11 Finite Difference Method1

# 11 Finite Difference Method1 - Fundamentals of...

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Fundamentals of Fundamentals of Nanoelectronics Nanoelectronics Prof. Supriyo Datta ECE 453 Purdue University Network for Computational Nanotechnology 09.20.2004 Lecture 11: Finite Difference Method Ref. Chapter 2.2

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Φ + = Φ ) ( 2 2 2 2 x U dx d m E h = N N E φ M M 2 1 2 1 x 1 2 3 n-1 n n+1 N a • As it was mentioned before all numerical methods for solving differential equations involve some scheme to covert the original equation into a matrix equation. What we do here is to discretize the lattice and then we write the second derivative as a difference equation. • Figure on the left shows the discrete points. As it can be seen corresponding to each lattice point there is a value for the wave function. This can also be viewed as sampling of a continuous function into discrete values. Remember that in order to be able to perform a numerical method we have to a have a finite number of equations so that we can solve them. To make things easier, for now we set U(x)=0. Note that the matrix for U can be written down easily and it can later be added to the matrix that represents the differential operator which we now we’ll try to find… n Time Independent Schrödinger Equation and Discretization 00:08
• What goes in the parenthesis is the discrete representation of the differential operator. • So to turn this differential equation to a difference equation, the most important step is to write the second derivative as a difference expression. We start with the difference equation for the first derivative… x 1 2 3 n-1 n n+1 N a n φ At each particular point the Schrödinger equation (after dropping U) can be written as: Now start from left and right everything at point n. The constants remain the same. For the wavefunction we right its value at that particular point. We will have: = m E n 2 2 h [] n n x x x x dx d m E = = Φ = 2 2 2 2 h From Derivatives to Differences 1 06:54

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Consider the lattice in the a vicinity of the nth point. In order to right the derivative of the function at n we need to know its values close to n. This can be done by calculating the amount by which the value of wave function changes as moving from one lattice point to another. This gives us the first derivative. The second derivative is of course the difference of the first derivatives. • The amount by which the function changes going from n-1 to n divided by the distance between the two points is the first derivative.
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## This note was uploaded on 01/12/2011 for the course ME 305 taught by Professor Wright,j during the Spring '10 term at Birla Institute of Technology & Science, Pilani - Hyderabad.

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11 Finite Difference Method1 - Fundamentals of...

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