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Last Time

• Bohr model of Hydrogen atom
• Wave properties of matter
• Energy levels from wave properties

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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 Particle in a box

• Particle confined to a fixed region of spacee.g.
  ball in a tube- ball moves only along length L
• Classically, ball bounces back and forth in tube.
• This is a classical state of the ball.
• Identify each state by speed, momentum(mass)x(sp
  eed), or kinetic energy.
• Classical any momentum, energy is
  possible.Quantum momenta, energy are quantized

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Classical vs Quantum

• Classical particle bounces back and forth.
• Sometimes velocity is to left, sometimes to right

• Quantum mechanics
• Particle represented by wave p mv h / ?
• Different motions waves traveling left and right
• Quantum wave function
• superposition of both at same time

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Quantum version

• Quantum state is both velocities at the same time
• Ground state is a standing wave, made equally of
• Wave traveling right ( p h/? )
• Wave traveling left ( p - h/? )
• Determined by standing wave condition Ln(?/2)

Quantum wave function superposition of both
motions.
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Different quantum states

• p mv h / ?
• Different speeds correspond to different ?
• subject to standing wave condition integer
  number of half-wavelengths fit in the tube.

Wavefunction
n1
n2
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Particle in box question

• A particle in a box has a mass m. Its energy is
  all kinetic p2/2m. Just saw that momentum in
  state n is npo. Its energy levels
• A. are equally spaced everywhere
• B. get farther apart at higher energy
• C. get closer together at higher energy.

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Particle in box energy levels

• Quantized momentum
• Energy kinetic
• Or Quantized Energy

nquantum number
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Question

• A particle is in a particular quantum state in a
  box of length L. The box is now squeezed to a
  shorter length, L/2.
• The particle remains in the same quantum state.
• The energy of the particle is now
• A. 2 times bigger
• B. 2 times smaller
• C. 4 times bigger
• D. 4 times smaller
• E. unchanged

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Quantum dot particle in 3D box
CdSe quantum dots dispersed in hexane(Bawendi
group, MIT) Color from photon absorption Determine
d by energy-level spacing

• Energy level spacing increases as particle size
  decreases.
• i.e

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Interpreting the wavefunction

• Probabilistic interpretation
• The square magnitude of the wavefunction ?2
  gives the probability of finding the particle at
  a particular spatial location

Wavefunction
Probability (Wavefunction)2
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Higher energy wave functions
n
p
E
Wavefunction
Probability
n3
n2

• n1

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Probability of finding electron

• Classically, equally likely to find particle
  anywhere
• QM - true on average for high n

Zeroes in the probability!Purely quantum,
interference effect
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Quantum Corral
D. Eigler (IBM)

• 48 Iron atoms assembled into a circular ring.
• The ripples inside the ring reflect the electron
  quantum states of a circular ring (interference
  effects).

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Scanning Tunneling Microscopy
Tip
Sample

• Over the last 20 yrs, technology developed to
  controllably position tip and sample 1-2 nm
  apart.
• Is a very useful microscope!

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Particle in a box, again
Particle contained entirely within closed tube.
Open top particle can escape if we shake hard
enough. But at low energies, particle stays
entirely within box. Like an electron in metal
(remember photoelectric effect)
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Quantum mechanics says something different!
Low energy Classical state

• Quantum Mechanics
• some probability of the particle penetrating
  walls of box!

Low energy Quantum state
Nonzero probability of being outside the box.
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Two neighboring boxes

• When another box is brought nearby, the electron
  may disappear from one well, and appear in the
  other!
• The reverse then happens, and the electron
  oscillates back an forth, without traversing
  the intervening distance.

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Question
Suppose separation between boxes increases by a
factor of two. The tunneling probability

1. Increases by 2
2. Decreases by 2
3. Decreases by lt2
4. Decreases by gt2
5. Stays same

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Example Ammonia molecule

• Ammonia molecule NH3
• Nitrogen (N) has two equivalent stable
  positions.
• Quantum-mechanically tunnels 2.4x1011 times per
  second (24 GHz)
• Known as inversion line
• Basis of first atomic clock (1949)

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Atomic clock question

• Suppose we changed the ammonia molecule so that
  the distance between the two stable positions of
  the nitrogen atom INCREASED.The clock would
• A. slow down.
• B. speed up.
• C. stay the same.

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Tunneling between conductors

• Make one well deeper particle tunnels, then
  stays in other well.
• Well made deeper by applying electric field.
• This is the principle of scanning tunneling
  microscope.

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Scanning Tunneling Microscopy
Tip, sample are quantum boxes Potential
difference induces tunneling Tunneling extremely
sensitive to tip-sample spacing
Tip
Sample

• Over the last 20 yrs, technology developed to
  controllably position tip and sample 1-2 nm
  apart.
• Is a very useful microscope!

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Surface steps on Si

• Images courtesy M. Lagally, Univ. Wisconsin

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Manipulation of atoms

• Take advantage of tip-atom interactions to
  physically move atoms around on the surface

• This shows the assembly of a circular corral by
  moving individual Iron atoms on the surface of
  Copper (111).
• The (111) orientation supports an electron
  surface state which can be trapped in the corral

D. Eigler (IBM)
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Quantum Corral
D. Eigler (IBM)

• 48 Iron atoms assembled into a circular ring.
• The ripples inside the ring reflect the electron
  quantum states of a circular ring (interference
  effects).

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The Stadium Corral
D. Eigler (IBM)

• Again Iron on copper. This was assembled to
  investigate quantum chaos.
• The electron wavefunction leaked out beyond the
  stadium too much to to observe expected effects.
  

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Some fun!
Carbon Monoxide man Carbon Monoxide on Pt (111)

• Kanji for atom (lit. original child)
• Iron on copper (111)

D. Eigler (IBM)
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Particle in box again 2 dimensions
Motion in y direction
Motion in x direction

• Same velocity (energy), but details of motion
  are different.

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Quantum Wave Functions

• 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)

Probability(2D)
Wavefunction
Probability (Wavefunction)2
One-dimensional (1D) case
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2D excited states
(nx, ny) (2,1)
(nx, ny) (1,2)
These have exactly the same energy, but the
probabilities look different. The different
states correspond to ball bouncing in x or in y
direction.
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Particle in a box

• What quantum state could this be?
• A. nx2, ny2
• B. nx3, ny2
• C. nx1, ny2

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Next higher energy state

• The ball now has same bouncing motion in both x
  and in y.
• This is higher energy that having motion only in
  x or only in y.

(nx, ny) (2,2)
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Three dimensions

• Object can have different velocity (hence
  wavelength) in x, y, or z directions.
• Need three quantum numbers to label state
• (nx, ny , nz) labels each quantum state (a
  triplet of integers)
• Each point in three-dimensional space has a
  probability associated with it.
• Not enough dimensions to plot probability
• But can plot a surface of constant probability.

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Particle in 3D box

• Ground state surface of constant probability
• (nx, ny, nz)(1,1,1)

2D case
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(121)
(112)
(211)
All these states have the same energy, but
different probabilities
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(222)
(221)
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The principal quantum number

• In Bohr model of atom, n is the principal
  quantum number.
• Arise from considering circular orbits.
• Total energy given by principal quantum number

• Orbital radius is

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Other quantum numbers?

• Hydrogen atom is three-dimensional structure
• Should have three quantum numbers
• Special consideration
• Coulomb potential is spherically symmetric
• x, y, z not as useful as r, ?, ?

Angular momentum warning!
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Sommerfeld modified Bohr model

• Differently shaped orbits

All these orbits have same energy but
different angular momenta Energy is same as Bohr
atom, but angular momentum quantization altered
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Angular momentum question

• Which angular momentum is largest?

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The orbital quantum number l

• In quantum mechanics, the angular momentum can
  only have discrete values ,
  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
• These states all have the same energy

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Orbital mag. moment

• Since
• Electron has an electric charge,
• And is moving in an orbit around nucleus
• it produces a loop of current,and hence a
  magnetic dipole field, very much like a bar
  magnet or a compass needle.
• Directly related to angular momentum

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Orbital magnetic dipole moment
Can calculate dipole moment for circular orbit
Dipole moment µIA
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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

Example For l1, m l -1, 0, or -1,
corresponding to three different directionsof
orbital bar magnet.
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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• Example For l2, m l -2, -1, 0, 1, 2
  corresponding to three different directionsof
  orbital bar magnet.

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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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Summary of quantum numbers

• n describes the energy of the orbit
• l describes the magnitude of angular momentum
• m l describes the behavior in a magnetic field
  due to the magnetic dipole moment produced by
  orbital motion (Zeeman effect).

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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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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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Electron spin orientations

• Spin up

• Spin down

Only two possible orientations
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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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Include spin

• We labeled the states by their quantum numbers.
  One quantum number for each spatial dimension.
• Now there is an extra quantum number spin.
• A quantum state is specified four quantum
  numbers
• An atom with several electrons filling quantum
  states starting with the lowest energy, filling
  quantum states until electrons are used.

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Quantum Number Question
How many different quantum states exist with
n2? 1 2 4 8
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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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Number of electrons
Which of the following is a possible number of
electrons in a 5g (n5, l4) sub-shell of an
atom? 22 20 17
l4, so 2(2l1)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 17 will fit. And there is room left
for 1 more !
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Putting electrons on atom

• Electrons are obey exclusion principle
• Only one electron per quantum state

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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

• Elements are arranged in the periodic table so
  that atoms in the same column have similar
  chemical properties.
• Quantum mechanics explains this by similar
  outer electron configurations.
• If not for Pauli exclusion principle, all
  electrons would be in the 1s state!

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Wavefunctions and probability

• Probability of finding an electron is given by
  the square of the wavefunction.

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Hydrogen atom Lowest energy (ground) state

• 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 wave around an atom

• Electron wave extends around circumference of
  orbit.
• Only integer number of wavelengths around orbit
  allowed.

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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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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)