Particle in a one-dimensional lattice

Particle in a one-dimensional lattice

In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside the lattice. This is an extension of the free electron model that assumes zero potential inside the lattice.

Contents

Problem definition

When talking about solid materials, the discussion is mainly around crystals - periodic lattices. Here we will discuss a 1D lattice of positive ions. Assuming the spacing between two ions is a, the potential in the lattice will look something like this:

Potential-actual.PNG

The mathematical representation of the potential is a periodic function with a period a. According to Bloch's theorem, the wavefunction solution of the Schrödinger equation when the potential is periodic, can be written as:

 \psi (x) = e^{ikx} u(x). \,\!

Where u(x) is a periodic function which satisfies:

 u(x+a)=u(x) \,\!

When nearing the edges of the lattice, there are problems with the boundary condition. Therefore, we can represent the ion lattice as a ring following the Born-von Karman boundary conditions. If L is the length of the lattice so that L >> a, then the number of ions in the lattice is so big, that when considering one ion, its surrounding is almost linear, and the wavefunction of the electron is unchanged. So now, instead of two boundary conditions we get one circular boundary condition:

 \psi (0)=\psi (L). \,\!

If N is the number of Ions in the lattice, then we have the relation: aN = L. Replacing in the boundary condition and applying Bloch's theorem will result in a quantization for k:

 \psi (0) = e^{ik \cdot 0} u(0) = e^{ikL} u(L) = \psi (L) \,\!
 u(0) = e^{ikL} u(N a) \rightarrow e^{ikL} = 1 \,\!
 \Rightarrow kL = 2\pi n \rightarrow k = {2\pi \over L} n \qquad \left( n=0, \pm 1, \pm 2, ..., \pm {N \over 2} \right). \,\!

Kronig–Penney model

The Kronig–Penney model (named after Ralph Kronig and William Penney) is a simple, idealized quantum-mechanical system that consists of an infinite periodic array of rectangular potential barriers.

The potential function is approximated by a rectangular potential:

Potential-approx.PNG

Using Bloch's theorem, we only need to find a solution for a single period, make sure it is continuous and smooth, and to make sure the function u(x) is also continuous and smooth.

Considering a single period of the potential:
We have two regions here. We will solve for each independently:

 For \quad -\frac {1}{2}(a-b) <x<\frac {1} {2}(a-b) : \,\!
{-\hbar^2 \over 2m} \psi_{xx} = E \psi \,\!
\Rightarrow \psi = A e^{i \alpha x} + A' e^{-i \alpha x} \quad \left( \alpha^2 = {2mE \over \hbar^2} \right) \,\!
 For \quad -\frac {1}{2}(a+b)<x<-\frac {1}{2}(a-b) : \,\!
{-\hbar^2 \over 2m} \psi_{xx} = (E+V_0)\psi \,\!
\Rightarrow \psi = B e^{i \beta x} + B' e^{-i \beta x} \quad \left( \beta^2 = {2m(E+V_0) \over \hbar^2} \right). \,\!

To find u(x) in each region, we need to manipulate the electron's wavefunction:

 \psi(0<x<a-b) = A e^{i \alpha x} + A' e^{-i \alpha x} = e^{ikx} \cdot \left( A e^{i (\alpha-k) x} + A' e^{-i (\alpha+k) x} \right) \,\!
 \Rightarrow u(0<x<a-b)=A e^{i (\alpha-k) x} + A' e^{-i (\alpha+k) x}. \,\!

And in the same manner:

 u(-b<x<0)=B e^{i (\beta-k) x} + B' e^{-i (\beta+k) x}. \,\!

To complete the solution we need to make sure the probability function is continuous and smooth, i.e.:

 \psi(0^{-})=\psi(0^{+}) \qquad \psi'(0^{-})=\psi'(0^{+}). \,\!

And that u(x) and u'(x) are periodic

 u(-b)=u(a-b) \qquad u'(-b)=u'(a-b). \,\!

These conditions yield the following matrix:

 \begin{pmatrix} 1 & 1 & -1 & -1 \\ \alpha & -\alpha & -\beta & \beta \\ e^{i(\alpha-k)(a-b)} & e^{-i(\alpha+k)(a-b)} & -e^{-i(\beta-k)b} & -e^{i(\beta+k)b} \\ (\alpha-k)e^{i(\alpha-k)(a-b)} & -(\alpha+k)e^{-i(\alpha+k)(a-b)} & -(\beta-k)e^{-i(\beta-k)b} & (\beta+k)e^{i(\beta+k)b} \end{pmatrix} \begin{pmatrix} A \\ A' \\ B \\ B' \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}. \,\!

For us not to have the trivial solution, the determinant of the matrix must be 0. This leads us to the following expression:

 \cos(k a) = \cos(\beta b) \cos[\alpha(a-b)]-{\alpha^2+\beta^2 \over 2\alpha \beta} \sin(\beta b) \sin[\alpha(a-b)]. \,\!

To further simplify the expression, we perform the following approximations:

 b \rightarrow 0 \ ; \ V_0 \rightarrow \infty \ ; \ V_0 b = \mathrm{constant} \,\!
 \Rightarrow \beta^2 b = \mathrm{constant} \ ; \ \alpha^2 b \rightarrow 0 \,\!
 \Rightarrow \beta b \rightarrow 0 \ ; \ \sin(\beta b) \rightarrow \beta b \ ; \ \cos(\beta b) \rightarrow 1. \,\!

The expression will now be:

 \cos(k a) = \cos(\alpha a)-P{\sin(\alpha a) \over \alpha a} \qquad \left( P={m V_0 ba\over \hbar^2} \right). \,\!

See also

External links

  • "The Kronig-Penney Model" by Michael Croucher, an interactive calculation of 1d periodic potential band structure using Mathematica, from The Wolfram Demonstrations Project.

Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Particle in a one-dimensional lattice (periodic potential) — In quantum mechanics, the particle in a one dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The problem can be simplified from the 3D infinite potential barrier (particle in a box) to a one dimensional… …   Wikipedia

  • Lattice Boltzmann methods — (LBM) is a class of computational fluid dynamics (CFD) methods for fluid simulation. Instead of solving the Navier–Stokes equations, the discrete Boltzmann equation is solved to simulate the flow of a Newtonian fluid with collision models such as …   Wikipedia

  • Lattice gauge theory — In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized onto a lattice. Although most lattice gauge theories are not exactly solvable, they are of tremendous appeal because they can be studied by… …   Wikipedia

  • Lattice gas automaton — Lattice gas automata (LGA) or lattice gas cellular automata (LGCA) methods are a series of cellular automata methods used to simulate fluid flows. It was the precursor to the lattice Boltzmann methods. From the LGCA, it is possible to derive the… …   Wikipedia

  • Dimensional analysis — In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions (usually mass,… …   Wikipedia

  • One-way quantum computer — The one way or measurement based quantum computer is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is one way because the… …   Wikipedia

  • subatomic particle — or elementary particle Any of various self contained units of matter or energy. Discovery of the electron in 1897 and of the atomic nucleus in 1911 established that the atom is actually a composite of a cloud of electrons surrounding a tiny but… …   Universalium

  • Phenomenology (particle physics) — Particle physics phenomenology is the part of theoretical particle physics that deals with the application of theory to high energy particle physics experiments. Within the Standard Model, phenomenology is the calculating of detailed predictions… …   Wikipedia

  • Delta potential — The delta potential is a potential that gives rise to many interesting results in quantum mechanics. It consists of a time independent Schrödinger equation for a particle in a potential well defined by a Dirac delta function in one dimension. For …   Wikipedia

  • Delta potential well — The Delta potential well is a common theoretical problem of quantum mechanics. It consists of a time independent Schrödinger equation for a particle in a potential well defined by a delta function in one dimension.DefinitionThe time independent… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”