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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.
2.
3.
4.
Commuting operators
operators, Heisenberg principle
Measurements and collapse of the wavefunction
Angular momentum and spherical harmonics
Electron in a central potential and radial
solutions
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
1
Simultaneous eigenfunctions of L2, Lz
LˆzYl m (θ , ϕ ) = m=Yl m (θ , ϕ )
Lˆ2Yl m (θ , ϕ ) = = 2l ( l +1) Yl m (θ , ϕ )
Yl m (θ , ϕ ) = Θlm (θ ) Φ m (ϕ )
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
An electron in a central potential
2
2
=
1
d
d
L̂
⎛
⎞
2
+ Vˆ (r )
Hˆ = −
⎜r
⎟+
2
2
2 μ r dr ⎝ dr ⎠ 2 μ r
G
ψ nlm (r ) = Rnlm (r )Ylm (ϑ ,
ϕ )
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
2
An electron in a central potential (III)
unl (r ) = r Rnl (r )
= 2 l (l + 1) Ze 2
Veff (r ) =
−
2
2μ r
4πε 0 r
⎡ =2 d 2
⎤
−
+V
(
r
)
eff
⎢
⎥ unl (r ) = Enl unl (r )
2
⎣ 2μ dr
⎦
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
The Radial Wavefunctions
for Coulomb V(r)
R10
r2R210
2
0.4
1
0
0.2
0
1
2
3
4
0
r
R20
0
1
2
3
4
r
r2R220
0.6
0.4
0.2
0
-0.2
2
6
10
r
0
2
6
10
r
0
2
6
10
r
r2R221
R21
0.12
0.08
0.04
0
R30
0.15
0.1
0.05
0
0
2
6
10
r
0.15
0.1
0.05
0
r2R230
0.4
0.08
0.2
4
0
-0.1
8
12
16
0.04
r
R31
0
0
4
8
12
16
r
0
4
8
12
16
r
0
4
8
12
16
r
r2R231
0.08
0.8
0.04
4
0
-0.04
8
12
16
0.4
r
0
r2R232
R32
0.04
0.8
0.02
0
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 am = (m/m) a0, where a0 is the
first Bohr radius.
Figures by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
3
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)
4
Good Quantum Numbers
• For an operator that does not depend on t:
•
d A
dt
=
d Ψ Â Ψ
dt
...
=
∂
∂ ˆ
∂
Ψ Aˆ Ψ + Ψ
A Ψ + Ψ Aˆ
Ψ = ...
∂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 = − 8πε 0 a0 n 2
n
n
• L̂2 ↔ Angular momentum quantum number l
l = 0,1,…,n-1
01
1 (a.k
( k.a. s, p, d… orbit
bitalls))
• L̂z ↔ Magnetic quantum number m
m = -l,-l+1,…,l-1,l
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
5
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
μ
Hˆ → Hˆ + B Lˆ ⋅ B = Hˆ + B Lˆz Bz
=
=
Image courtesy Teresa Knott. Used with Permission.
G
μ
μ
Hˆ → Hˆ + B ( Lˆ + 2 Sˆ ) ⋅ B = Hˆ + B (L̂z + 2 Sˆz ) Bz
=
=
Goudsmit and Uhlenbeck
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
6
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
G via
μ
B
ˆ) ⋅ B
ˆ
H→
Hˆ +
( Lˆ + 2 S
=
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Spin Eigenvalues/Eigenfunctions
• Norm (s integer → bosons
bosons, half
half-integer
integer → fermions)
Ŝ 2 Ψ spin = = 2 s ( s +1) Ψ spin
• Z-axis projection (electron is a fermion with s=1/2)
=
Sˆz Ψ spin = ± Ψ spin
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)
7
Pauli Exclusion Principle
We can’t
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
6p
5d
HIGH ENERGY
6s
Auf-bau
4d
5s
4p
3d
4s
LOW ENERGY
4f
5p
3p
3s
2p
2s
1s
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
8