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3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
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3.23 Fall 2007 – Lecture 5
THE HYDROGEN
ECONOMY
3.23 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)
Last time
1. Commuting operators, Heisenberg principle
2. Measurements and collapse of the
wavefunction
3. Angular momentum and spherical harmonics
4. Electron in a central potential and radial
solutions
3.23 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)
Simultaneous eigenfunctions of
L2 , L z
m
m
ˆ
LzYl (θ , ϕ ) = m=Yl (θ , ϕ )
2 m
2
m
ˆ
L Yl (θ , ϕ ) = = l ( l + 1) Yl (θ , ϕ )
Yl
m
(θ , ϕ ) = Θ (θ ) Φ m (ϕ )
m
l
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
An electron in a central
potential
2
ˆ
L
= 1 d ⎛ 2 d ⎞
ˆ
ˆ (r )
H =−
r
V
+
+
⎜
⎟
2
2
2 μ r dr ⎝ dr ⎠ 2μ r
2
G
ψ nlm (r ) = Rnlm (r )Ylm (ϑ , ϕ )
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
An electron in a central
potential (III)
unl (r ) = r Rnl (r )
= l (l + 1) Ze
Veff (r ) =
−
2
2μ r
4πε 0 r
2
⎡ = d
⎤
(
)
(
)
(
)
V
r
u
r
E
u
r
+
=
eff
nl
nl
nl
⎢−
⎥
2
⎣ 2μ dr
⎦
2
2
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
2
The Radial
Wavefunctions
for Coulomb V(r)
Z
R10
r2R210
2
0.4
1
0
0.2
0
1
2
3
4
R20
2
6
r
r2R221
0
2
6
10
4
8
12
16
0
4
r
0
2
6
10
r
0
2
6
10
r
0
4
8
12
16
r
0
4
8
12
16
r
0
4
8
12
16
r
r R 31
2
2
0.08
0.8
0.04
4
0
-0.04
8
12
16
0.4
r
0
rR
2
R32
0.04
Figures by MIT OpenCourseWare.
3
0.04
r
R31
X
2
0.08
0
-0.1
Y
1
r2R230
0.2
Thickness dr
0.15
0.1
0.05
0
r
0.4
r
0.15
0.1
0.05
0
10
R21
R30
0
r2R220
0.6
0.4
0.2
0
-0.2
0.12
0.08
0.04
0
0
r
0.8
0.02
0
2
32
0.4
0
4
8
12
16
r
0
Radial functions Rnl(r) and radial distribution functions r2R2nl(r) for
atomic hydrogen. The unit of length is aµ = (m/µ) a0, where a0 is the
first Bohr radius.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Solutions in a Coulomb
Potential
5d
4f
5g
Images removed; please see any visualization of the 5d, 4f, and 5g hydrogen orbitals.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
The Full Alphabet Soup
http://www.orbitals.com/orb/orbtable.htm
Courtesy of David Manthey. Used with permission.Source: http://www.orbitals.com/orb/orbtable.htm
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Good Quantum Numbers
• For an operator that does not depend on t:
d A
dt
...
=
d Ψ Aˆ Ψ
dt
=
∂
∂ ˆ
∂
Ψ Aˆ Ψ + Ψ
Ψ = ...
A Ψ + Ψ Aˆ
∂t
∂t
∂t
1 ⎡ˆ ˆ⎤
=
A, H ⎦
⎣
ih
• Then, if it commutes with the Hamiltonian, its
expectation value does not change with time
(it’s a constant of motion – if we are in an
eigenstate, that quantum number will remain
constant)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Three Quantum Numbers
• Ĥ ↔ Principal quantum number n
(energy, accidental degeneracy)
e2
Z2
Z2
Z2
= − (13.6058 eV ) 2 = − (1 Ry ) 2
En = −
2
8πε 0 a0 n
n
n
• L̂ ↔ Angular momentum quantum
number l
l = 0,1,…,n-1 (a.k.a. s, p, d… orbitals)
Lˆ z ↔ m = -l,-l+1,…,l-1,l
•
Magnetic quantum number m
2
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
How do you measure angular momentum ?
G
• Coupling to a (strong !) magnetic field B
Image removed due to copyright restrictions.
Please see any experimental setup for observing the Zeeman Effect.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Right experiment – wrong theory (Stern-Gerlach)
G
μ
μB ˆ
B
ˆ
ˆ
ˆ
ˆ
H →H+
L⋅B = H +
Lz Bz
=
=
Image courtesy Teresa Knott. Used with Permission.
G
μ
μB ˆ
B
ˆ
ˆ
ˆ
ˆ
ˆ
( L + 2S ) ⋅ B = H +
( Lz + 2Sˆz ) Bz
H →H+
=
=
Goudsmit and Uhlenbeck
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Spin
• Dirac derived the relativistic extension of
Schrödinger’s equation; for a free particle he
found two independent solutions for a given
energy
• There is an operator (spin S) that commutes
with the Hamiltonian and that can only have
two eigenvalues
• In a magnetic field, the spin combines with the
angular momentum, and they couple via
G
μ
Hˆ → Hˆ + B ( Lˆ + 2Sˆ ) ⋅ B
=
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Spin
Eigenvalues/Eigenfunctions
• Norm (s integer → bosons, half-integer →
fermions)
2
ˆ2
S Ψ spin = = s ( s + 1) Ψ spin
=
Sˆ z Ψ spin = ± Ψ spin
2
• Z-axis projection (electron is a fermion with
s=1/2)
• Spin-orbital: product of the “space”
wavefunction and the “spin” wavefunction
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Pauli Exclusion Principle
We can’t have two electrons in the same
quantum state →
Any two electrons in an atom cannot have
the same 4 quantum numbers n,l,m,ms
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
ENERGY LEVELS OF THE ELECTRONS ABOUT THEIR NUCLEI
Auf-bau
6p
5d
HIGH ENERGY
6s
5p
4d
5s
4p
4s
LOW ENERGY
4f
3d
3p
3s
2p
2s
1s
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)