MTE 3 Results - physics.wisc.edu · 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

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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