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The solutions of the Schrödinger equation form a complete basis. This means that any wavefunction can be written as a linear combination of the solutions. Take, for example, the function
(Strictly speaking this is not an acceptable wavefunction as explained on page 299 of Atkins. However, the function does illustrate how to write an arbitrary wavefunction as a linear combination of a set of functions that are normalized and orthogonal: i.e., an orthonormal set.) This function can be written as
| j(x) = |
infinity
å
n = 1
|
cn |
é
ê
ê
ê
ë |
æ
ú
Ö
|
|
sin(nx) |
ù
ú
ú
ú
û |
. |
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You will recognize on the right-hand-side the solutions of the Schrödinger equation for a particle in a box with walls at x = 0 and x = p. The task is to determine the coefficients cn.
a) Show that the coefficients can be determined by
| cn = |
ó
õ |
p
0
|
dx |
æ
ú
Ö
|
|
sin(nx)j(x). |
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(Hint: use the fact that the solutions of the Schrödinger equation form an orthonormal set.)
b) Show that c2m-1 = 0 with m = 1,2,3,.... (Hint: compare the integral over [0,p/2] with the one over [p/2,p].
c) Show that c4m = 0 with m = 1,2,3,.... (Hint: compare the integral over [0,p/4] with the one over [p/4,p/2], and the one over [p/2,3p/4] with the one over [3p/4,p].)
d) Show that
with m = 1,2,3,....
e) Plot
|
M
å
m = 1
|
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4
|
sin[(4m-2)x], |
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for several values of M and compare the result with j(x).
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