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Bohr model of Hydrogen atom
Wave properties of matter
Energy levels from wave properties
Thu. Nov. 29 2007 Physics 208, Lecture 25 2
Hydrogen atom energies
Zero energy
n=1
n=2
n=3
n=4
€
E1 = −13.6
12 eV
€
E2 = −13.6
22 eV
€
E3 = −13.6
32 eV
En
erg
y
€
En = −13.6
n2 eV
Quantized energy levels: Each corresponds to
different Orbit radius Velocity Particle wavefunction Energy
Each described by a quantum number n
Thu. Nov. 29 2007 Physics 208, Lecture 25 3
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(speed), or kinetic energy.
Classical: any momentum, energy is possible.Quantum: momenta, energy are quantized
L
Thu. Nov. 29 2007 Physics 208, Lecture 25 4
Classical vs Quantum
Classical: particle bounces back and forth. Sometimes velocity is to left, sometimes to right
L
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
Thu. Nov. 29 2007 Physics 208, Lecture 25 5
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 L=n(/2) :
€
=2LOne half-wavelength
€
p =h
λ=
h
2L
momentum
L
Quantum wave function: superposition of both motions.
€
ψ x( ) =2
Lsin
2π
λx
⎛
⎝ ⎜
⎞
⎠ ⎟
Thu. Nov. 29 2007 Physics 208, Lecture 25 6
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.
€
=LTwo half-wavelengths
€
p =h
λ=
h
L= 2po
momentum
€
=2LOne half-wavelength
€
p =h
λ=
h
2L≡ po
momentumn=1
n=2
€
ψ x( ) =2
Lsin
2π
λx
⎛
⎝ ⎜
⎞
⎠ ⎟Wavefunction:
Thu. Nov. 29 2007 Physics 208, Lecture 25 7
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. It’s energy levels
A. are equally spaced everywhere
B. get farther apart at higher energy
C. get closer together at higher energy.
Thu. Nov. 29 2007 Physics 208, Lecture 25 8
Particle in box energy levels
Quantized momentum
Energy = kinetic
Or Quantized Energy
€
E =p2
2m=
npo( )2
2m= n2Eo
€
En = n2Eo
€
p =h
λ= n
h
2L= npo
En
erg
yn=1n=2
n=3
n=4
n=5
n=quantum number
Thu. Nov. 29 2007 Physics 208, Lecture 25 9
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
Thu. Nov. 29 2007 Physics 208, Lecture 25 10
Quantum dot: particle in 3D box
Energy level spacing increases as particle size decreases.
i.e
CdSe quantum dots dispersed in hexane(Bawendi group, MIT)
Color from photon absorption
Determined by energy-level spacing
Decreasing particle size
€
En +1 − En =n +1( )
2h2
8mL2−
n2h2
8mL2
Thu. Nov. 29 2007 Physics 208, Lecture 25 11
Interpreting the wavefunction
Probabilistic interpretationThe square magnitude of the wavefunction ||2 gives the
probability of finding the particle at a particular spatial location
Wavefunction Probability = (Wavefunction)2
Thu. Nov. 29 2007 Physics 208, Lecture 25 12
Higher energy wave functions
n=1
n=2
n=3
Wavefunction Probability
L
€
h
2L€
2h
2L€
3h
2L
€
h2
8mL2€
22 h2
8mL2€
32 h2
8mL2
n p E
Thu. Nov. 29 2007 Physics 208, Lecture 25 13
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
Thu. Nov. 29 2007 Physics 208, Lecture 25 14
Quantum Corral
48 Iron atoms assembled into a circular ring. The ripples inside the ring reflect the electron quantum states of a
circular ring (interference effects).
D. Eigler (IBM)
Thu. Nov. 29 2007 Physics 208, Lecture 25 15
Scanning Tunneling Microscopy
Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart.
Is a very useful microscope!
Tip
Sample
Thu. Nov. 29 2007 Physics 208, Lecture 25 16
Particle in a box, againL
Wavefunction Probability = (Wavefunction)2
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)
Thu. Nov. 29 2007 Physics 208, Lecture 25 17
Quantum mechanics says something different!
Quantum Mechanics:some probability of the
particle penetrating walls of box!
Low energy Classical state
Low energy Quantum state
Nonzero probability of being outside the box.
Thu. Nov. 29 2007 Physics 208, Lecture 25 18
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.
Thu. Nov. 29 2007 Physics 208, Lecture 25 19
Question
‘low’ probability
‘high’ probability
Suppose separation between boxes increases by a factor of two. The tunneling probability
A. Increases by 2
B. Decreases by 2
C. Decreases by <2
D. Decreases by >2
E. Stays same
Thu. Nov. 29 2007 Physics 208, Lecture 25 20
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)
N
H
H
H
Thu. Nov. 29 2007 Physics 208, Lecture 25 21
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.
N
H
H
H
Thu. Nov. 29 2007 Physics 208, Lecture 25 22
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.
Thu. Nov. 29 2007 Physics 208, Lecture 25 23
Scanning Tunneling Microscopy
Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart.
Is a very useful microscope!
Tip
Sample
Tip, sample are quantum ‘boxes’
Potential difference induces tunneling
Tunneling extremely sensitive to tip-sample spacing
Thu. Nov. 29 2007 Physics 208, Lecture 25 24
Surface steps on Si
Images courtesy M. Lagally, Univ. Wisconsin
Thu. Nov. 29 2007 Physics 208, Lecture 25 25
Manipulation of atoms Take advantage of tip-atom interactions to
physically move atoms around on the surface
D. Eigler (IBM)
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
Thu. Nov. 29 2007 Physics 208, Lecture 25 26
Quantum Corral
48 Iron atoms assembled into a circular ring. The ripples inside the ring reflect the electron quantum states of a
circular ring (interference effects).
D. Eigler (IBM)
Thu. Nov. 29 2007 Physics 208, Lecture 25 27
The Stadium Corral
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.
D. Eigler (IBM)
Thu. Nov. 29 2007 Physics 208, Lecture 25 28
Some fun!
Kanji for atom (lit. original child)
Iron on copper (111)
D. Eigler (IBM)
Carbon Monoxide man
Carbon Monoxide on Pt (111)
Thu. Nov. 29 2007 Physics 208, Lecture 25 29
Particle in box again: 2 dimensions
Same velocity (energy), but details of motion are different.
Motion in x direction Motion in y direction
Thu. Nov. 29 2007 Physics 208, Lecture 25 30
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
Thu. Nov. 29 2007 Physics 208, Lecture 25 31
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.
Thu. Nov. 29 2007 Physics 208, Lecture 25 32
Particle in a box
What quantum state could this be?
A. nx=2, ny=2
B. nx=3, ny=2
C. nx=1, ny=2
Thu. Nov. 29 2007 Physics 208, Lecture 25 33
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)
Thu. Nov. 29 2007 Physics 208, Lecture 25 34
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.
Thu. Nov. 29 2007 Physics 208, Lecture 25 35
Particle in 3D box Ground state
surface of constant probability
(nx, ny, nz)=(1,1,1)
2D case
Thu. Nov. 29 2007 Physics 208, Lecture 25 36
(211)
(121)
(112)
All these states have the same energy, but different probabilities
Thu. Nov. 29 2007 Physics 208, Lecture 25 37
(221)
(222)
Thu. Nov. 29 2007 Physics 208, Lecture 25 38
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
€
En = −13.6
n2 eV
•Orbital radius is
€
rn = n2ao
Thu. Nov. 29 2007 Physics 208, Lecture 25 39
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!
Thu. Nov. 29 2007 Physics 208, Lecture 25 40
Sommerfeld: modified Bohr model Differently shaped orbits
Big angular momentumSmall
angular momentum
All these orbits have same energy…… but different angular momenta
Energy is same as Bohr atom, but angular momentum quantization altered
Thu. Nov. 29 2007 Physics 208, Lecture 25 41
Angular momentum question
Which angular momentum is largest?
Thu. Nov. 29 2007 Physics 208, Lecture 25 42
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-1ℓ=0, most ellipticalℓ=n-1, most circular
These states all have the same energy
€
L = h l l +1( )
Thu. Nov. 29 2007 Physics 208, Lecture 25 43
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
Current
electron
Orbital magnetic moment
Thu. Nov. 29 2007 Physics 208, Lecture 25 44
Orbital magnetic dipole moment
Dipole moment
µ=IA
Current =
€
chargeperiod
=e
2πr /v=
ev
2πr
Area =
€
πr2
Can calculate dipole moment for circular orbit
€
μ =evr
2=
eh
2mmvr /h( ) ≡ μB L /h( )
€
μB ≡eh
2m= 0.927 ×10−23 A ⋅m2
= 5.79 ×10−5eV /Tesla
€
L = h l l +1( )
€
μ =μB l l +1( ) magnitude of orb. mag. dipole moment
In quantum mechanics,
Thu. Nov. 29 2007 Physics 208, Lecture 25 45
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 = 2ℓ+1
Example: For ℓ=1, m ℓ = -1, 0, or -1, corresponding to three different directionsof orbital bar magnet.
N
S
m ℓ = +1
S
N
m ℓ = -1ℓ=1 gives 3 states: m ℓ = 0
S N
Thu. Nov. 29 2007 Physics 208, Lecture 25 46
Question
For a quantum state with ℓ=2, how many different orientations of the orbital magnetic dipole moment are there?
A. 1B. 2C. 3D. 4E. 5
Thu. Nov. 29 2007 Physics 208, Lecture 25 47
Example: For ℓ=2, m ℓ = -2, -1, 0, +1, +2 corresponding to three different directionsof orbital bar magnet.
Thu. Nov. 29 2007 Physics 208, Lecture 25 48
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
€
E = −13.6
n2eV −
r μ •
r B
= −13.6
n2eV − μ zB
= −13.6
n2eV − ml μB B
This means that spectral lines will splitin a magnetic field
Thu. Nov. 29 2007 Physics 208, Lecture 25 49
Thu. Nov. 29 2007 Physics 208, Lecture 25 50
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).
Thu. Nov. 29 2007 Physics 208, Lecture 25 51
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.
N
S
Thu. Nov. 29 2007 Physics 208, Lecture 25 52
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 It’s size is intrinsic to the electron,
but the direction is variable.
The ‘bar magnet’ can point in different directions.
Thu. Nov. 29 2007 Physics 208, Lecture 25 53
Electron spin orientations
Spin up
Spin down
N
S
N
S
Only two possible orientations
Thu. Nov. 29 2007 Physics 208, Lecture 25 54
Thu. Nov. 29 2007 Physics 208, Lecture 25 55
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.
Thu. Nov. 29 2007 Physics 208, Lecture 25 56
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.
€
n, l , ml , ms( )
Thu. Nov. 29 2007 Physics 208, Lecture 25 57
Quantum Number Question
How many different quantum states exist with n=2?
1248
l = 0 :ml = 0 : ms = 1/2 , -1/2 2
statesl = 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
2s2
2p6
There are a total of 8 states with n=2
Thu. Nov. 29 2007 Physics 208, Lecture 25 58
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.
Thu. Nov. 29 2007 Physics 208, Lecture 25 59
Number of electrons
Which of the following is a possible number of electrons in a 5g (n=5, l=4) sub-shell of an atom?
222017
l=4, so 2(2l+1)=18. In detail, ml = -4, -3, -2, -1, 0, 1, 2, 3, 4and ms=+1/2 or -1/2 for each.
18 available quantum states for electrons
17 will fit. And there is room left for 1 more !
Thu. Nov. 29 2007 Physics 208, Lecture 25 60
Putting electrons on atom
Electrons are obey exclusion principle Only one electron per quantum state
Hydrogen: 1 electron one quantum state occupied
occupiedunoccupied
n=1 states
Helium: 2 electronstwo quantum states occupied
n=1 states
Thu. Nov. 29 2007 Physics 208, Lecture 25 61
Other elements: Li has 3 electrons
n=1 states, 2 total, 2 occupiedone spin up, one spin down
n=2 states, 8 total, 1 occupied
€
n =1
l = 0
ml = 0
ms = +1/2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
n =1
l = 0
ml = 0
ms = −1/2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
n = 2
l = 0
ml = 0
ms = +1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
€
n = 2
l = 0
ml = 0
ms = −1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
€
n = 2
l =1
ml = 0
ms = +1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
€
n = 2
l =1
ml = 0
ms = −1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
€
n = 2
l =1
ml =1
ms = +1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
€
n = 2
l =1
ml =1
ms = −1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
€
n = 2
l =1
ml = −1
ms = +1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
€
n = 2
l =1
ml = −1
ms = −1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
Thu. Nov. 29 2007 Physics 208, Lecture 25 62
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)
Electron Configurations
Thu. Nov. 29 2007 Physics 208, Lecture 25 63
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!
H1s1
Be2s2
Li2s1
N2p3
Mg3s2
Na3s1
C2p2
B2p1
Ne2p6
F2p5
O2p4
P3p3
Si3p2
Al3p1
Ar3p6
Cl3p5
S3p4
H1s1
He1s2
Thu. Nov. 29 2007 Physics 208, Lecture 25 64
Wavefunctions and probability Probability of finding an electron is given by the
square of the wavefunction.Probability large here
Probability small here
Thu. Nov. 29 2007 Physics 208, Lecture 25 65
Hydrogen atom: Lowest energy (ground) state
Spherically symmetric. Probability decreases
exponentially with radius. Shown here is a surface
of constant probability
€
n =1, l = 0, ml = 0
1s-state
Thu. Nov. 29 2007 Physics 208, Lecture 25 66
n=2: next highest energy
€
n = 2, l =1, ml = 0
€
n = 2, l =1, ml = ±1
€
n = 2, l = 0, ml = 0
2s-state
2p-state2p-state
Same energy, but different probabilities
Thu. Nov. 29 2007 Physics 208, Lecture 25 67
€
n = 3, l =1, ml = 0
€
n = 3, l =1, ml = ±1
3s-state
3p-state
3p-state
€
n = 3, l = 0, ml = 0
n=3: two s-states, six p-states and…
Thu. Nov. 29 2007 Physics 208, Lecture 25 68
…ten d-states
€
n = 3, l = 2, ml = 0
€
n = 3, l = 2, ml = ±1
€
n = 3, l = 2, ml = ±2
3d-state 3d-state3d-state
Thu. Nov. 29 2007 Physics 208, Lecture 25 69
Wave representingelectron
Electron wave around an atom
Wave representingelectron
Electron wave extends around circumference of orbit.
Only integer number of wavelengths around orbit allowed.
Thu. Nov. 29 2007 Physics 208, Lecture 25 70
Emitting and absorbing light
Photon is emitted when electron drops from one quantum state to another
Zero energy
n=1
n=2
n=3
n=4
€
E1 = −13.6
12 eV
€
E2 = −13.6
22 eV
€
E3 = −13.6
32 eV
n=1
n=2
n=3
n=4
€
E1 = −13.6
12 eV
€
E2 = −13.6
22 eV
€
E3 = −13.6
32 eV
Absorbing a photon of correct energy makes electron jump to higher quantum state.
Photon absorbed hf=E2-E1
Photon emittedhf=E2-E1
Thu. Nov. 29 2007 Physics 208, Lecture 25 71
The wavefunction
Wavefunction = = |moving to right> + |moving to left>
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.
Thu. Nov. 29 2007 Physics 208, Lecture 25 72
Silicon
7x7 surface reconstruction
These 10 nm scans show the individual atomic positions
Thu. Nov. 29 2007 Physics 208, Lecture 25 73
Particle in box wavefunction
x=0 x=L
€
ψ x( )2dx = Prob. Of finding particle in region dx
about x
Particle is never here
€
ψ x < 0( ) = ?
Particle is never here
€
ψ x > L( ) = ?
Thu. Nov. 29 2007 Physics 208, Lecture 25 74
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)