Quantum Interference and Duality
Kiyohide NOMURA
Department of Physics

December 21, 2016

1 / 49Quantum Physics(Mechanics)▶ Basic notion of Quantum Physics: ”Wave-Particle Duality”▶

Light (electromagnetic wave)▶

Light as wave
Interference, Diﬀraction, Polarization▶ Light as a particle: Photon
Photoelectric eﬀect, Compton eﬀect

2 / 49Quantum Physics(Mechanics)▶ Basic notion of Quantum Physics: ”Wave-Particle Duality”▶

Light (electromagnetic wave)▶

Light as wave
Interference, Diﬀraction, Polarization▶ Light as a particle: Photon
Photoelectric eﬀect, Compton eﬀect▶

Electron▶

Electron as a particle
Mass-to-charge ratio
Elementary electric charge (Millikan’s oil drop experiment)▶ Electron as wave
Davisson-Germer experiment (1923-1927)

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Light as wave (Interference)
Interference: a phenomenon in which two waves superpose to forma resultant wave of greater, lower, or the same amplitude
MOVIE: Interference of waves from two point sources.

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Light as wave (Interference)
Interference: a phenomenon in which two waves superpose to forma resultant wave of greater, lower, or the same amplitude
MOVIE: Interference of waves from two point sources.

Figure: Left: constructive interference; Right: destructive interference

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Light as wave (Interference of thin film)

Figure: Interference pattern on a soap bubble

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Light as wave (Newton’s ring)

Another (and quantitative) example of interference:
Newton’s ring (1717)

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Light as wave (Newton’s ring)

Another (and quantitative) example of interference:
Newton’s ring (1717)
an interference pattern created by placing a very slightly convex
curved glass (lens) on an optical flat glass.

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Light as wave (Newton’s ring)
When viewed with monochromatic light, Newton’s rings appear asa series of concentric, alternating bright and dark rings centered at
the point of contact between the two surfaces.

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Light as wave (Newton’s ring)7 / 49

Light as wave (Newton’s ring)

R: the radius of curvature of the glass lensd: the vertical distance between the glass lens and the flat glass
R2 = (R − d)2 + r2 = R2 − 2Rd + d2 + r2()
d2
∴ r = 2Rd 1 −≈ 2Rd(∵ d ≪ R)
2R(1)
(2)
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Light as wave (Newton’s ring)▶ Reflection at the glass-air boundary causes no phase shift▶ Reflection at the air-glass boundary causes a half-cycle phase(π) shiftThus, when the distance 2d is mλ(λ: the wavelength) , the twowaves interfere destructively.
The radius r of the N th dark ring is given by√
r = mλR,(m = 0, 1, 2, · · · )(3)

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Light as wave: Double Slit

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Light as wave: Double Slit

Figure: Up Single slit Down: Double Slit distance between slits 0.7mm
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Light as wave: Diﬀraction
Diﬀraction: various phenomena which occur when a wave
encounters an obstacle or a slit

Figure: diﬀraction pattern from a slit of width four wavelengths with anincident plane wave

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Light as wave: Diﬀraction

MOVIE: diﬀraction pattern from a slit of width equal to five times the wa

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Light as wave: Diﬀraction

Figure:14 / 49

Light as wave: Diﬀraction

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Light as wave: Bragg’s law (X ray diﬀraction)

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Light as wave: Bragg’s law (X ray diﬀraction)

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Light as wave: Bragg’s law (X ray diﬀraction)

2d sin θ = nλ(n = 1, 2, · · · )(4)▶ λ: the wavelength of incident wave.
▶ d: separation between planes of lattice points▶ θ: the scattering angle
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Light as wave: Bragg’s law (X ray diﬀraction)

Figure: Laue pattern

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Light as wave: Bragg’s law (X ray diﬀraction)

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Light as a particle: Photoelectric eﬀect

Figure: the production of electrons or other free carriers when light isshone onto a material

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Light as a particle: Photoelectric eﬀect

Ekin = hν − W(5)▶ h: Planck’s constant (6.62606957 × 10−34 m2 kg/s)▶ ν: the frequency of the incident photon
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Light as a particle: Photoelectric eﬀect
The work function W (which gives the minimum energy required
to remove a delocalised electron from the surface of the metal) is
diﬀerent between materials.

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Light as a particle: Photoelectric eﬀect

Internal photoemission▶ Solar cell
▶ CCD
▶ photosynthesis in plants

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Light as a particle: Photoelectric eﬀect
Theory of the solar cell

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Light as a particle: Photoelectric eﬀect
CCD

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Light as a particle: Compton scattering
scattering between a photon (X-ray or gamma ray) and a charged
particle (electron)`λ′ − λ =h(1 − cos θ)me c(6)▶ λ: the initial wavelength, λ′ : the wavelength after
scattering,λ′ > λ▶ θ: the scattering angle▶ h: the Planck constant (6.62606957(29) × 10−34 m2 kg/s)▶ me : the electron rest mass (9.10938291(40) × 10−31 kg)▶ c: the speed of light (2.99792458 × 108 m/s)
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Light as a particle: Compton eﬀect
Energy quanta relation(quantum)
E = hν

(7)

The relation of the wavelength λ ,frequency ν , and light speed cc = λν(8)The relation between the energy E and the momentum p of theelectromagnetic wave
E = c|p|(9)
We obtain the relation between the momentum and the wavelengthp=hλ(10)28 / 49

Electron as a particle (Mass-to-charge ratio)

F = q(E + v × B)

(11)

dv
dt

(12)

(Lorentz force law)
F = ma = m(Newton’s second law of motion)
Combining the two previous equations yields:(mq)
a=E+v×B(13)29 / 49

Electron as a particle (Mass-to-charge ratio)

Figure: The cathode ray tube by Thomson

Cathode rays were emitted from the cathod C, passed through slits
A (the anode) and B (grounded), then through the electric field
generated between plates D and E, finally impacting the surface at
the far end.

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Electron as a particle (Mass-to-charge ratio)

Figure: Crookes tube

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Electron as a particle (Mass-to-charge ratio)

Figure: Crookes tube with the electric field

The cathode ray was deflected by the electric field
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Electron as a particle (Mass-to-charge ratio)

Figure: Crookes tube with the electric field

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Electron as a particle (Mass-to-charge ratio)

Figure: Crookes tube under the magnetic field
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Electron as a particle (Charge)
Millikan’s oil drop experiment(1909)
cover

severalthousandvolts

oilspraymicroscope
duniform electric eld

Figure: Millikan’s oil drop experiment

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Electron as a particle (Charge)

Figure: Millikan’s oil drop experiment

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Electron as a particle (Charge)

Figure: Millikan’s oil drop experimentelectric charge −1.602176565(35) × 10−19 C
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Electron as a wave

Figure: electron diﬀraction by Ta2 O5
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Electron as a wave

Figure: left: diﬀraction by X-ray right: diﬀraction by electron (Al film)

The similarity of the two diﬀraction patterns means that theelectron has a wave character as X-ray.
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Electron as a wave (de Broglie wave)

L. de Broglie proposed (1924) that the electron (in general matter)with momentum p behaves as a wave with wavelength
λ=hp(14)from the analogy with the photon.

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Electron as a wave
Double slit experimentA. Tonomura, 1982

Electrons pass through a device called the ”electron biprism”,
which consists of two parallel plates and a fine filament at thecenter
The filament is thinner than 1 micron.
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Electron as a wave

1. At the beginning of the experiment, we can see that brightspots (electrons) begin to appear here and there at random
positions (Fig. 2 (a) and (b))

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Electron as a wave

1. At the beginning of the experiment, we can see that brightspots (electrons) begin to appear here and there at random
positions (Fig. 2 (a) and (b))
2. Electrons are detected one by one as particles
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Electron as a wave

1. At the beginning of the experiment, we can see that brightspots (electrons) begin to appear here and there at random
positions (Fig. 2 (a) and (b))
2. Electrons are detected one by one as particles
3. Clear interference fringes can be seen in the last scene of theexperiment after 20 minutes (Fig. 2(d)).

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Electron as a wave

MOVIE: the electron double slit experiment

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Electron as a waveSupplement for double slit experiment
1. Electron source: Field-emission gun
more coherent and with up to three orders of magnitude
greater current density or brightness than can be achieved
with conventional thermionic emitters
2. These electrons were accelerated to 50,000 V (50keV), thespeed is about 40 % of the speed of the light, i. e., it is120,000 km/second.
3. There is no more than one electron in the microscope at onetime, since only 10 electrons are emitted per second.
4. No interaction between electrons.

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Electron as a waveSupplement for double slit experiment
1. Electron source: Field-emission gun
more coherent and with up to three orders of magnitude
greater current density or brightness than can be achieved
with conventional thermionic emitters
2. These electrons were accelerated to 50,000 V (50keV), thespeed is about 40 % of the speed of the light, i. e., it is120,000 km/second.
3. There is no more than one electron in the microscope at onetime, since only 10 electrons are emitted per second.
4. No interaction between electrons.
Single electron is enough to create a quantum interference!

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Neutron as a wave

Figure: A double-slit interference pattern made with neutrons.

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Neutron as a wave

Figure: neutron interferrometer

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Molecule as a wave: Phthalocyanine

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Molecule as a wave: Phthalocyanine

Figure:47 / 49

Molecule as a wave: Phthalocyanine

Figure:48 / 49

Molecule as a wave: Phthalocyanine

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