Solving Hamiltonian with chain of charge centers?

[Gerenuk]

I was thinking of how to solve the single particle Hamiltonian
$$H=...+\sum_i \frac{1}{\vec{r}-\vec{r}_i}$$
where $\vec{r}_i=i\cdot\vec{a}$
Transforming it into second quantization k-space I had terms like
$$H=...+\sum_G...c^\dag_{k+G}c_k$$
But it seems to me that for the method of trial wavefunctions any wavefunction $\psi$ gives zero matrix elements?!
$$<\psi|c^\dag_{k+G}c_k|\psi>=<c_{k+G}\psi|c_k\psi>=0$$
Is there anything wrong? How would I solve a potential from equally spaced chain of static point charges with a single electron moving?

 

[azntoon]

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.