Page 52 - Science at the nanoscale

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RPS: PSP0007 - Science-at-Nanoscale
June 9, 2009
Brief Review of Quantum Mechanics
42
L
z
Z
y
x
L
y
L
x
Figure 3.4.
Three-dimensional potential box.
the box, ψ(x, y, z) = 0, inside the box

(3.28)
ψ(x, y, z) = D sin (k x x) sin k y y sin (k z z)
where D is the normalisation constant and
n x π
k x =
L x
n y π
(3.29)
k y =
L y
n z π
k z =
L z
Hence the particle is now described by a set of integer quantum
numbers (n x , n y , n z ). The energy of the particle with mass m is ch03
given by
" 2 2 2 #
2
2
h h 2 2 2 i h n x n y n z
E = 2 k + k + k z = 2 + 2 + 2 (3.30)
y
x
8mπ 8m L x L y L z
A few interesting cases follow from the above relations.
Case 1: L x = L y = L z = L. Here the energy of the particle simpli-
fies to
2
2
2
(n + n + n )h 2
x
y
z
E = (3.31)
8mL 2