Last Time

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Last Time
Quantum dots (particle in box)
Quantum tunneling

• 3-dimensional wave functions

This weeks honors lecture Prof. Brad
Christian, Positron Emission Tomography
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Exam 3 results

• Exam average 76
• Average is at B/BC boundary

Course evaluations Rzchowski Thu, Dec.
6 Montaruli Tues, Dec. 11
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3-D particle in box summary

• Three quantum numbers (nx,ny,nz) label each state
• nx,y,z1, 2, 3 (integers starting at 1)
• Each state has different motion in x, y, z
• Quantum numbers determine
• Momentum in each direction e.g.
• Energy
• Some quantum states have same energy

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Question

• How many 3-D particle in box spatial quantum
  states have energy E18Eo?
• A. 1
• B. 2
• C. 3
• D. 5
• E. 6

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3-D Hydrogen atom

• Bohr model
• Restricted to circular orbits
• Found 1 quantum number n
• Energy , orbit
  radius

• From 3-D particle in box, expect that
• H atom should have more quantum numbers
• Expect different types of motion w/ same energy

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Modified Bohr model

• Different orbit shapes

1. A, B, C
2. C, B, A
3. B, C, A
4. B, A, C
5. C, A, B

These orbits have same energy, but different
angular momenta Rank the angular momenta from
largest to smallest
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Angular momentum is quantized orbital quantum
number l

• Angular momentum quantized ,
  l is the orbital quantum number
• For a particular n, l has values 0, 1, 2, n-1
• l0, most elliptical
• ln-1, most circular

For hydrogen atom, all have same energy
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Orbital mag. moment

• Orbital charge motion produces magnetic dipole
• Proportional to angular momentum

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Orbital mag. quantum number ml

• Directions of orbital bar magnet quantized.
• Orbital magnetic quantum number
• m l ranges from - l, to l in integer steps
  (2l1) different values
• Determines z-component of L
• This is also angle of L

For example l1 gives 3 states
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Question

• For a quantum state with l2, how many different
  orientations of the orbital magnetic dipole
  moment are there?A. 1B. 2C. 3D. 4E. 5

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Summary of quantum numbers
For hydrogen atom

• n describes energy of orbit
• l describes the magnitude of orbital angular
  momentum
• m l describes the angle of the orbital angular
  momentum

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Hydrogen wavefunctions

• Radial probability
• Angular not shown
• For given n, probability peaks at same place
• Idea of atomic shell
• Notation
• s l0
• p l1
• d l2
• f l3
• g l4

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Full hydrogen wave functions Surface of
constant probability

• Spherically symmetric.
• Probability decreases exponentially with radius.
• Shown here is a surface of constant probability

1s-state
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n2 next highest energy
2s-state
2p-state
2p-state
Same energy, but different probabilities
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n3 two s-states, six p-states and
3p-state
3s-state
3p-state
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ten d-states
3d-state
3d-state
3d-state
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Electron spin
New electron propertyElectron acts like a bar
magnet with N and S pole. Magnetic moment fixed
but 2 possible orientations of magnet up and
down
Described by spin quantum number ms
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Include spin

• Quantum state specified by four quantum numbers
  
• Three spatial quantum numbers (3-dimensional)
• One spin quantum number

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Quantum Number Question
How many different quantum states exist with
n2? A. 1 B. 2 C. 4 D. 8
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Question
How many different quantum states are in a 5g
(n5, l 4) sub-shell of an atom? A. 22 B.
20 C. 18 D. 16 E. 14
l 4, so 2(2 l 1)18. In detail, ml -4, -3,
-2, -1, 0, 1, 2, 3, 4and ms1/2 or -1/2 for
each. 18 available quantum states for electrons
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Putting electrons on atom

• Electrons obey Pauli exclusion principle
• Only one electron per quantum state (n, l, ml, ms)

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Atoms with more than one electron

• Electrons interact with nucleus (like hydrogen)
• Also with other electrons
• Causes energy to depend on l

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Other elements Li has 3 electrons
n2 states, 8 total, 1 occupied
n1 states, 2 total, 2 occupiedone spin up, one
spin down
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Electron Configurations
Atom Configuration
H 1s1
He 1s2
1s shell filled
(n1 shell filled - noble gas)
Li 1s22s1
Be 1s22s2
2s shell filled
B 1s22s22p1
etc
(n2 shell filled - noble gas)
Ne 1s22s22p6
2p shell filled
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The periodic table

• Atoms in same column have similar chemical
  properties.
• Quantum mechanical explanation similar outer
  electron configurations.

Na3s1
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Excited states of Sodium

• Na level structure
• 11 electrons
• Ne core 1s2 2s2 2p6(closed shell)
• 1 electron outside closed shell Na Ne3s1
• Outside (11th) electron easily excited to other
  states.

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Emitting and absorbing light
Zero energy
n4
n4
n3
n3
n2
n2
Photon emittedhfE2-E1
Photon absorbed hfE2-E1
n1
n1
Absorbing a photon of correct energy makes
electron jump to higher quantum state.

• Photon is emitted when electron drops from one
  quantum state to another

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Optical spectrum

• Optical spectrum of sodium
• Transitions from high to low energystates
• Relatively simple
• 1 electronoutside closed shell

Na
589 nm, 3p -gt 3s
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How do atomic transitions occur?

• How does electron in excited state decide to make
  a transition?
• One possibility spontaneous emission
• Electron spontaneously drops from excited state
• Photon is emitted

lifetime characterizes average time for
emitting photon.
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Another possibility Stimulated emission

• Atom in excited state.
• Photon of energy hf?E stimulates electron to
  drop.
• Additional photon is emitted,
• Same frequency,
• in-phase with stimulating photon

One photon in,two photons out light has been
amplified
?E
hf?E
Before
After
If excited state is metastable (long lifetime
for spontaneous emission) stimulated emission
dominates
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LASER

• Light Amplification by Stimulated Emission of
  Radiation

Atoms prepared in metastable excited
states waiting for stimulated emission Called
population inversion (atoms normally in ground
state) Excited states stimulated to emit photon
from a spontaneous emission. Two photons out,
these stimulate other atoms to emit.
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Ruby Laser

• Ruby crystal has the atoms which will emit
  photons
• Flashtube provides energy to put atoms in excited
  state.
• Spontaneous emission creates photon of correct
  frequency, amplified by stimulated emission of
  excited atoms.

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Ruby laser operation
3 eV
2 eV
Metastable state
1 eV

• PUMP

Ground state
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The wavefunction

• Wavefunction ? moving to rightgt moving
  to leftgt
• The wavefunction is an equal superposition of
  the two states of precise momentum.
• When we measure the momentum (speed), we find one
  of these two possibilities.
• Because they are equally weighted, we measure
  them with equal probability.

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Silicon

• 7x7 surface reconstruction
• These 10 nm scans show the individual atomic
  positions

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Particle in box wavefunction
Prob. Of finding particle in region dx about x
Particle is never here
Particle is never here
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Making a measurement

• Suppose you measure the speed (hence, momentum)
  of the quantum particle in a tube. How likely
  are you to measure the particle moving to the
  left?
• A. 0 (never)
• B. 33 (1/3 of the time)
• C. 50 (1/2 of the time)

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Interaction with applied B-field

• Like a compass needle, it interacts with an
  external magnetic field depending on its
  direction.
• Low energy when aligned with field, high energy
  when anti-aligned
• Total energy is then

This means that spectral lines will splitin a
magnetic field
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Orbital magnetic dipole moment
Can calculate dipole moment for circular orbit
Dipole moment µIA
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Electron magnetic moment

• Why does it have a magnetic moment?
• It is a property of the electron in the same way
  that charge is a property.
• But there are some differences
• Magnetic moment has a size and a direction
• Its size is intrinsic to the electron, but the
  direction is variable.
• The bar magnet can point in different
  directions.

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Additional electron properties

• Free electron, by itself in space, not only has a
  charge, but also acts like a bar magnet with a N
  and S pole.
• Since electron has charge, could explain this if
  the electron is spinning.
• Then resulting current loops would produce
  magnetic field just like a bar magnet.
• But
• Electron in NOT spinning.
• As far as we know, electron is a point particle.

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Spin another quantum number

• There is a quantum associated with this
  property of the electron.
• Even though the electron is not spinning, the
  magnitude of this property is the spin.
• The quantum numbers for the two states are
• 1/2 for the up-spin state
• -1/2 for the down-spin state
• The proton is also a spin 1/2 particle.
• The photon is a spin 1 particle.

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Orbital mag. moment
Make a question out of this

• Since
• Electron has an electric charge,
• And is moving in an orbit around nucleus
• produces a loop of current,and a magnetic dipole
  moment ,
• Proportional to angular momentum

magnitude of orb. mag. dipole moment
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Orbital mag. quantum number ml

• Possible directions of the orbital bar magnet
  are quantized just like everything else!
• Orbital magnetic quantum number
• m l ranges from - l, to l in integer steps
• Number of different directions 2l1

For example l1 gives 3 states
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Particle in box quantum states
n
p
E
Wavefunction
Probability
n3
n2

• n1

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Particle in box energy levels

• Quantized momentum
• Energy kinetic
• Or Quantized Energy

nquantum number
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Hydrogen atom energies

• Quantized energy levels
• Each corresponds to different
• Orbit radius
• Velocity
• Particle wavefunction
• Energy
• Each described by a quantum number n

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Quantum numbers

• Two quantum numbers

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Pauli Exclusion Principle

• Where do the electrons go?
• In an atom with many electrons, only one electron
  is allowed in each quantum state (n, l, ml, ms).
• Atoms with many electrons have many atomic
  orbitals filled.
• Chemical properties are determined by the
  configuration of the outer electrons.

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Atomic sub-shells

• Each