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The solution of the Schrödinger equation for the particle in a box with walls at x = 0 and x = p are given by
| yn(x) = cnsin(nx), with n = 1,2,3,..., |
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and cn a coefficient that takes care of the normalization.
a) Show that the wavefunctions yn are normalized if cn = Ö{2/p}.
b) Show that the wavefunctions yn are orthogonal. (Two wavefunctions are called orthogonal if the integral of their product equals zero. So here you have to show that
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ó
õ |
p
0
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dx yn(x)ym(x) = 0 |
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if n and m differ. Orthogonality is not something specific for a particle in a box. Different solutions of a Schrödinger equation are always orthogonal.)
To simplify the calculation of integrals of trigonometric functions it often helps to write these functions in terms of complex exponentials. For example,
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| = |
ó
õ |
b
a
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dx |
é
ê
ë |
1
2
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(eix+e-ix) |
ù
ú
û |
2
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| = |
1
4
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ó
õ |
b
a
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dx[ei2x+e-i2x+2] |
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| = |
1
4
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é
ê
ë |
1
2i
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(ei2b-ei2a) - |
1
2i
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(e-i2b-e-i2a) +2(b-a) |
ù
ú
û |
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| = |
1
4
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é
ê
ë |
1
2i
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(ei2b-e-i2b) - |
1
2i
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(ei2a-e-i2a) +2(b-a) |
ù
ú
û |
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| = |
1
4
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[sin2b-sin2a]+ |
1
2
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(b-a) |
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