1. Exam 3 covers Lecture, Readings, Discussion, HW, Lab
Exam 3 is tonight, Thurs. Dec. 3, 5:30-7 pm, 145 Birge
Magnetic dipoles, dipole moments, and torque
Magnetic flux, Faraday effect, Lenz’ law
Inductors, inductor circuits
Electromagnetic waves:
Wavelength, freq, speed
E&B fields, intensity, power, radiation pressure
Polarization
Modern Physics (quantum mechanics)
Photons & photoelectric effect
Bohr atom: Energy levels, absorbing & emitting photons
Uncertainty principle
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2. From Last time… De Broglie wavelength Uncertainty principle
Wavefunction of a particle

3. Spatial extent of ‘wave packet’
x = spatial spread of ‘wave packet’
Spatial extent decreases as the spread in included wavelengths increases.
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4. Wavefunction
Quantify this by giving a physical meaning to the wave that describing the particle.
This wave is called the wavefunction.
Cannot be experimentally measured!
But the square of the wavefunction is a physical quantity.
It’s value at some point in space is the probability of finding the particle there!
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5. The wavefunction 2  Particle has a wavefunction (x) x xdx
Very small x-range
= probability to find particle in infinitesimal range dx about x
Larger x-range
= probability to find particle between x1 and x2
Entire x-range
particle must be somewhere
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6. Probability P(heads)=0.5 P(tails)=0.5 P(1)=1/6 P(2)=1/6 etc Heads
0.0
P(heads)=0.5
P(tails)=0.5 1
Probability
0.5
0.0 2 3 4 5 6
1/6
P(1)=1/6
P(2)=1/6
etc
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7. Discrete vs continuous
6-sided die, unequal prob
0.5
Loaded die
Probability
1/6
0.0 1 2 3 4 5 6
Infinite-sided die, all numbers between 1 and 6
“Continuous” probability distribution 1
P(x)
0.5
0.0 2 3 4 5 6
1/6 x
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8. Example wavefunction 2  What is P(-2<x<-1)? What is (0)?
= fractional area under curve -2 < x < -1
= 1/8 total area -> P = 0.125
What is (0)?
= entire area under 2 curve = 1
= (1/2)(base)(height)=22(0)=1
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9. Wavefunctions Each quantum state has different wavefunction
Wavefunction shape determined by physical characteristics of system.
Different quantum mechanical systems
Pendulum (harmonic oscillator)
Hydrogen atom
Particle in a box
Each has differently shaped wavefunctions
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10. Quantum ‘Particle in a box’
Particle confined to a fixed region of space e.g. ball in a tube- ball moves only along length L
Classically, ball bounces back and forth in tube.
A classical ‘state’ of the ball.
State indexed by speed, momentum=(mass)x(speed), or kinetic energy.
Classical: any momentum, energy is possible. Quantum: momenta, energy are quantized L
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11. 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 wavefunction:
superposition of both at same time
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12. Quantum version Quantum state is both velocities at the same time
Superposition waves is standing wave, made equally of
Wave traveling right ( p = +h/ )
Wave traveling left ( p = - h/ )
Determined by standing wave condition L=n(/2) :
One half-wavelength
momentum L
Quantum wave function: superposition of both motions.
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13. 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:
n=1
momentum
One half-wavelength
n=1
n=2
n=2
Two half-wavelengths
momentum
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14. 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.
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15. Particle in box energy levels
Quantized momentum
Energy = kinetic
Or Quantized Energy
n=quantum number
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16. Zero-point motion Lowest energy is not zero
Confined quantum particle cannot be at rest
Always some motion
Consequency of uncertainty principle
p cannot be zero
p not exactly known
p cannot be exactly zero
Comment here about right-hand-side of uncertainty relation.
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17. 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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18. Quantum dot: particle in 3D box
CdSe quantum dots dispersed in hexane (Bawendi group, MIT)
Color from photon absorption
Determined by energy-level spacing
Decreasing particle size
Energy level spacing increases as particle size decreases.
i.e
Video of making CdSe nanocrystals:
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19. Interpreting the wavefunction
Probability 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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20. Higher energy wave functionsL n p E
Wavefunction
Probability
n=3
n=2
n=1
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21. 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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22. 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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23. Particle in box again: 2 dimensions
Motion in x direction
Motion in y direction
Same velocity (energy), but details of motion are different.
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24. Quantum Wave Functions
Probability (2D)
Ground state: same wavelength (longest) in both x and y
Need two quantum #’s, one for x-motion one for y-motion
Use a pair (nx, ny)
Ground state: (1,1)
Wavefunction
Probability = (Wavefunction)2
One-dimensional (1D) case
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25. 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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26. 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
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27. Next higher energy state
Ball has same bouncing motion in x and in y.
Is higher energy than state with motion only in x or only in y.
(nx, ny) = (2,2)
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28. 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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29. Particle in 3D box Ground state surface of constant probability
(nx, ny, nz)=(1,1,1)
2D case
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30. All these states have the same energy, but different probabilities
(121)
(112)
(211)
All these states have the same energy, but different probabilities
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31. (222)
(221)
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32. 3-D particle in box: summary
Three quantum numbers (nx,ny,nz) label each state
nx,y,z=1, 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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