2) waves and particleshome.iiserb.ac.in/~sebastian/material/quantphys19/week3_waves.pdf · what is...
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3Week PHY 106 Quantum Physics Instructor: Sebastian Wüster, IISER Bhopal, 2018
These notes are provided for the students of the class above only. There is no warranty for correctness, please contact me if you spot a mistake.
2) Waves and Particles revision of movie:Quantum physics is essentially all about “things that ought to be particles are also waves” and “things that ought to be waves are also particles”.
Thus, let’s make sure we are all on the same page regarding waves…..
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What is a wave?
Definition of wave:A perturbation of some property is transported through a medium, without transport of the medium itself
2.1) Introduction to wave mechanics
Book: A.P. French, “Vibrations and waves”
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Examples:
Sound wave
Seismic wave
Waves
Rope waves
transverse wave
Elm wave
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Gravitational wave
Traffic waves
Spin waves
Waves
Water wave
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y(x) = A sin ( 2πλ
x)
X
y
amplitudewavelength
2.1.1) Waves, frequencies, wavelengths
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Waves
y(x, t) = A sin ( 2πλ
(x − Vt))
X
y
amplitudewavelength
wave velocity
v t
(5)
Form of progressive/travelling wave
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Waves
y(x, t) = A sin ( 2πλ
(x − Vt))
X
y
t = 0
t =λ4
1V
t =λ2
1V
Argument of sin is called phaseV here is the phase velocity
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k is the wave number
y(x, t) = A sin (kx − ωt)
ω is the angular frequency
(6)
Waves
y(x, t) = A sin ( 2πλ
(x − Vt))Rewrite wave form:
k =2πλ
(7)
ωk
= V (8)
is the frequency ω = 2πν (9)ν(unit 1/m)
(unit Hz = 1/s)
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Waves
k is the wave number
y(x, t) = A sin (kx − ωt)
ω is the angular frequencyis the frequency
k =2πλ
ωk
= V
ω = 2πν
Relation between frequency, wave length or wave number and phase velocity of any wave
(7)(6)
(8) (10)ν λ = V
(9)
(unit m/s)
ν
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Wave velocities ωk
= V (8) (10)ν λ = V
Examples:sound in solid
V =Yρ
≈ 5000 m/s
ν = 440 Hz λ = 11.4 m
gravitational wavesν = 440 Hz λ = 681 km
V = c = 299792458 m/s
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Wave velocities ωk
= V (8) (10)ν λ = V
Examples II:V = 500 km/h
ν = 3.3 /h λ = 151 km
water wave (tsunami)
light wave (elm) V = c = 299792458 m/s
ν = 4.2 × 1014 Hzλ = 700 nm
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Is there a general equation that governs wave behavior?
2.1.2) The wave equation
y(x, t) = A sin (kx − ωt)We see:
∂2
∂x2y(x, t) = − k2 A sin (kx − ωt) = − k2 y(x, t)
∂2
∂t2y(x, t) = − (−ω)2 A sin (kx − ωt) = − ω2 y(x, t)
(11)
(12)
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Wave equation Any function:
y(x, t) = f(x − Vt)
Chain rule:∂2
∂x2y(x, t) = (1)2 f′�′�(x − Vt)
∂2
∂t2y(x, t) = (−V)2 f′�′�(x − Vt)
Fulfills wave-equation:∂2
∂x2y(x, t) =
1V2
∂2
∂t2y(x, t)
(13)
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The wave equation
∂2
∂t2y(x, t) = − ω2 y(x, t)
∂2
∂x2y(x, t) = − k2 y(x, t)
ωk
= VWith: (8)
(11) (12)
y(x, t) = −1
ω2
∂2
∂t2y(x, t)
kω
=1V
∂2
∂x2y(x, t) =
1V2
∂2
∂t2y(x, t)
General wave equation
(13)
•Change in time causes change in space and vice versa.
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Wave equation moves to the right with velocity V!!y(x, t) = f(x − Vt)
t=0t=Δtt=2Δt
V Δt
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Wave equation moves to the right with velocity V!!y(x, t) = f(x − Vt)
y(x, t) = f(x + vt)• Moves to the left with velocity V, also fulfills wave equation
•Can be generalized to 2D, 3D
•There are many wave-equations, one for each medium.
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Superposition principle
The wave equation is linear. That means any combination of waves is also a solution
∂2
∂x2y(x, t) =
1V2
∂2
∂t2y(x, t)
∂2
∂x2w(x, t) =
1V2
∂2
∂t2w(x, t)
let:
∂2
∂x2 [y(x, t) + w(x, t)] =1
V2
∂2
∂t2 [y(x, t) + w(x, t)]
Then:
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2.1.3) Standing waves What happens if we combine two identical waves travelling in opposite directions?
https://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html
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2.1.3) Standing waves What happens if we combine two identical waves travelling in opposite directions?
Using (6), we can write this as:y(x, t) = A sin (kx − ωt) + A sin (−kx − ωt)
Animation from: https://www.youtube.com/
watch?v=ic73oZoqr70
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Standing waves
y(x, t) = A sin (kx − ωt) + A sin (−kx − ωt)
Trigonometric identitysin(α ± β) = sin(α)cos(β) ± cos(α)sin(β) (14)
y(x, t) = A[sin(kx)cos(ωt) − cos(kx)sin(ωt)+sin(−kx)cos(ωt) − cos(−kx)sin(ωt)]⏟−sin(kx) ⏟cos(kx)
y(x, t) = − 2A cos(kx) sin(ωt)
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Standing waves
y(x, t) = A cos(kx) sin(ωt)Formula for some standing wave
(15)
X
y
t =3π2ω
λ =2πk
t =π
2ωt =
πω
just before
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Standing waves
X
y
L0 λ =2πk
Boundary condition: y(0,t) = y(L, t) = 0
L = nλ2
Resonance condition for standing wave(16)λ =
2Ln
n = 1,2,3…
(16b)Now(*) let’s add:
* Q: Eq. (15) is an example that does not fulfill Eq. (16b). Find another example that does.
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Standing waves
Antenna
Examples:
http://whatmusicreallyis.com/research/physics/
09/12/2018 wmri_isolate_overharm.svg
file:///Users/sebastianwuester/writing/Lectures_Courses/IISERB/Phy106_QuantPhys/LectureNotes/Images/wmri_isolate_overharm.svg 1/1
Musical instrumentsBackreflected string wave
Micro-wave ovencurrent charge
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Interference
Standing wave: example where superimposed waves always cancel at anti-node:
2.1.4) Phenomena characteristic for waves
Superposition principle: Waves taking different paths get added.
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Interference
Circular waves on a water surface
Usually (2D, 3D) more options:
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Interference Usually (2D, 3D) more options:
Two circular waves: strengthen cancelhttps://www.youtube.com/watch?v=ovZkFMuxZNc
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Interference
Waves can show interference
-strengthening in certain directions/ at certain times: constructive interference
-weakening in certain directions/ at certain times: destructive interference
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Diffraction
Waves can turn around corners:
https://www.youtube.com/watch?v=BH0NfVUTWG4
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Diffraction
Decompose wave into lots of spherical waves:
Could see this from 2D wave equation
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Diffraction equal phase fronts
Slit smaller than wavelength: emits circular waves going in ALL directions
Slit larger than wavelength: waves destructively interfere if direction not almost forward (tutorial, waves and optics course)
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Diffraction and Interference
Double slit interference
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Diffraction and Interference Double slit interference
Xf
equal phase fronts
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Diffraction and Interference Double slit interference
d screen
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Diffraction and Interference Double slit interference
d screenL
r1
r2
y1(r1, t) =Ar1
sin(kr1 − ωt)
y2(r2, t) =Ar2
sin(kr2 − ωt)
X
z
Don’t worry about 1/r prefactors (energy conservation, see optics course later)
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Diffraction and Interference Double slit interference
d screenL
r1
r2
y1(r1, t) =Ar1
sin(kr1 − ωt)
y2(r2, t) =Ar2
sin(kr2 − ωt)
z
Fig. 2
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Diffraction and Interference Double slit interference
d screenL
r1
r2
y(z, t) = y1(r1(z), t) + y2(r2(z), t)
z
wave amplitude at position z
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Diffraction and Interference Double slit interference
d screenL
r1
r2z
I(z) = |y(z, t) |2 = y21 + y2
2 + 2y1y2
mean wave intensity at position z
( is time average)⋯
θ
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Diffraction and Interference
double slit interference pattern
I(θ) ≈ I0 cos(πdsin θ
λ) (17)
zθ
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Diffraction and Interference
d
r1
r2
θ
screen
z
Intensity
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Diffraction and Interference
d
r1
r2
Path difference
screen
z
Intensity
r2-r1
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Diffraction and Interference
d
r1
r2
screen
z
Intensity
Path difference r2-r1=λ
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Diffraction and Interference
d
r1
r2
screen
z
Intensity
Path difference r2-r1=λ/2
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Diffraction and Interference
double slit interference pattern
I(θ) ≈ I0 cos2(πdsin θ
λ) (17)
zθ
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Diffraction and Interference Examples:
Colors reflected from CD
Water in bay
VLA Radio Astronomy
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2.1.5) Electromagnetic waves
You will learn in Electro-magnetism lecture:
Changing magnetic field causes electric field (induction)
Changing electric field causes magnetic field
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Electromagnetic waves
Electric field
travelling direction of wave (transverse wave)
Magnetic field
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Electromagnetic waves Electric field
Electromagnetic wave equation:
Magnetic field
(18)( ∂2
∂x2+
∂2
∂y2+
∂2
∂z2 ) E(r, t) =1c2
∂2
∂t2E(r, t)
Speed of light (vacuum) c =1μ0ϵ0
(19)
c = 29 97 92 458 m/s
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Electromagnetic waves (10)ν λ = c