0 10 20 30 40 50 30 40 50 60 70 80 90 100 Phy208 Exam 3 # SCORES SCORE MTE 3 Results 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% Final Mon. May 12, 12:25-2:25, Ingraham B10 Get prepared for the Final! 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) 3 Atomic Physics 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 HONOR LECTURE ! 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. Classical: particle bounces back and forth. Sometimes velocity is to left, sometimes to right 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 From last week: particle in a box Energy n=1 n=2 n=3 n=4 n=5 E n = p 2 2m = n 2 h 2 8mL 2 " # $ % & ’ Summary of quantum information Energy is quantized 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 #"p x $ 1/ L % 0!
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Phy208 Exam 3
# S
CO
RE
S
SCORE
MTE 3 Results
Average 79.75/100std 12.30/100
A 19.9% AB 20.8%B 26.3%BC 17.4%C 13.1%D 2.1%F 0.4%
FinalMon. May 12, 12:25-2:25, Ingraham B10
Get prepared for the Final!
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)
3
Atomic Physics
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
HONOR LECTURE
! 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.
Classical: particle bounces back and forth. Sometimes velocity is to left, sometimes to right
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
From last week: particle in a box
Energy
n=1
n=2
n=3
n=4
n=5!
En =p2
2m= n
2 h2
8mL2
"
# $
%
& '
Summary of quantum information
Energy is quantized
the larger the box the lower the energyof the particle in the box
A quantum particle in a box cannot beat rest! Fundamental state energy is not zero:En=1 = 0.38 eV for an electronin a quantum well of L = 1 nm Consequence of uncertainty principle:
!
"x = L#"px $1/L % 0!
Classical/Quantum Probability
Similar when n $%
n=3
n=2
n=1
Tunneling: nonzero probability of escaping the box. Tunneling Microscope: tunneling electron current from sample to probesensitive to surface variations
Probability(2D)
(nx, ny) = (2,1) (nx, ny) = (1,2)
Particle in a 2D box
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)
x
y
Same energybut different probability in space
! Ground state
surface of constant
probability
! (nx, ny, nz)=(1,1,1)
All these states have the same energy, but different probabilities
(211) (121) (112)
Particle in 3D box
(221)
(222)
!
px =h
"nx
= nxh
2L
!
E =px
2
2m+py
2
2m+pz
2
2m= E
onx
2 + ny
2 + nz
2( )
!
E = Eonx
2 + ny
2 + nz
2( )nx,ny,nz( ) = 4,1,1( ), 1,4,1( ), (1,1,4)
With increasing energy...
same for y,z
quantum states with same nx, ny, nz have same E
Eg: how many 3D particle states have 18E0?
!Bohr model fails describing atoms heavier than H
!Does it violate the Heisenberg uncertainty principle? A) YES
B) No
! Schrödinger: Hydrogen atom is 3D structure.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
Other quantum numbers? H-atom
radius and energy of electron cannot be exactly known at the same time!
!
En
= "13.6
n2
eV
!
rn
= n2ao
Bohr
!
L = h l l +1( )
Quantization of angular momentum
" 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
!
Lz
= mlh
m ℓ ranges from - ℓ, to ℓ integer
values=> (2ℓ+1) different values
!
r µ
!
µ =e
T"r2 =
ev
2"r"r2 =
e
2m(mvr) =
e
2mL
!
r µ = µ
Bl l +1( )
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 current
e-
The electron moving on the orbit is like a current thatproduces a magnetic momentum µ=IA
Current
electron
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
s: ℓ=0
p: ℓ=1
d: ℓ=2
f: ℓ=3
g: ℓ=4
“atomic shells”
! n : describes energy of orbit
! !"describes the magnitude of orbital angular momentum
! m ! describes the angle of the orbital angular momentum
For hydrogen atom:
Summary of quantum numbers
!
Lz
= mlh
!
L = h l l +1( )
!
En
= "13.6
n2
eV
! Spherically symmetric.
! Probability decreases
exponentially with radius.
! Shown here is a surface
of constant probability
!
n =1, l = 0, ml
= 0
3D Surfaces of constant prob. for H-atom
Electron cloud: probability density in 3D of electronaround the nucleus
!
P(r," ,#)dV = $(r," ,#)2dV
!
n = 2, l =1, ml
= 0
!
n = 2, l =1, ml
= ±1
2s-state
2p-state2p-state
Same energy, but different probabilities
Next highest energy: n = 2
!
n = 2, l = 0, ml
= 0
!
n = 3, l =1, ml
= 0
!
n = 3, l =1, ml
= ±1
3p-state
!
n = 3, l = 0, ml
= 0
n = 3: 2 s-states, 6 p-states and...
3s-state
3p-state
!
n = 3, l = 2, ml
= 0
!
n = 3, l = 2, ml
= ±1
!
n = 3, l = 2, ml
= ±2
3d-state 3d-state3d-state
...10 d-states Radial probability
! For 1s, 2p, 3d, rpeak = a0, 4a0, 9a0
! These are the Bohr orbit radii!
! most probable distance of electron
from nucleus!
! They behave like Bohr orbits
because for states with same E,
larger angular momentum
corresponds to more spherical orbits,
orbits are elliptical for small "
!
"n,l,m
(r,# ,$) = Rn,l(r)Y
l,m(# ,$)
Radial Angular
!
rn
= n2ao
New electron property:
Electron acts like a
bar magnet with N and S pole.
Magnetic moment fixed…
…but 2 possible orientations
of magnet: up and down
Electron spin
z-component of spin angular momentum
!
Sz
= msh
Described byspin quantum number ms
! Quantum state specified by four quantum numbers:
! Three spatial quantum numbers (3-dimensional)
! One spin quantum number
!
n, l, ml, m
s( )
How many different quantum states exist with n=2?
A. 1
B. 2
C. 4
D. 8
! = 0 :
ml = 0 : ms = 1/2 , -1/2 2 states
2s2
! = 1 :
ml = +1: ms = 1/2 , -1/2 2 states
ml = 0: ms = 1/2 , -1/2 2 states
ml = -1: ms = 1/2 , -1/2 2 states
2p6
All quantum numbers of electrons in atoms
! Electrons obey Pauli exclusion principle
! Only one electron per quantum state (n, !, m!, m