Page 64 - Science at the nanoscale

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RPS: PSP0007 - Science-at-Nanoscale
June 9, 2009
Brief Review of Quantum Mechanics
54
Angular momentum along z direction operator
h
Y
L z Y = m
(3.65)
l
2π
where the complete form for Eq. (3.64) can be written as
2
2
2
∂ Y

1
1
h

h
∂
sin θ
−
= −l(l + 1)
2
2
2
∂θ
4π
sin(θ) ∂θ
4π
(3.66)
On the other hand, the radial function satisfies the following
equation:
2
2
2

Ze
h

∂
2 ∂
l(l + 1)
−
+
−
R(r) −
2
2
r ∂r
r
∂r
8mπ
(3.67)
Once the solutions to the wavefunction are obtained, we can
2
plot |ψ| and this gives the probability density distribution; the
probability of finding the electron in any region is equal to an
integral of the probability density over the region. Depending
on the quantum numbers of the electron, we can classify differ-
ent wavefunction for the electron in different states. States corre-
sponding to different l are in different orbitals. Table 3.1 gives a
summary of the various states for the hydrogen atom.
Table 3.2 Mathematical equations for the
various spherical harmonic functions.
Angular Function
Y l,m l
= 1/ 4π
Y 0,0
√ √ ∂Y 2 + sin (θ) ∂φ 2 4πε 0 r R(r) = ER(r) Y ch03
Y 1,0 = 3/4π cos(θ)
√
Y 1,1 = − 3/8π sin(θ)e iφ
√
Y 1,−1 = 3/8π sin(θ)e −iφ
√
1 2
Y 2,0 = 5/4π(3 cos (θ) − 1)
2
√
Y 2,1 = − 15/8π sin(θ) cos(θ)e iφ
√
Y 2,−1 = 15/8π sin(θ) cos(θ)e −iφ
√
1 2 i2φ
Y 2,2 = 15/2π sin (θ)e
4
1 √ 2 −i2φ
Y 2,−2 = 15/2π sin (θ)e
4