^\dag_{k+G}c_k[/tex]
But it seems to me that for the method of trial wavefunctions any wavefunction [itex]\psi[/itex] gives zero matrix elements?!
[tex]<\psi|c^\dag_{k+G}c_k|\psi>=<c_{k+G}\psi|c_k\psi>=0[/tex]
Is there anything wrong? How would I solve a potential from equally spaced chain of static point charges with a single electron moving?
 
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  • #2
This is a problem in Quantum Mechanics that requires a quantum mechanical solution. One of the most common approaches to solving such problems is through the use of trial wavefunctions. The idea is to construct a trial wavefunction, which is a linear combination of orthogonal single particle states, and use it to calculate the expectation value of the Hamiltonian. By doing this, one can obtain a variational estimate of the energy of the system.The problem you have mentioned can be solved using a single particle trial wavefunction of the form\psi=\sum_k c_k \phi_kwhere \phi_k are the single particle states, and c_k are the coefficients. The expectation value of the Hamiltonian can then be calculated as<\psi|H|\psi>=\sum_{k,G} c^*_k c_{k+G} <\phi_k|H|\phi_{k+G}>In this case, since the Hamiltonian contains a sum over all static point charges, we will need to evaluate the matrix elements of the form <\phi_k|\frac{1}{\vec{r}-\vec{r}_i}|\phi_{k+G}>. These will generally be non-zero, depending on the form of the single particle states and the position of the static point charges. Once the expectation value of the Hamiltonian is calculated, one can minimize this with respect to the coefficients c_k in order to find the lowest energy state.
 
  • #3


There are a few things to consider when trying to solve this Hamiltonian. First, it is important to note that the potential term in the Hamiltonian, which involves the positions of the charge centers, will depend on the specific arrangement of charges in the chain. Therefore, the solution to this Hamiltonian will also depend on the specific arrangement of charges.

One approach to solving this Hamiltonian could be to use perturbation theory, where the potential term is treated as a perturbation to the kinetic energy term. This could potentially provide a way to approximate the solution for a given arrangement of charges.

Another approach could be to use numerical methods, such as solving the Schrodinger equation using finite difference or finite element methods. This would allow for the solution to be obtained for any given arrangement of charges, as long as it can be represented in a numerical grid.

Regarding the issue of trial wavefunctions giving zero matrix elements, it is important to consider the specific form of the wavefunction being used. It is possible that the form of the trial wavefunction is not suitable for this particular Hamiltonian, and a different form may need to be used. Additionally, the specific arrangement of charges could also affect the form of the wavefunction that is most appropriate for this system.

In summary, solving this Hamiltonian with a chain of charge centers may require a combination of analytical and numerical methods, as well as careful consideration of the specific arrangement of charges and the form of the trial wavefunction.
 

1. What is a Hamiltonian and how is it related to chain of charge centers?

The Hamiltonian is a mathematical operator used in quantum mechanics to describe the total energy of a physical system. In the context of a chain of charge centers, the Hamiltonian represents the total energy of the system, taking into account the interactions between the charge centers.

2. How do you solve a Hamiltonian with a chain of charge centers?

The solution to a Hamiltonian with a chain of charge centers involves using mathematical techniques such as perturbation theory, variational methods, or numerical methods like matrix diagonalization. The approach used will depend on the complexity of the system and the desired level of accuracy.

3. What is the significance of solving a Hamiltonian with a chain of charge centers?

Solving a Hamiltonian with a chain of charge centers allows us to understand the behavior and properties of a system of interacting charges. This can have applications in fields such as materials science, condensed matter physics, and quantum computing.

4. Are there any simplifications or assumptions made when solving a Hamiltonian with a chain of charge centers?

Yes, there are often simplifications and assumptions made when solving a Hamiltonian with a chain of charge centers. These can include assuming a certain geometry or symmetry of the system, neglecting certain interactions, or using approximations to make the calculations more manageable.

5. Can the results from solving a Hamiltonian with a chain of charge centers be experimentally verified?

Yes, the results from solving a Hamiltonian with a chain of charge centers can be experimentally verified through techniques such as spectroscopy, which can measure the energy levels and properties of a system. These experiments can confirm the accuracy of the theoretical calculations and provide further insight into the system.

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