1 Numerical Approximation to Integrals and Expectation

Transcription

1 1 Numerical Approximation to Integrals and Expectation Values Physics 21 Columbia University Copyright A. J. Millis All rights reserved 1.1 Overview These notes cover one approach to the numerical evaluation of expectation values in one dimensional quantum mechanics. Consider a world confined to the range <x<. We wish to approximately evaluate integrals of the form <O>= Z ψ (x) b Oψ(x) (1) with O b some operator and ψ(x) a normalized wave function satisfying the boundary conditions ψ() = and ψ() =. We approximate ψ(x) by its values ψ n at the N points x n = n N+1 : ψ n = ψ(x n ). The operator O b is then represented by an N N matrix and the integral becomes a discrete sum. 1.2 Approximate Evaluation of Integrals General For more details and discussion of better algorithms see Numerical Recipes, Chapter. The simplest ( trapezoidal rule ) approximation to an integral over the finite range is obtained by dividing the range into intervals, assuming the function varies linearly over each interval, and doing the integration analytically in each integral. This gives Z f(x) n= f(x n+1 )+f(x n ) 2 In our problem, the function vanishes at x =and x = so f(x )=f(x N+1 )= and Eq 2 becomes Z f(x) (2) f(x n ) (3) The error in this approximation is of the order of N 2,meaningthatfor large N, if you double the number of points the error gets smaller by a factor of. 1

2 1.2.2 Normalization One integral wich arises frequently in quantum mechanics is the normalization integral Z norm = φ (x)φ(x) () where φ(x) is a wave function which we would like to normalize. Applying Eq 3 we obtain norm φ (x n )φ(x n ) (5) Observe that if we represent the set of N numbers φ(x n ) by a vector of N entries (hereishowthevectorforn =3) v φ = φ(x 1) φ(x 2 ) (6) φ(x 3 ) then the summation in Eq 5 is just the dot product of v φ and its complex conjugate vφ φ (x n )φ(x n )=vφ v φ (7) Thus we may write norm v φ v φ (8) This formula is useful because mathematica has a convenient set of commands for taking the dot product of two vectors Expectation Values Consider first the expectation value of the x operator Z <x>= ψ (x)xψ(x) (9) Applying Eq 3 gives (recall x n = n N+1 ). <x> ψ (x n )x n ψ(x n ) (1) As shown in class, the function x n ψ(x n ) is the result of multiplying the vector n v ψ by the diagonal matrix X whose nn entry is N+1. For the case N =3we have X = 2 (11) 3 2

3 We may therefore write the summation in Eq 1 as the dot product of the vector vψ with the vector X v ψ obtained by multiplying the matrix X by the vector v ψ i.e. <x> v ψ X v ψ (12) Now consider the expectation value of the momentum operator Z < bp >= ψ (x) ~ i dψ(x) Numerical evaluation requires an approximate value for the function dψ(x) at the points x n. A suitable approximation is (13) dψ(x) x=x n ψ(x n+1) ψ(x n 1 ) = ψ(x n+1) ψ(x n 1 ) x n+1 x 2 n 1 N+1 (1) In other words, the derivative of ψ at point x n is determined by the values of ψ at the points x n+1 and x n 1. This relation may be encoded as the matrix relation v ~ i dψ(x) P v ψ (15) so that the vector v dψ(x) (set of numbers giving the value of ~ i time the derivative dψ/ at the points x n ) is obtained by multiplying the vector v ψ by the matrix P. Bearing in mind that ψ(x )=ψ(x N+1 )=we have (I write P here for the case N =) i P = 5~ i i 2 i i (16) i Thus < bp > v ψ P v ψ (17) We argued in class that the second derivative should be approximated as d 2 ψ(x) 2 ψ(x n+1) 2ψ(x n )+ψ(x n 1 ) (x n+1 x n ) 2 (18) 2 d 2 ψ 2 so that we should write the matrix T corresponding to ~ i as (here I write it for N =) 2 1 T = 25~ (19) 1 2 3

4 Taking the second derivative (times ~ i 2) ought to be the same thing as applying P twice. However, you can verify from Eqs 16 and 19 that this is not the case, for N =5. As an optional homework problem I ask you to show that as N the difference between using the matrix T defined from Eq 18 and the matrix obtained from the square of P goes to zero.

5 plotting.nb file:///c:/millis/quantumcourse/21_3/foo.html 1 of 2 9/8/23 6:8 PM Defining functions and making plots in mathematica Defining a function--say a simple sine function this function may be evaluated A function of two arguments Plotting a function Combining plots

6 plotting.nb file:///c:/millis/quantumcourse/21_3/foo.html 2 of 2 9/8/23 6:8 PM Converted by Mathematica September 8, 23