Discretization of 1D continuous Hamiltonian

Continuous single-particle Hamiltonian in 1D

Hamiltonian operator of single particle in 1D in a coordinate representation:

$\hat{H}=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+U(x)$.

is a usual sum of the kinetic energy differential operator and potential energy (multiplicative) operator.

Discrete space and finite differences

Discretization means that real axis is now replaced with an infinite chain of lattice points:

located at $x=ja$, where $j$ is integer and $a$ is the lattice spacing. The Hamiltonian operator acting on function F(x) in this space gives:

$[\widehat{HF}{]}_{x=ja}=-{\left[\frac{{\hslash }^2}{2m}\frac{d^{2}F}{d{x}^2}\right]}_{x=ja}+U_j{F}_j$,

where

$F_j\iff F(x=ja)$ and $U_j\iff U(x=ja)$. We now use the method of finite differences to approximate the derivative operators. Assuming $a$ is small, we can approximate the first derivative by:

${\left[\frac{dF}{dx}\right]}_{x=(j+1/2)a}\to \frac{1}{a}[F_{j+1}-F_j]$,

and the second derivative by:

${\left[\frac{d^{2}F}{d^{2}x}\right]}_{x=ja}\to \frac{1}{a}({\left[\frac{dF}{dx}\right]}_{x=(j+1/2)a}-{\left[\frac{dF}{dx}\right]}_{x=(j-1/2)a})\to \frac{1}{a^2}[F_{j+1}-2F_j+F_{j-1}]$.

With this approximation we can rewrite $[\hat{H}F{]}_{x=ja}$ as

$[\hat{H}F{]}_{x=ja}=(U_j+2t)F_j-t{F}_{j-1}-t{F}_{j+1}$,

where $t\equiv \frac{{\hslash }^2}{2m{a}^2}$.

Matrix representation of 1D Hamiltonian in discrete space

Thus, the action of the Hamiltonian operator $\hat{H}$ on function $F(x)$ in discrete space is given by the following matrix equation:

$[\hat{H}F(x){]}_{x=ja}=\sum _i{H}_{ji}{F}_i=[\mathbf{𝐇}\cdot \mathbf{𝐅}{]}_j$

where the matrix elements $H_{ji}$ are non-zero only on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal:

$\mathbf{𝐇}=\left(\begin{array}{ccccc}\dots & -t& 0& 0& 0\\ -t& U_{-1}+2t& -t& 0& 0\\ 0& -t& U_0+2t& -t& 0\\ 0& 0& -t& U_1+2t& -t\\ 0& 0& 0& -t& \dots \\ \end{array}\right)$

Such matrices are called tridiagonal in the linear algebra theory. Each site is linked to its nearest neighbor by the element $t$, while the diagonal elements are given by the potential energy plus $2t$. The discrete values of the function $F$ at lattice sites form a vector in this representation:

$\mathbf{𝐅}=\left(\begin{array}{c}⋮\\ F_{-1}\\ F_0\\ F_1\\ ⋮\\ \end{array}\right)$

Energy-momentum dispersion relation for a discrete lattice

For a uniform 1D wire with a constant potential $U(x)=U_0$, the eigenfunctions are plane waves with parabolic energy-momentum dispersion:

${\mathrm{\Psi }}_k(x)=\exp (ikx)$, $E=U_0+\frac{{\hslash }^2{k}^2}{2m}$.

The corresponding eigenfunctions and dispersion for a discrete lattice are:

${\mathrm{\Psi }}_k(x_j)=\exp (ik{x}_j),x_j=ja$, $E=U_0+2t[1-\cos (ka)]$,

as the solution of the discretized Schrodinger equation:

$\mathbf{𝐇}\cdot \mathrm{\Psi }=E\mathrm{\Psi }\iff (U_0+2t){\mathrm{\Psi }}_j-t{\mathrm{\Psi }}_j-t{\mathrm{\Psi }}_{j+1}=E{\mathrm{\Psi }}_j$,

As we let the lattice spacing tend to zero, $a\to 0$, we recover the usual parabolic relationship. Also, the velocity is given by:

$v=\frac{1}{\hslash }\frac{\partial E}{\partial k}=2at\sin (ka)$,

which reduces to the usual result $v=\hslash k/m$ in the limit $a\to 0$.

How good is discrete approximation in practical calculations?

In practical calculations, we can mimic parabolic dispersion in the limit in which cosine function can be approximated by the first two terms of its Taylor series expansion, which imposes the following condition

$\cos (ka)\approx 1-\frac{1}{2}{k}^2{a}^2\Rightarrow ka\ll 1\Rightarrow \frac{\lambda }{2\pi }\gg a$,

This has a clear physical interpretation: We can use the discrete approximation only if the number of discrete points within one electron wavelength $\lambda =\frac{2\pi }{k}$ is sufficiently large so that we have enough discrete points within one wavelength to recover the spatial form of the function.