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Lecture 4, p 1
Lecture 4:
y
L
d
θ
Spectra of atoms reveal the
quantum nature of matter
Take a plastic grating from the bin as you enter class.
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View Full Document Lecture 4, p 2
Today’s Topics
*
Derivations in Appendix
(also in Young and Freeman, 36.2 and 36.4)
c
SingleSlit Diffraction
*
c
Multipleslit Interference
*
c
Diffraction Gratings
c
Spectral Resolution
c
Optical Spectroscopy
c
Interference + Diffraction
Lecture 4, p 3
Review of 2Slit Interference
• Only the phase difference matters.
Phase difference is due to source phases and/or path difference.
• In a more complicated geometry
(see figure on right)
,
one must calculate the total
path from source to screen.
• If the amplitudes are equal
⇒
Use trig identity: A = 2A
1
cos(
φ
/2).
• Phasors:
Phasors are amplitudes.
Intensity is the square of the phasor length.
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View Full Document Lecture 4, p 4
Phasors
A
1
α
α
A
φ
A
1
Lets find the resultant amplitude of
two waves using phasors.
Suppose the amplitudes are the same.
Represent
each wave by a vector with magnitude (A
1
) and
direction (
φ
).
One wave has
φ
= 0.
Isosceles triangle:
α=φ/2
.
So,
This is identical to our previous result !
More generally, if the phasors have different amplitudes
A
and
B:
C
2
= A
2
+ B
2
+ 2AB cos
φ
1
2
cos
2
=
φ
A
A
Here
φ
is the
external
angle.
φ
A
C
B
•
See the supplementary slide.
•
See text: 35.3, 36.3, 36.4.
•
See Physics 212 lecture 20.
•
Phasors make it easier to
solve other problems later.
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View Full Document Phasors for 2
Phasors for 2


Slits
Slits
c
Plot the phasor diagram for different
φ
:
A
1
A
1
φ
A
φ
=
45
°
8
/
λ
δ
=
A
1
A
1
φ
φ
=
90
°
4
/
=
A
1
A
1
φ
φ
=
135
°
8
/
3
=
A
1
A
1
φ
=
180
°
φ
2
/
=
A
1
A
1
A
φ
=
360
°
=
A
1
A
1
A
φ
=
0
0
=
A
1
A
1
φ
φ
=
225
°
8
/
5
=
A
1
A
1
φ
φ
=
270
°
4
/
3
=
A
1
A
1
φ
φ
=
315
°
8
/
7
=
I
0
4I
1
φ
0
2
π
2
π
θ
λ
/d

λ
/d
y
(λ
(λ
(λ
(λ
/d)L
(
λ
/d)L
*Smallangle approx.
assumed here
Q:
What happens when a plane wave meets a small aperture?
A:
The result depends on the ratio of the wavelength
λ
to the size
of
the aperture, a:
Huygens’ principle
A Consequence of Superposition
λ
>> a
Similar to a wave from a point source.
This effect is called
diffraction
.
λ
<< a
The transmitted wave is concentrated in the
forward direction, and at near distances the
wave fronts have the shape of the aperture.
The wave eventually spreads out.
We will next study what happens when waves pass through
one slit. We will use
Huygens’ principle
(1678):
All points on a wave front (
e.g.
, crest or trough) can be treated
as point sources of secondary waves with speed, frequency,
and phase equal to the initial wave.
Wavefront at
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This note was uploaded on 01/19/2012 for the course PHYS 214 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Staff
 Quantum Physics, Diffraction

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