Transcript Last Time… - UW-Madison Department of Physics

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
Thurs. Dec. 3, 2009
Phy208 Lect. 26
1
From Last time…
De Broglie wavelength
Uncertainty principle
Wavefunction of a particle
Spatial extent of ‘wave
packet’
x
• x = spatial spread of ‘wave packet’
• Spatial extent decreases as the spread in
included wavelengths increases.
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Phy208 Lect. 26
3
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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Phy208 Lect. 26
4
The wavefunction

Particle has a wavefunction (x)
2

2(x)
x
dx
Very small x-range
x
2 x dx = probability to find particle in
infinitesimal range dx about x
x2
Larger x-range

Entire x-range
2

 xdx = probability to find particle
x1

between x1 and x2
2

 xdx  1 particle must be somewhere

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Phy208 Lect. 26
5
Probability
Probability
0.5
P(heads)=0.5
P(tails)=0.5
0.0
Heads
Tails
Probability
Pheads  Ptails 1
0.5

P(1)=1/6
P(2)=1/6
1/6
0.0
etc
1 2 3 4 5 6
P1  P2  P3  P4  P5  P6 1
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Phy208 Lect. 26
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Discrete vs continuous
6-sided die, unequal prob
0.5
1/6
0.0


Loaded die
Infinite-sided die, all
numbers between 1 and 6
“Continuous” probability
distribution
1 2 3 4 5 6
0.5
P(x)
Probability

6

 P(x)dx 1
0.0
0 1 2 3 4
x
1
Thurs. Dec. 3, 2009
1/6
Phy208 Lect. 26
5 6
7
Example wavefunction


What is P(-2<x<-1)?

1
2

 xdx = fractional area under curve -2 < x < -1
2

2
= 1/8 total area -> P = 0.125
What is (0)?

2

 xdx = entire area under 2 curve = 1

Thurs. Dec. 3, 2009
= (1/2)(base)(height)=22(0)=1  0 1/ 2
Phy208 Lect. 26
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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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Phy208 Lect. 26
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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
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
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Phy208 Lect. 26
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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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Phy208 Lect. 26
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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) :
  2L
One halfwavelength
L
Quantum wave function:
superposition of both motions.
Thurs. Dec. 3, 2009
momentum
h h
p 
 2L
Phy208 Lect. 26

 x  
2 2 
sin x 
L   
12

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.
  2L/n
  2L
n=1 One
halfwavelength
n=2
L
Two halfwavelengths
Thurs. Dec. 3, 2009
Wavefunction:  x  
2  2 
sinn
x 
L   
momentum
h h
p 
 po
 2L
n=1

n=2
Phy208 Lect. 26

momentum
h h
p    2po
 L
13
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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Phy208 Lect. 26
14
Particle in box energy levels
Quantized momentum
h
h
p n
 npo

2L
 Energy = kinetic
2
2
npo 
p

E

 n2Eo

2m
2m


Or Quantized Energy
Energy

n=5
n=4
En  n2Eo
n=3
n=quantum number
n=2
n=1
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Phy208 Lect. 26
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Zero-point motion


Lowest energy is not zero
Confined quantum particle cannot be at rest


Always some motion
Consequency of uncertainty principle xp  /2



p cannot be zero
p not exactly known
p cannot be exactly zero
Thurs. Dec. 3, 2009

Phy208 Lect. 26
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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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Phy208 Lect. 26
17
Quantum dot: particle in 3D box
CdSe quantum dots
dispersed in hexane
(Bawendi group, MIT)
Color from photon
absorption
Decreasing particle size


Determined by energylevel spacing
Energy level spacing increases
as particle size decreases.
i.e
2
E n 1  E n
Thurs. Dec. 3, 2009
n  1 h 2



8m L2
n 2h 2
8m L2
Phy208 Lect. 26
18
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
Thurs. Dec. 3, 2009
Probability = (Wavefunction)2
Phy208 Lect. 26
19
Higher energy wave functions
L
n
p
n=3
h
3
2L
E
Probability
2
h
32
8mL2
n=2  2 h
2L
h2
2
8mL2
 h
2L
h2
8mL2
 n=1
Wavefunction
Thurs. Dec. 3, 2009
2
Phy208 Lect. 26
20
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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Phy208 Lect. 26
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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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Phy208 Lect. 26
22
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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Phy208 Lect. 26
23
Quantum Wave Functions
Probability
(2D)
Wavefunction
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)
Probability = (Wavefunction)2
One-dimensional (1D) case
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Phy208 Lect. 26
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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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Phy208 Lect. 26
25
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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Phy208 Lect. 26
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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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Phy208 Lect. 26
27
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.
Thurs. Dec. 3, 2009
Phy208 Lect. 26
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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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Phy208 Lect. 26
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(121)
(112)
(211)
All these states have the same
energy, but different probabilities
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30
(222)
(221)
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Phy208 Lect. 26
31
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 px 

h
n
 nx
x
h
2L

Momentum in each direction: e.g.

2
2
p
p
p
 y  z  E o n x2  n y2  n z2 
Energy: E 
2m 2m 2m
2
x
Some quantum states have same energy

Thurs. Dec. 3, 2009
Phy208 Lect. 26
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