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Phy208 Exam 3
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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 levelsQuantum-mechanical tunneling and the scanning tunneling microscope Start Particle in 2D,3D boxes
This Lecture:More on Particle in 2D,3D boxesOther quantum numbers than n: angular momentumH-atom wave functionsPauli 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/λ 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=1n=2
n=3
n=4
n=5€
En =p2
2m= n2 h2
8mL2
Summary of quantum informationEnergy 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:
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Δx = L⇒Δpx ≈1/L ≠ 0!
Classical/Quantum Probability
Similar when n →∞
n=3
n=2
n=1Tunneling: 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-motionone for y-motion
Use a pair (nx, ny)Ground state: (1,1)
x
y
Same energybut different probability in space
Ground statesurface of constantprobability
(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)
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px =hλnx
= nxh2L
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E =px2
2m+py2
2m+pz2
2m= Eo nx
2 + ny2 + nz
2( )
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E = Eo nx2 + ny
2 + nz2( )
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, ϑ, φ
★ 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!
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En = −13.6n2 eV
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rn = n2ao Bohr
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L = h l l +1( )
Quantization of angular momentum
ℓ is the orbital quantum number
States with same n, have same energy and can have ℓ = 0,1,2,...,n-1 orbital quantum numberℓ =0 orbits are most ellipticalℓ =n-1 most circular
The z component of the angular momentum must also be quantized
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Lz = mlh
m ℓ ranges from - ℓ, to ℓ integer
values=> (2ℓ+1) different values
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r µ
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µ =eTπr2 =
ev2πr
πr2 =e2m
(mvr) =e2m
L
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r µ = µB l l +1( )
The experiment: Stern and GerlachIt 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
Orbitalmagneticdipole
For a quantum state with ℓ = 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 ℓ describes the magnitude of orbital angular momentum m ℓ describes the angle of the orbital angular momentum
For hydrogen atom:
Summary of quantum numbers
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Lz = mlh
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L = h l l +1( )
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En = −13.6n2 eV
Spherically symmetric. Probability decreases
exponentially with radius. Shown here is a surface
of constant probability
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n =1, l = 0, ml = 0
3D Surfaces of constant prob. for H-atom
Electron cloud: probability density in 3D of electronaround the nucleus
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P(r,ϑ ,ϕ)dV = Ψ(r,ϑ ,ϕ) 2dV
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n = 2, l =1, ml = 0
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n = 2, l =1, ml = ±1
2s-state2p-state
2p-state
Same energy, but different probabilities
Next highest energy: n = 2
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n = 2, l = 0, ml = 0
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n = 3, l =1, ml = 0
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n = 3, l =1, ml = ±1
3p-state
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n = 3, l = 0, ml = 0
n = 3: 2 s-states, 6 p-states and...
3s-state3p-state
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n = 3, l = 2, ml = 0
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n = 3, l = 2, ml = ±1
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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 ℓ
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Ψn,l,m (r,ϑ ,ϕ) = Rn,l (r)Yl,m (ϑ ,ϕ)Radial Angular
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rn = n2ao
New electron property:Electron acts like abar 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
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Sz = msh
Described byspin quantum number ms
Quantum state specified by four quantum numbers:
Three spatial quantum numbers (3-dimensional)
One spin quantum number
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n, l, ml , ms( )
How many different quantum states exist with n=2?
A. 1B. 2C. 4D. 8
ℓ = 0 :ml = 0 : ms = 1/2 , -1/2 2 states2s2
ℓ = 1 :ml = +1: ms = 1/2 , -1/2 2 statesml = 0: ms = 1/2 , -1/2 2 statesml = -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, ℓ, mℓ, ms)
Hydrogen: 1 electron one quantum state occupied
occupiedunoccupied
n=1 states
Helium: 2 electronstwo quantum states occupied
n=1 states
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n =1,l = 0,ml = 0,ms = +1/2( )
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n =1,l = 0,ml = 0,ms = +1/2( )
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n =1,l = 0,ml = 0,ms = −1/2( )
Pauli exclusion principle
Atom Configuration
H 1s1
He 1s2
Li 1s22s1
Be 1s22s2
B 1s22s22p1
Ne 1s22s22p6
1s shell filled
2s shell filled
2p shell filled
etc
(n=1 shell filled -noble gas)
(n=2 shell filled -noble gas)
Building Atoms
H1s1
He1s2
The periodic table
Atoms in same columnhave ‘similar’ chemical properties.
Quantum mechanical explanation:similar ‘outer’ electron configurations.
Be2s2
Li2s1
N2p3
C2p2
B2p1
Ne2p6
F2p5
O2p4
Mg3s2
Na3s1
P3p3
Si3p2
Al3p1
Ar3p6
Cl3p5
S3p4
As4p3
Ge4p2
Ga4p1
Kr4p6
Br4p5
Se4p4
Sc3d1
Y3d2
8 moretransition
metals
Ca4s2
K4s1