Download Atomic Physics

MTE 3 Results
Phy208 Exam 3
50
Get prepared for the Final!
40
# SCORES
Final
Mon. May 12, 12:25-2:25, Ingraham B10
Average 79.75/100
std 12.30/100
A 19.9%
AB 20.8%
B 26.3%
BC 17.4%
C 13.1%
D 2.1%
F 0.4%
30
20
10
0
30
40
50
60
70
80
SCORE
90
Remember Final counts 25% of final grade!
It will contain new material and MTE1-3
material
(no alternate exams!!! but notify SOON any
potential and VERY serious problem you have
with this time)
100
Atomic Physics
HONOR LECTURE
!
Previous Lecture:
Particle in a Box, wave functions and energy levels
Quantum-mechanical tunneling and the scanning
tunneling microscope
Start Particle in 2D,3D boxes
!
This Lecture:
More on Particle in 2D,3D boxes
Other quantum numbers than n: angular momentum
H-atom wave functions
Pauli exclusion principle
PROF. R. Wakai (Medical Physics)
Biomagnetism
Biomagnetism deals with the registration and analysis of magnetic fields which
are produced by organ systems in the body.
3
From last week: particle in a box
Summary of quantum information
Energy is quantized
Sometimes velocity is to left, sometimes to right
En =
Quantum mechanics:
Particle is a wave: p = mv = h/!
!
standing wave: superposition of waves traveling left and right => integer
number of wavelengths in the tube
" h2 %
p2
= n 2$
2'
2m
# 8mL &
the larger the box the lower the energy
of the particle in the box
A quantum particle in a box cannot be
at rest!
Fundamental state energy is not zero:
En=1 = 0.38 eV for an electron
in a quantum well of L = 1 nm
Consequence of uncertainty principle:
"x = L # "px $ 1/L % 0!
!
Energy
Classical: particle bounces back and forth.
n=5
n=4
n=3
n=2
n=1
Classical/Quantum Probability
n=3
Probability
(2D)
x
Ground state: same wavelength
(longest) in both x and y
Need two quantum #’s,
one for x-motion
one for y-motion
Use a pair (nx, ny)
Ground state: (1,1)
n=2
Tunneling: nonzero probability
of escaping the box. Tunneling
Microscope: tunneling electron
current from sample to probe
sensitive to surface variations
n=1
y
Particle in a 2D box
Same
energy
but different
probability in
space
Similar when n $%
(nx, ny) = (2,1)
Particle in 3D box
(nx, ny) = (1,2)
With increasing energy...
(222)
(221)
!
!
Ground state
surface of constant
probability
(nx, ny, nz)=(1,1,1)
px =
h
h
= nx
"n x
2L
same for y,z
!
(211)
(121)
(112)
E=
p2
p2
px2
+ y + z = E o (n x2 + n y2 + n z2 ) quantum states with
2m 2m 2m
same nx, ny, nz have
same E
(
2
manyo 3Dx
Eg: howE = E n + n + n
All these states have the same energy, but different
probabilities
!
)
2
2
particle
y
zstates
have 18E0?
(n ,n ,n ) = (4,1,1), (1,4,1), (1,1,4)
x
y
z
!
Other quantum numbers? H-atom
Quantization of angular momentum
!Bohr model fails describing atoms heavier than H
!Does it violate the Heisenberg uncertainty principle?
A) YES
B) No
13.6
eV rn = n 2 ao Bohr
n2
radius and energy of electron cannot be exactly
known at the same time!
! Hydrogen atom is 3D structure.
! Schrödinger:
L = h l(l + 1)
En = "
!
" is the orbital quantum number
States with same n, have same energy and can have " =
0,1,2,...,n-1 orbital quantum number
" =0 orbits are most elliptical
" =n-1 most circular
The z component of the angular
momentum must also be quantized
!
Should have 3 quantum numbers.
!Coulomb potential (electron-proton interaction) is spherically
symmetric.
x, y, z not as useful as r, !, #
! Modified H-atom should have 3 quantum numbers
Lz = m l h
m ℓ ranges from - ℓ, to ℓ integer
values=> (2ℓ+1) different values
!
The experiment: Stern and Gerlach
It is possible to measure the number of possible values of Lz
respect to the axis of the B-field produced by the electron
r
current
Orbital
magnetic
dipole
For a quantum state with " = 2, how many different orientations
of the orbital angular momentum respect to the z-axis are
there?
!
A. 1
B. 2
C. 3
D. 4
E. 5
µ
electron
Current
e-
!
!
!
!
!
!
s: ℓ=0
p: ℓ=1
d: ℓ=2
f: ℓ=3
g: ℓ=4
The electron moving on the orbit is like a current that
produces a magnetic momentum µ=IA
r
e
ev 2
e
e
µ = "r 2 =
"r =
(mvr) =
L
µ = µB l(l + 1)
T
2"r
2m
“atomic shells”
2m
!
!
Summary of quantum numbers
3D Surfaces of constant prob. for H-atom
Electron cloud: probability density in 3D of electron
around the nucleus
2
P(r," ,# )dV = $(r," ,# ) dV
!
!
For hydrogen atom:
!
!
!
En = "
13.6
!
eV
n : describes energy of orbit
n2
!"describes the magnitude of orbital angular momentum
m ! describes the angle of the orbital angular momentum
!
L = h l(l + 1)
Spherically symmetric.
Probability decreases
exponentially with radius.
Shown here is a surface
of constant probability
Lz = m l h
n = 1, l = 0, ml = 0
!
!
!
!
Next highest energy: n = 2
2s-state
n = 2, l = 0, ml = 0
2p-state
3p-state
3s-state
2p-state
n = 2, l = 1, ml = 0
!
n = 3: 2 s-states, 6 p-states and...
3p-state
n = 3, l = 0, ml = 0
n = 2, l = 1, ml = ±1
n = 3, l = 1, m l = 0
Same energy, but different probabilities
!
!
!
!
!
n = 3, l = 1, ml = ±1
...10 d-states
Radial probability
"n,l,m (r,# ,$ ) = Rn,l (r)Yl,m (# ,$ )
Angular
Radial
!
!
3d-state
3d-state
3d-state
n = 3, l = 2, ml = 0
n = 3, l = 2, ml = ±1
n = 3, l = 2, ml = ±2
!
rn = n 2 ao
!
!
!
!
!
For 1s, 2p, 3d, rpeak = a0, 4a0, 9a0
These are the Bohr orbit radii!
!
most probable distance of electron
from nucleus!
They behave like Bohr orbits
because for states with same E,
larger angular momentum
corresponds to more spherical orbits,
orbits are elliptical for small "
All quantum numbers of electrons in atoms
Electron spin
!
New electron property:
Electron acts like a
bar magnet with N and S pole.
Magnetic moment fixed…
Quantum state specified by four quantum numbers:
(n, l, ml , ms )
!
Three spatial quantum numbers (3-dimensional)
!
One spin quantum number
!
…but 2 possible orientations
of magnet: up and down
How many different quantum states exist with n=2?
A. 1
B. 2
C. 4
D. 8
Described by
spin quantum number ms
z-component of spin angular momentum Sz = msh
! = 0 : 2s2
ml = 0 : ms = 1/2 , -1/2
! = 1 : 2p6
ml = +1: ms = 1/2 , -1/2
ml = 0: ms = 1/2 , -1/2
ml = -1: ms = 1/2 , -1/2
2 states
2 states
2 states
2 states
!
Pauli exclusion principle
!
!
Building Atoms
Electrons obey Pauli exclusion principle
Only one electron per quantum state (n, !, m!, ms)
unoccupied
Atom
Configuration
H
1s1
He
1s2
Li
1s22s1
Be
1s22s2
B
1s22s22p1
1s shell filled
occupied
Hydrogen: 1 electron
n=1 states
one quantum state occupied
(n = 1,l = 0,ml = 0,ms = +1/2)
!
Helium: 2 electrons
two quantum states occupied
(n = 1,l = 0,ml = 0,ms = +1/2)
(n = 1,l = 0,ml = 0,ms = "1/2)
!
!
n=1 states
Ne
etc
(n=1 shell filled noble gas)
2s shell filled
1s22s22p6 2p shell filled
(n=2 shell filled noble gas)
The periodic table
!
!
Atoms in same column
have ‘similar’ chemical properties.
Quantum mechanical explanation:
similar ‘outer’ electron configurations.
H
1s1
Li
2s1
Na
3s1
K
4s1
Be
2s2
Mg
3s2
Ca
4s2
Sc
3d1
Y
3d2
8 more
transition
metals
B
C
N
O
F
2p1 2p2 2p3 2p4 2p5
Al
Si
P
S
Cl
3p1 3p2 3p3 3p4 3p5
Ga Ge As Se Br
4p1 4p2 4p3 4p4 4p5
He
1s2
Ne
2p6
Ar
3p6
Kr
4p6