Hi sage supporters,
I am attempting to verify some properties of the quantum mechanics
"particle in a box" problem.
integral() is returning the wrong results for <x> and <x^2>.
I can't figure out what I might be doing wrong.
To find <x>:
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| Sage Version 4.1.2, Release Date: 2009-10-14 |
| Type notebook() for the GUI, and license() for information. |
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sage: var('a, n, x')
(a, n, x)
sage: assume(a > 0)
sage: assume(n, 'integer')
sage: integral(a/2 * x * sin(n*pi*x/a)^2,x,0,a).simplify_full()
1/8*a^3
The result should be a/2, which can almost be verified by inspection,
but I have worked out the integral by hand also, and I am confident that
a/2 is the correct result.
To find <x^2>:
sage: integral(a/2 * x^2 * sin(n*pi*x/a)^2,x,0,a).simplify_full()
1/24*(2*pi^2*a^4*n^2 - 3*a^4)/(pi^2*n^2)
I believe that the correct result is 1/3*a^2.
However, sage does perform the indefinite integrals correctly:
sage: integral(x*sin(x)^2,x)
1/4*x^2 - 1/4*x*sin(2*x) - 1/8*cos(2*x)
(which I used to hand-calculate <x>, getting the result a/2.)
and
sage: integral(x^2*sin(x)^2,x)
1/6*x^3 - 1/8*(2*x^2 - 1)*sin(2*x) - 1/4*x*cos(2*x)
I am running sage 4.1.2 on OS X 10.4 (compiled myself from source).
Thanks in advance for any insights!
Jim Clark
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