Quantum Mechanics Course Zeemansplitting
Quantum Mechanics Course Zeemansplitting
Quantum Mechanics Course Zeemansplitting
• Periodic table
– Relationship to quantum numbers n, l, m
– Trends in radii and ionization energies
Schrödinger Equation: Coordinate Systems
Potential
1D Cartesian Kinetic energy Total energy
ℏ 2 d 2ψ ( x )
energy
− 2
+ V ( x )ψ ( x ) = Eψ ( x )
2m dx
Eigenfunctions:
ψ nlm (r , θ , φ ) = R n (r ) Y lm (θ , φ )
Laguerre Spherical
Polynomials Harmonics
2
− z E0 2
1 ke 2
n=1
l=0
n=2
l=1
Wave Functions: Angular Component
l=0 s-orbital
l=1 p-orbitals
l = 2 d-orbitals
http://cwx.prenhall.com/bookbind/pubbooks/giancoli3/chapter40/multiple3/deluxe-content.html
What is the relationship between the number of zero crossings for the radial
component of the wave function and the quantum numbers n and l?
Wave Functions: Angular & Radial Components
l = 0 s-orbitals
From: http://pcgate.thch.uni-bonn.de/tc/people/
hanrath.michael/hanrath/HAtomGifs.html
l=1 p-orbitals
100
l=2 d-orbitals
200 211 210 211
l=3 f-orbitals
P = ∫ψ ∗ψ (r ,θ ,φ )dV
∞ π 2π
= ∫ ∫ ∫ ψ ∗ψ r 2 sin θ dϕ dθ dr
0 0 0
Volume Element dV
∞
P = 4π ∫ψ ∗ψ r 2 dr for spherically symmetricψ
0
∞
P = ∫ P (r ) dr where P ( r ) = ( 4π r 2 )ψ ∗ψ
0
m=0
m=1
http://webphysics.davidson.edu/faculty/dmb/hydrogen/default.html
Can you draw the radial probability functions for the 2s to 3d wave functions?
Probability Density: Cross Sections
http://cwx.prenhall.com/bookbind/pubbooks/giancoli3/chapter40/multiple3/deluxe-content.html
Rank the states (1s to 3d) from smallest to largest for the electron’s most
PROBABLE radial position.
For which state(s) do(es) the most probable value(s) of the electron's position
agree with the Bohr model?
Probability Density: Problem
e− r ao
e−2r ao
where Ψ100 (r ) = and [ Ψ100 (r )]2
=
π a1.5
o π ao3
− ao ao
e
P100 (ao )∆r = (π ao2 )
π a3
(0.05ao )
o
after substitution of r , Ψ100 , and ∆r
Eigenvalues: L = l ( l + 1) ℏ
• Z-component of L
∂
Lzψ ( r ,θ ,φ ) = −iℏ ψ = mℏψ
ˆ
∂φ
Eigenvalues: Lz = mℏ
Orbital Momentum L: Vector Diagram
For l = 2, find the magnitude of the angular momentum L and the possible m
values. Draw a vector diagram showing the orientations of L with the z axis.
qv 2 q qL
Remember that µ = i A → π r = ( vr ) = where L = mvr
2π r 2 2m
Orbital l = 0, 1, 2, … Spin: s = ½
−gL µB − gs µB
µl = L = l ( l + 1) g L µ B µs = s = s ( s + 1) g s µ B
ℏ ℏ
z-component µlz = − ml g L µ B z-component µ sz = −ms g s µ B ≈ ± µ B
where µ B =
eℏ
= 5.79 × 10 −5 eV
and g L , g s = gyromagnetic ratios
2me T
Zeeman Effect: Splits m values
l=0 m=0
U = − µ ⋅ Bext ⇒ U = − µlz B Different energies for
different ml values!
assume z
direction
“Anomalous” Zeeman Effect: More Lines??
Zeeman ????
ml = 1
l = 1 ml = 0
ml = –1
Add external
magnetic field
l = 0 ml = 0
Why are there more energy levels than expected from the Zeeman effect?
• Electron’s spin magnetic moment µs interacts with internal B field caused by its
orbital magnetic moment µl and separates energy levels.
Spin down: µs
L, Bl
Low Energy
e- l=0
v j = 1/2
S p+ s = 1/2
Angular Momentum Addition: L + S gives J
• Special Case: L+S
Vectors Quantum Numbers
J = L+S j = l + s, l − s
J = j ( j + 1)ℏ m j = − j , − j + 1,... j − 1, j
Example: l = 1, s = ½
j = 1 + 12 = 3
2
and j = 1 − 12 = 1
2
m j = − 32 ,− 12 , 12 , 32 and m j = − 12 , 12
j = 3/2 j = 1/2
“Anomalous” Zeeman Effect: Spin-Orbit + Zeeman
Spin-Orbit Zeeman
mj = 3/2
mj = 1/2
mj = –1/2
j = 3/2 mj = –3/2
l = 1 s = 1/2
mj = 1/2
mj = –1/2
j = 1/2
mj = 1/2
l = 0 s = 1/2
j = 1/2 mj = –1/2
dB
Fz = µ z
• A magnetic force dz deflects atoms up or down by an
amount that depends on its magnet moment and the B field gradient.
• For hydrogen (mlz = 0), two lines are observed (spin up, spin down).
– Since l = 0, this experiment gave direct evidence for the
existence of spin.
Case of ml = -1, 0, 1
What is the “story” of this experiment? Did Stern & Gerlach know about
Stern-Gerlach Experiment: Problem
The angular momentum of the yttrium atom in the ground state is
characterized by the quantum number j = 5/2. How many lines would
you expect to see if you could do a Stern-Gerlach experiment with
yttrium atoms?
Remember that in the Stern-Gerlach experiment all of the atoms with
different mj values are separated when passing through an
inhomogeneous magnetic field, resulting in the presence of distinct
lines.
How many lines would you expect to see if the beam consisted of
atoms with l = 1 and s = 1/2?
ke 2
Multi-electron Atoms Vint =
r2 − r1
• Orbitals “fill” in table as follows: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p . .
– Only ONE electron per state (n, l, ml , ms ) - Pauli Exclusion Rule!
– Why is 4s filled before 3d? ⇒ 4s orbital has a small bump near
origin and “penetrates” shielding of core electrons better than 3d
orbital, resulting in a larger effective nuclear charge and lower
energy.
Noble Gas
Group VI
Group IV
Group III
Alkali
Halogen
Group V
l = 0 (s) Periodic Table
n
l = 1 (p)
1
2 l = 2 (d)
3
4
5
6
7
l = 3 (f)
Periodic Table: Trends for Radii and Ionization Energies
Maxima = alkali metals
• Effective atomic radii decrease
across each row of table.
– Why? Effective nuclear charge
increases and more strongly
attracts outer electrons,
decreasing their radius.
d-orbital
l=2
l = 1 m = ±1
f-orbital n=3
l=3
l = 0,1,2