# Simulating the evolution of a wavepacket through a crystal lattice

I am interested simulating the evolution of an electronic wave packet through a crystal lattice which does not exhibit perfect translational symmetry. Specifically, in the Hamiltonian below, the frequency of each site $\omega_n$ is not constant.

Suppose the lattice is specified by a certain tight-binding Hamiltonian $$H = \sum_n \omega_n a^\dagger_n a_n + t \sum_{<n>} a^\dagger_n a_{n+1} +\text{all nearest neighbor interactions} + \text{h.c}.$$ We prepare a wavepacket, and for simplicity, we express the wavepacket in the fock basis of each lattice site $$| \psi \rangle = \sum_i |b_1\rangle |b_2\rangle \ldots |b_n\rangle.$$ Thus, there are $b_1$ electrons in the $1$st lattice site. Of course, electrons are fermions and $b_1$ may be either $0$ or $1$.

Suppose we treat this problem purely quantum mechanically. Then we will need to prepare a vector of length $2^n$, which is computationally intractable for any significant $n$.

I am interested in physical techniques that may be employed to simplify this problem. Is it possible to attempt the problem in a semiclassical manner?

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Our FAQ actually disavowes computational questions. With your permission I will migrate this to the new Scientific Computation beta site. Of course, you can ask about the physics here not withstanding that you are planning a computational attack, but this seems to be a implementation question. Or have I mistaken your intent? – dmckee♦ Dec 24 '12 at 21:19

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I am more interested in the physical techniques that can be used to simplify the problem and hence, make it computationally viable. As we know, quantum mechanical simulations on classical computers are often intractable as the computational steps required increase exponentially with the degrees of freedom in the system. – flamearchon Dec 24 '12 at 21:25

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At the end of the day, I would like to numerically time-step through some differential equation. The question is which differential equation do I solve! – flamearchon Dec 24 '12 at 21:29

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Ah...thank you for the clarification. This certainly should remain here. – dmckee♦ Dec 24 '12 at 21:45
@flamearchon For the exact method, you either use eigenvalue or direct evolution, and you dont have the symmetry in the Hamiltonian. The other method should only be approximation. If you get the answer, please post here. – hwlau Dec 28 '12 at 7:18
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## 1 Answer

If you use tight-binding Hamiltonian, it is reasonable to start not from semiclassical, but one-particle approximation. In that case, you have an amplitude (complex number) at each site, the state is complex vector of length $n$, Hamiltonian is $n\times n$ (sparse) matrix and the problem of time evolution and/or eigenstates (for one particle state) is solvable for relatively large lattices.

If you are interested in many particle physics, you may build a model on top of these oneparticle states. The details are dependent on what exactly you wish to compute.

Unfortunately, I do not know a reference with rigorous transfer from one formulation to another.

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Yes, usually we dont want to get the exact wavefunction. I think the ground state energy is usually one want to compute. Do you have any idea how to do that with some approximation? – hwlau Jan 6 at 9:34
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