Physics 214 Lecture 1

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Physics 214
Waves and Quantum Physics
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Welcome to Physics 214

• Content Waves and Quantum Physics
• Format Active Learning (Learn from
  Participation)
• Textbook Young and Freedman -- Assignments on
  Syllabus page
• Lectures (presentations, demonstrations, ACTs)
• Discussion Sections (group problem solving) --
  start this week
• Labs (hands-on interactions with the phenomena)
  Submit Prelab to instructor at 1st session
  (next week)

Bring your calculator!
Ask the Professor (from the Homepage) An
opportunity for you to get answers in lecture to
your questions. This Thursday only 2 bonus
points for filling in the Ask the Professor
survey.
James Scholar Students Email rmartin_at_uiuc.edu
within 1 week. Note Make sure you do not have
a lab or discussion section before your Tuesday
lecture.
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WWW and Grading Policy

• This course makes extensive use of the
  World-Wide-Web
• http//online.physics.uiuc.edu/courses/phys214
• Here you will find general course announcements,
  the course syllabus, the course description
  information, the lecture slides, lab information,
  the homework assignments (which you will actually
  submit to be graded on the Web), sample exams,
  and the official gradebook.
• Faculty Lectures Richard Martin
  Labs Tai Chiang
• Discussion Mike Weissman
• Grading Policy
• See Course Description on Web for CAREFUL
  STATEMENT!!
• Your final grade for Physics 214 will be based
  your total score
• final exam (350 pts),
• midterm exam (200 pts),
• four labs (150 pts total), Begin in Week 2.
  Prelabs due then.
• six homeworks (150 pts total), and
• five quizzes (highest four 150 pts total)
• Rough guidelines for letter grade minima are
• A(960) A(935) A-(910)
• B(885) B(855) B-(830)
• C(800) C(770) C-(740)
• D(700) D(660) D-(620) F(lt620)

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What is 214 all about?(1)

• Many physical phenomena of great practical
  interest to engineers, chemists, biologists,
  physicists, etc. were not in 211212.
• Wave phenomena
• Classical waves (brief review)
• Sound, electromagnetic waves, waves on a string,
  etc.
• Traveling waves, standing waves
• Interference and the principle of superposition
• Constructive and destructive interference
• Amplitudes and intensities

Interference!

• Interferometers
• Colors of a soap bubble, . . .
• Precise measurements, e.g., Michelson
  Interferometer

• Diffraction
• Optical Spectroscopy - diffraction gratings
  (butterfly wings!)
• Optical Resolution - diffraction-limited
  resolution of lenses,

• Quantum Physics - See next slide.
• Particles act like waves -- Waves act like
  particles
• Completely different from classical physics!

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What is 214 all about? (2)

• Quantum Physics
• Particles act like waves!
• Particles (electrons, protons, nuclei, atoms, . .
  . ) interfere like classical waves, i.e.,
  wave-like behavior
• Particles have only certain allowed energies
  like waves on a piano

• Explains nature of chemical bonds, structure of
  molecules, solids, metals, semiconductors, . . .
• One equation (the Schrodinger equation)
  forelectron waves explains all these effects

• Lasers, Superconductors, . . .

• Quantum tunneling
• Particles can tunnel through walls!

• Waves act like particles!
• When you observe (detect) a wave, you find
  quanta, i.e., particle-like behavior
• Instead of a continuous intensity, the result is
  a probabilityof finding quanta!

• Probability and uncertainty are part of nature!

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Today

• Examples of Waves
• Superposition
• Holds exactly for electromagnetic waves in vacuum
  and quantum waves
• Very good approximation for sound, waves on a
  string, etc.
• Wave Equations
• Equations that describe waves key point is that
  solutions of linear wave equations obey the
  superposition principle
• Amplitude and intensity
• Start interference continued next time

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Waves are all around us

• Familiar examples of waves

Earth
Light from stars that has traveled billions of
years traversing the universe and observed on
earth
Sound waves travelingthrough the air

• Pressure waves in the air
• Characterized by
• Frequency (pitch)
• Wavelength
• Speed 330 m/s (approx)

• Electromagnetic waves in vacuum
• Characterized by
• Frequency (pitch)
• Wavelength
• Speed c (constant of nature)

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Act 1

• Examples of Waves In this course we will
  emphasize certain types of waves 1. Sound
  waves 2. Electromagnetic waves3. Waves on a
  string 4. Matter waves (later in the
  course)
• What are some other examples of waves

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Act 1 - Solution

• Examples of Waves In this course we will
  emphasize certain types of waves 1. Sound
  waves 2. Electromagnetic waves3. Waves on a
  string 4. Matter waves (later in the course)
  
• What are some other examples of waves5.
  Examples by students in class6. Water waves,
  waves on a drumhead, . . . We will have simple
  illustrations in general more complicated
  speed depends on depth of water

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Waves and Wave Equations

• Electromagnetic waves
• Maxwells equations (Physics 212), which lead to
  wave equations for the electric and magnetic
  fields

• General form of wave equation for waves with
  constant speed

See appendix fordefinition of apartial
derivativedenoted by ?h/ ?x, ?h2/ ?x2, etc.
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Waves and Wave Equations

• Sound Waves in air
• Pressure waves (derived from Newtons laws
  Physics 211) p(x,t)

Snapshot at a time t

• General form of wave equation for waves with
  constant speed

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Waves and Wave Equations

• Waves on a string under tension (piano, violin, .
  . . )
• Transverse displacement of the string y(x,t)

Snapshot at a time t The waves obey a wave
equation with a speed n determined by the
tension see next slide Note You cannot tell
if the wave is traveling to the left or right or
is a standing wave from the snapshot at one time
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Where does the wave equation come from?

• A simple example waves on a string under tension
  T. T pulls on each segment.

This segment is pulled down on each side
accelerates downward
Force due to tension
This segment is pulled equally up and down
The net force (up-down) on each segment of
length dx (and mass rdx) depends on the
curvature there. From Newtons Laws
curvature
acceleration
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A Wave Pulse

• A wave pulse is non-zero only in a finite length
  example of a pulse moving to the right Note a
  pulse always moves it cannot be a standing wave!

This segment is pulled up on each side
accelerates upward, i.e. its vertical motion is
increasing
This segment is pulled up on each side
accelerates upward, i.e. its vertical motion is
slowing
x
Force due to tension
This segment is pulled down on each side
accelerate downward
This segment is pulled equally up and down
Same wave equation!
x
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Superposition
Key point for this course!
See appendix fordefinition of apartial
derivativedenoted by ?h/ ?x, ?h2/ ?x2, etc.
If h1 and h2 are solutions, then c1h1c2h2 is a
solution!
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Act 2

• Here are 4 examples of equations

• Which ones obey the principle of superposition

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Act 2 - Solution

• Here are 4 examples of equations

• Note This is the form of the Schrodinger
  equation!

• Which ones obey the principle of superposition
• a, b, and c are all linear in h
• In d the last term involves h2 is non-linear
  if h1 and if h2 aresolutions, h1 h2 is not a
  solution

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Superposition of Waves

• Q What happens when two waves of the same type
  collide?
• A Because of superposition,
• the two waves just pass through each other
  unchanged! The wave at the end is just the sum
  of whatever would have become of the two parts at
  the beginning
• Superposition is an exact property for
• Electromagnetic waves in vacuum
• Matter waves in quantum mechanics (later)
• Established by experiment
• In this course we will assume other waves obey
  the principle of superposition This is a very
  good approximation for many cases
• Sound in air, waves on a string,

Sum of waves 1 and 2
Movie (super_pulse)
Movie (super_pulse2)
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Act 3

• If you added the two sinusoidal waves shown in
  the top plot, what would the result look like ?

(a)
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Act 3 - Solution

• The correct answer is (b) The sum of two or more
  sines or cosines having the same frequency is
  just another sine or cosine with the same
  frequency.

How do we know? Add graphically or use trig
identities
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Adding Waves - Superposition

• Superposition works for ANY linear equation,
• It does NOT require that the solutions all travel
  with the same speed
• It will hold for quantum matter waves too, which
  have wavelength-dependent speed.

Two waves withdifferent speeds
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Mathematical Form of Constant-Speed Waves
y

• Suppose we have some function y f(x)

x
0

• Let d vt Then
• f(x - vt) will describe the same shape
  moving to the right with speed v.
• f(x vt) will describe the same shape moving
  to the left with speed v.

v
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We can have all sorts of waveforms, but thanks to
superposition, if we find a nice simple set of
solutions, easy to analyze, we can just write
more complicated solutions as sums of simple
ones.
Wave Forms

• pulses caused by a brief disturbance of the
  medium

• various wavepackets

But we focus on harmonic waves simple
sinusoidal waves extending forever
These harmonic waves are useful ingredients
because they have the simplest behavior in
prisms, filters, diffraction gratings. It is a
mathematical fact that any reasonable waveform
can be represented as a combination of various
harmonic waves, i.e., sines and cosines. This is
the topic of Fourier Analysis (and very useful
for signal processing!)
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The Harmonic Waveform (in 1-D)
Y(x,t) for a fixed time t
x
Review from Physics 211/212. For more detail see
Lectures 26 and 27 in the 211 website
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Wave Properties

• Period The time T for a point on the wave to
  undergo one complete oscillation.

Period T
For a fixed position x
Amplitude A
t
Movie (twave)

• Speed The wave moves one wavelength ? in one
  period T so its speed is v ??/ T.

• Frequency f 1/T cycles/second.
• Angular frequency w 2p f radians/second

Movie (tspeed)
Careful Remember the factor of 2p
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Example 1

• What is the amplitude, A, of this wave?

• What is the period, T, of this wave?

• If this wave moves with a velocity v 18 m/s,
  what is the wavelength, l, of the wave?

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Example 1

• What is the amplitude, A, of this wave?

T 0.1 s

• What is the period, T, of this wave?

v fl l/T

• If this wave moves with a velocity v 18 m/s,
  what is the wavelength, l, of the wave?

l vT 1.8 m
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Act 4
The speed of sound in air is a bit over 300 m/s,
and the speed of light in air is about
300,000,000 m/s. Suppose we make a sound wave
and a light wave that both have a wavelength of 3
meters. 1. What is the ratio of the frequency of
the light wave to that of the sound wave?
(a) About 1,000,000 (b) About 0.000001 (c)
About 1000
2. What happens to the frequency if the light
passes under water?
(a) Increases (b) Decreases (c) Stays
the same
3. What happens to the wavelength if the light
passes under water?
(a) Increases (b) Decreases (c) Stays
the same
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Act 4 - Solution
The speed of sound in air is a bit over 300 m/s,
and the speed of light in air is about
300,000,000 m/s. Suppose we make a sound wave
and a light wave that both have a wavelength of 3
meters. 1. What is the ratio of the frequency of
the light wave to that of the sound wave?
(a) About 1,000,000 (b) About 0.000001 (c)
About 1000
2. What happens to the frequency if the light
passes under water?
(a) Increases (b) Decreases (c) Stays
the same
3. What happens to the wavelength if the light
passes under water?
(a) Increases (b) Decreases (c) Stays
the same
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Act 4 - Solution
The speed of sound in air is a bit over 300 m/s,
and the speed of light in air is about
300,000,000 m/s. Suppose we make a sound wave
and a light wave that both have a wavelength of 3
meters. 1. What is the ratio of the frequency of
the light wave to that of the sound wave?
(a) About 1,000,000 (b) About 0.000001 (c)
About 1000
Why does the wavelength change but not the
frequency?
The frequency does not change because the time
dependence of the wave is the same everywhere.
In the water the relation of wavelength and
frequency is different and l decreases.
Question Do we see frequency or wavelength?
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More Harmonic Plane Wave notation

• Consider a wave that is harmonic in x and has a
  wavelength of ?.

If the amplitude is maximum atx 0, this has
the functional form

• Now, if this is moving to the right with speed v
  it will be described by

• We defined the wave number

• k radians/wavelength, e.g. in radians/meter
• Radians are dimensionless, so k has dimensions of
  1/length. There are 2p radians (360?) in one
  wavelength or one period.

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Adding Waves - Superposition

• Example Suppose we have two waves with the same
  amplitude A1 and angular frequency ?. Then their
  wave numbers k are also the same. Suppose that
  one starts at phase ? after the other

y1 A1 cos(k x - ? t) and y2 A1
cos(k x - ? t ?)
Spatial dependence of 2 waves at t
0 Resultant wave
Trig identity
Amplitude Oscillation
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Act 5
A harmonic wave moving in the positive x
direction can be described by the equation
y(x,t) A cos ( kx - wt ). Which of the
following equations describes a harmonic wave
moving in the negative x direction?
(a) y(x,t) A sin (kx - wt) (b) y(x,t) A cos
(kx wt) (c) y(x,t) A cos (-kx wt)
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Act 5 - Solution
A harmonic wave moving in the positive x
direction can be described by the equation
y(x,t) A cos ( kx - wt ). Which of the
following equations describes a harmonic wave
moving in the negative x direction?
(a) y(x,t) A sin (kx - wt) (b) y(x,t) A cos
(kx wt) (c) y(x,t) A cos (-kx wt)
In order to keep the argument zero, if t
increases, x must decrease.
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Standing waves

• The formula y()(x,t) Acos(kx - wt) describes
  a wave of amplitude A moving in the x direction.

• The formula y(-)(x,t) Acos(kx wt) describes a
  wave of amplitude A moving in the -x direction.

• The sum of two waves with equal wavelength and
  amplitude traveling in x and x directions is a
  standing wave. If each traveling wave has
  amplitude A1, then y()(x,t) y(-)(x,t)
  A1cos(kx - wt) A1cos(kx wt) 2A1cos(kx)
  cos(wt)

where we used the identity
A 2A1

• This is a standing wave At each point x, y(x,t)
  A cos(kx) cos(wt) oscillates with frequency w.
  At each time t, y(x,t) A cos(kx) cos(wt)
  varies in space with wavelength l 2p/k.

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Standing waves Resonance
More in lecture 4

• A standing wave is the solutionfor a wave
  confined to a regionExamples
• Wave on a string with fixed ends
• Sound wave in a tube with closed ends (what
  happens with open ends?)

• Resonance describes the fact that a small driving
  force at the resonance frequency w can cause a
  large response
• Demonstration

Motor provides a smallforce near one end
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Harmonic Wave Summary

• A wave of amplitude A moving in the x direction
  is described by the formula y()(x,t) A cos(kx
  - wt)

• A wave of amplitude A moving in the -x direction
  is described by the formula y(-)(x,t) A cos(kx
  wt)

• A standing wave of amplitude A is described by
  the formula y(x,t) A cos(kx) cos(wt)

• In all cases, at each point x, the wave
  oscillates with values y(x,t) simple harmonic
  motion of angular frequency ?.

• The difference in the three cases is the relation
  of y(x1 ,t1 ) and y(x2 ,t2 ) at different points
  x1 ? x2 and t1 ? t2

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Amplitude and Intensity

• In the first part of 214 we will deal primarily
  with sound waves and electromagnetic waves (radio
  frequency, microwaves, light).
• How bright is the light? How loud is the sound?

Amplitude, A Intensity, I
SOUND WAVE peak differential pressure, po
power transmitted/area (loudness) EM WAVE
peak electric field, Eo power
transmitted/area (brightness)
Power transmitted is proportional to the square
of the amplitude.
Why? Think of waves of masses on springs What
are the KE and PE?
Transmitted power per unit area. E.g. (W/m2)

• We will rarely (if ever) calculate the magnitudes
  of p or E, and we will generally calculate ratios
  of intensities, so we can simplify our analysis
  and write

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Act 6
Pulses 1 and 2 pass through the same place at the
same time.. Pulse 2 has four times the peak
intensity of pulse 1, i.e., I2 4 I1.
1. What is the maximum possible total combined
intensity, Imax?
(a) 4 I1 (b) 5 I1 (c) 9 I1
2. What is the minimum possible intensity, Imin?
(a) 0 (b) I1 (c) 3 I1
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Act 6 - Solution
Pulse 2 has four times the peak intensity of
pulse 1, i.e., I2 4 I1.
1. What is the maximum possible intensity, Imax?
(a) 4 I1 (b) 5 I1 (c) 9 I1
2. What is the minimum possible intensity, Imin?
(a) 0 (b) I1 (c) 3 I1
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Act 6 - Solution
Pulse 2 has four times the peak intensity of
pulse 1, i.e., I2 4 I1.
1. What is the maximum possible intensity, Imax?
(a) 4 I1 (b) 5 I1 (c) 9 I1
2. What is the minimum possible intensity, Imin?
(a) 0 (b) I1 (c) 3 I1
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Next time Interference of wavesConsequence of
superposition

• Read Young and Freeman Sections 35.1, 35.2, and
  35.3
• Check the test your understanding questions
• Work problems on the two slides to prepare for
  Lecture 2.

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Problem for next time path length-dependent phase

• Each speaker alone produces intensity I1 1W/m2
  at the listener, and f 900 Hz.

Sound velocity v 330 m/s
d 3 m
I I1 1 W/m2
r1 4 m
Drive speakers in phase. Compute the intensity I
at the listener in this case
Hint f 2p(d/l) with d ? r2 - r1 Do you
see why?
Procedure 1) Compute path-length difference
d 2) Compute wavelength l 3) Compute phase
difference f 4) Write formula for resultant
amplitude A 5) Compute the resultant intensity,
I A2
Check your solution next lecture.
Answer I ?? W/m2
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Home Exercise 1

• Assume two waves with the same amplitude A1 1
  cm and angular frequency ?. They differ only in
  phase ?
• The resultant wave is

y1 A1cos(k x - ? t) and y2
A1cos(k x - ? t ?)

• Determine the amplitude A of the resultant wave
  in the following cases (you complete the table)

f (degrees) f (radians) A (cm) Rough
Drawing 450 p/4 1.85 900
1800 3600 300