# Momentum space representation for finite lattices - continued

• QuantumLeak
In summary, the conversation discusses the topic of basis changes and boundary conditions in relation to diagonalizing the Hamiltonian. The speaker also raises the question of how to accurately model a finite size chain and the use of different representations for this purpose.
QuantumLeak
I have been banned, maybe my nickname was not so kind. I let the topic continue here. I report my last comment:

"Ok, I got the point. thanks for replying!
It's just a change of basis that under boundary condition diagonalize the Hamiltonian. But then a subtle point:

In order for k-representation to be a good basis change (i.e. orthogonality and completeness properties) I guess that one has to choose the k values AS if there were periodic boundary conditions (i.e. k=2pi*n/L). Right?

Then another last point:
how do you really model a finite size chain? A finite size chain has zero boundary condition. In this case I use the k representation with k chosen as if there were periodic boundary condition and just then reconstruct the physical wavefunction by making a wavepacket that is zero on the borders?
"

A finite object in real space will need an infinite number of Fourier components in reciprocal space to fully describe it.

When you have nanometer scale objects with a few 100s or so atoms in each direction, then it is not so obvious which representation will give you a more accurate picture.

## What is momentum space representation?

Momentum space representation is a mathematical tool used in physics to describe the properties of a physical system in terms of momentum rather than position. It is often used to analyze systems that have periodic or repeating structures, such as finite lattices.

## What is a finite lattice?

A finite lattice is a discrete, periodic structure that consists of a finite number of points or nodes connected by edges. It is often used to model physical systems, such as crystals or electronic systems, in which the atoms or particles are arranged in a regular pattern.

## Why is momentum space representation useful for finite lattices?

Momentum space representation allows for a more efficient and simplified analysis of finite lattices. It can reveal important information about the physical properties of the system, such as energy levels and wave functions, that may not be easily observable in position space representation.

## How is momentum space representation related to Fourier transforms?

Momentum space representation is closely related to Fourier transforms, which are mathematical operations that convert a function from one representation to another. In the case of finite lattices, momentum space representation is achieved by taking the Fourier transform of the position space representation.

## What are the limitations of using momentum space representation for finite lattices?

One limitation is that momentum space representation is only applicable to systems with periodic structures. Additionally, it may not accurately capture the behavior of systems with strong interactions between particles, as it assumes that the particles are non-interacting. It also requires a good understanding of mathematical concepts, such as Fourier transforms, to interpret the results correctly.

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