Download Atomic and Molecular Physics for Physicists Ben-Gurion University of the Negev

Ben-Gurion University of the Negev
Atomic
Atomic and
and Molecular
Molecular Physics
Physics
for
for Physicists
Physicists
Ron Folman
Chapter 4: Wave-particle duality in nature: Photons and massive particles.
(photo-electric effect, diffraction and interference, de-Broglie,
beam splitter, Mach-Zehnder).
Main References: Corney A. Atomic and Laser
Spectroscopy, Oxford UP, 1987; QC 688.C67
1987; Chapters in Modern Physics, Open
University
Exercises:
Dudi Moravchik.
www.bgu.ac.il/atomchip
www.bgu.ac.il/nanofabrication
www.bgu.ac.il/nanocenter
First, summary of last week:
• Quantum mechanics was born out of several experimental problems such as
the UV catastrophe, and the quantized absorption and emission spectrum of the atoms
(light and Frank-Hertz)
• First good predictions were achieved when one assumed quantization of the
atomic oscillations in the bulk (Plank), and quantization of the electron angular
momentum (Bohr).
Wave-particle duality: Photons and massive particles.
(photo-electric effect, diffraction and interference, de-Broglie,
beam splitter, Mach-Zehnder).
This week we will deal with another problem in our classical understanding:
what is a particle?
In the first week we found that the double slit interference experiment works also
with single particles e.g. electrons. That means that a single particle behaves like
a wave going through both slits.
Thoughts: 1927
N. Bohr
Reality:1994
M. Heiblum
Other examples of particles behaving like waves
come from the scattering of particles off crystals.
neutrons
electrons
What about waves behaving like particles? This surprise came even earlier.
The photoelectric effect (Hertz 1887 – first observation, Einstein 1905 – model,
Nobel 1921, Millikan 1914 – accurate data, Nobel 1923).
L=1meter
Light source P=1W
Metal with ionization
energy Ei=1eV
Lets calculate how much time we need to wait until the first electrons are emitted:
S_atom=πr2=π (10-10)2
S_lightsphere=4πL2=4π 12
T= Ei/[P(S_atom/S_lightsphere)]=1.6 10-19 / [1(10-20)/4]=64sec
In reality, its immediate upon the arrival of the light!
The experiment
What the experiment teaches us:
1. Every metal has a different cut
off frequency.
2. Above the cutoff frequency, the
current is dependent on the light
intensity but not its frequency.
3. The maximal kinetic energy of
the emitted electrons is a linear
function of the frequency and is
Independent of the light intensity.
Different light intensities
Independent of intensity
Cutoff frequency
V0
Einstein’s model:
Photon energy = hν
Emax
eV0=Emax=hν - Φ
Prove from Millikan’s graph that the h
value suggested by Plank from
the black body radiation experiment
is correct also for Einstein’s model.
Φ
Back to massive particles: electrons, neutrons, atoms, molecules, etc.
How do we quantify this wave particle duality?
De-Broglie had a great idea:
Even a massive particle is a wave and its wave length is
λ_dB = h / p
Note, that if the kinetic energy Ek = 1/2 m v2 = 1/2 KB T
where T is the temperature, then
p=mv=m sqrt(K T /m)=sqrt(mKT) and hence λ_dB=h / sqrt(mKT)
Intel is going into quantum chips, BUT in an uncontrolled way!
Prove that: λ_dB=7nm (electron @ 300K).
Already now in Intels chips: T=2nm wire thickness and W=130nm width (193nm light)
In 10 years: 10nm width (100GHz)!
Prove that when the size of the confining potential is of the order of λ_dB,
only the bottom modes are populated and hence the propagation is quantum.
Assume the harmonic potential relations of ∆E=hν, ν=hbar/(m W2)
Beam splitting and interferometers – a test case for the wave particle duality:
A beam splitter is a device that creates a superposition
R
T
Where T=Transmission and R=reflection are complex amplitudes
( TR RT )
defined in the basis of H= ( 1 ) for horizontal going and V= ( 0 ) for vertical going
0
1
A beam splitter U may be described by a 2x2 matrix U=
V
H
H
V
Prove that since a quantum operator needs to be unitary, for a 50%/50% beam
Splitter i.e. |R|=|T|, the reflected beam gains
a π/2 phase difference relative to the transmitted beam i.e. R=Teiπ/2.
〰
Simple interferometer – The Mach-Zehnder
B
〰
A
Incoming single photon
100% mirror
Prove by adding phases and then by applying the matrix formalism that 100% of
the photons will arrive at detector A
Side note about interference:
Interference happens when a certain event is space and time has at least two
Indistinguishable histories
(remember the photon tagging with polarization in the second lecture)
Simplest interference calculation in QM: P=|A1+A2|^2
Home work:
1. Show that if a phase shifter (adding phase α) is introduced into one of the
arms of the MZ from the previous slide, the photon detection probability in
the bright detector goes like ½ (1+cos2α)
2. Explain why two photons meeting at a beam splitter always go together to
One of the sides (a process called 2-photon interference), and never
one there and one there. (note that the two photons come from the same source
And thus have the same phase).
Incoming single photon
Do they have to arrive at the
mirror at the same time?
〰
Exotics
Interaction free measurement – using the duality
or the non local nature of the photon particle
(A. Elitzur & L. Vaidman) to measure things without
interacting with them!
B
〰
A
Problem: you want to
verify that you have a certain
photosensitive material in your
bottle. Note that its so small that
only light can see it, but since its
photosensitive, and light will
destroy it!
Incoming single photon
100% mirror
Prove that we can detect the presence of the photosensitive material without
destroying it!! what maximal percentage will we be able to detect without destruction?
Realization: P. Kwiat, H. Harald & A. Zeilinger, Scientific American 275, 52 (1996).
For next week:
Please study the mathematical formalism of waves (unit 4 & chapter 1 of unit 5 in the
open university book), before the next lesson.
You should know: running waves, standing waves, border conditions, wave equation,
Transform Fourier.