the quantum for poets - physics &...

physics and poets, Fall 2018 !1

The Quantum for PoetsQuantum (“wave”) Mechanics

Benjamin Arizmendi


single slit diffraction of water wave

diffraction is hallmark of wave propagation

A wave is an oscillatory (periodic) disturbance moving (propagating) through a medium


Diffraction pattern of red laser by small hole

Maxwell’ equations (1862) predicted value of c; exp. verified by Hertz (1887)

Light is an electro-magnetic wave


Double Slit diffraction, Young 1801

diffraction is hallmark of wave propagation


h is Plank’s constant introduced to explain “black body spectrum” (recall CMB is a black body spectrum)

h, a tiny number in macroscopic units h= 6.6 x 10-34 Joules-second = 4.1 x 10 -15 eV-second

Fit to spectrum shape (h,T)

peak ~ 1/T

Atoms of black body cavity can be excited only in multiples of energy “quanta” E~frequency~1/(wavelength). Higher energy states less likely to be excited.

classical UV catastrophe


Photo-electric effectHertz, 1887

• if light frequency f< fmin, no PE • fmin depends on material • if f>fmin, PE emitted < nano-second • rate of PE proportional to light intensity • max. PE kinetic energy depends on f, not intensity

ejection of electrons (photo-electrons, PE) by light from metal plate

Maxwell’s EM wave, light energy proportional to intensity and independent of frequency.


Einstein 1905 Nobel prize 1921

light energy is absorbed in quanta, photons with E= hf = hc/(wave length)

h= 6.6 x 10-34 Joules-second = 4.1 x 10 -15 ev-second

multiplied by speed of light, hc = 1240 eV nm

Planck’s constant is a tiny number in macroscopic units

yellow 550 nm light photon has 2.25 eV first excited state of hydrogen is 10.2 eV


Scattering of light in atmosphere: Rayleigh scattering

scattering does not change wavelength scattered intensity ~ 1/(wavelength)4

shorter wavelength light (blue) more scattered than longer (red)

When you look at the sky, you are looking at the scattered light.

Light from setting sun going through more atmosphere has blue scattered out of sun-beam, so appears red.

Why is the sky blue?


Compton Effect17 keV x-rays, 1923

λ = c/f is wavelength

photon has energy hf and momentum hf/c

transfers energy and momentum to electron in collision

recoiling electron

scattered photon

λ = hc/E = 1240 eV-nm/17000 eV = 0.07 nm

incident photon


helium light from “spectral tube” separated by diffraction grating shows discrete emission lines

A gas-filled tube through which an electrical current is passed

All atoms show similar emission lines.

all light comes from jiggled (accelerated)

electrons

accelerated electrons emit light


The NucleusTheorized that most of mass of atom was concentrated in small core.

Experiment with Geiger & Marsden 1909 experimentally verified that atom had was concentration of positively charged “stuff” (nucleus) at the center with essentially the entire mass of the atoms.

Determined nuclear size ~ 1/1000 atomic size.

The atom is almost entirely empty space!

How then do atoms occupy space?

We now know that the nucleus is made of positively charged protons and neutral neutrons of approximately equal mass.


Bohr Atom

Quantized energy levels from ad-hoc “rule” that electron angular momentum is “quantized” as integer times Plank’s

constant

Calculates atomic spectra and gets them approximately correct


Davisson & Germer, diffraction of electrons (1927)

Electrons can be both particles and waves!


Ad-hoc Bohr quantization rule, not really a theory. More detailed features of atomic spectra not explained. Other problems cannot be addressed (e.g. scattering)

Classically orbiting electrons should continuously radiate and the electron should come to rest on the nucleus.

Problems…


Schrödinger Wave equation/Heisenberg Matrix Mechanics

A complete math (theory) addressing all non-relativistic physical problems


Δp

Lens

Δximage

ΔxΔp≥h/4π

Heisenberg microscope

At every instant a grain of sand has a definite position and

velocity. This is not the case with an electron.— Born


size of the hydrogen atom

density of points = probability to find electron

ΔxΔp=h/4π

Δx

E= KE+PE Δx smaller >KE Δx larger > PE

Δx = 0.05 nm Bohr Radius

electron (-e) attracted to proton (+e) atom is exactly neutral!


Neutron on a Table

gravity pulls neutron

down ΔZ≌10 μm

experiment has been done, 2002

E= KE+PE Δz smaller >KE Δz larger > PE


Pauli Exclusion Principle for identical, spin-1/2 particles

(e.g. electrons)

periodic table atoms occupy space

size


Max Born, by Kati Szilagyi

Probability Waves— Born rule


diffraction pattern builds up one

particle at a time

Probability Waves

electron has probability to go through either slit


Attempt to measure “which slit” washes out diffraction pattern


ject suggests the local particle picture in the source and onthe screen, but a wave model for the unobserved propagationof the object. Mathematically we describe the state of theparticle during the propagation as a superposition of states,in particular of position states, that are classically mutuallyexclusive. A classical object will either take one or the otherpath. A quantum object cannot be said to do that. The intrin-sic information content of the quantum system itself is insuf-ficient to allow such a description—even in principle.7 Wealso find the duality between objective randomness and de-terminism. The pattern on the screen is well determined forthe ensemble, but the detection point of a single object iscompletely unpredictable in all experiments.All of these ‘‘quantum mysteries’’ imply that in an experi-

ment the possibility of having a position is often the onlyobjective reality in contrast to the property of having a well-defined position.These reasons are why Richard Feynman emphasized that

the double-slit experiment is at the heart of quantummechanics:8 ‘‘In reality, it contains the only mystery, the ba-sic peculiarities of all of quantum mechanics.’’ We mightsuggest that another central issue of quantum physics,namely entanglement, is missing in this example. However,it turns out to be an essential ingredient if we consider howwe could diffuse which-path information to theenvironment—a phenomenon leading to loss of coherencebetween the neighboring paths in the double-slit experiment.The fact that the wave nature of matter is a cornerstone of

quantum mechanics, but that this very feature completelyescapes perception in our everyday life, is one of the remark-able properties of this theory. The smallness of Planck’s con-stant and therefore of the de Broglie wavelength of a macro-scopic object is certainly largely responsible for thenonobservability of quantum effects in the classical world.However, it is interesting to ask whether there are limits toquantum physics and how far we can push the experimentaltechniques to visualize quantum effects in the mesoscopicworld for objects of increasing size, mass, and complexity.We shall therefore briefly review the experimental efforts

in this field throughout the last century. Soon after Louis deBroglie’s proposed wave hypothesis for material particles,matter wave phenomena were experimentally verified for

electrons,5 atoms and dimers,9 and neutrons.10,11 Young’sdouble-slit experiment with matter waves was then done byJonsson for electrons,12 by Zeilinger and collaborators forneutrons,13 by Carnal and Mlynek for atoms,14 and bySchollkopf and Toennies for small molecules and noble gasclusters.15,16

Further advances in matter wave physics with atoms weremade possible by sophisticated techniques exploiting the in-teraction between atoms and light. Already in 1975 ideaswere put forward for slowing and cooling of atoms usinglight scattering.17,18 The rapid progress of this field was rec-ognized by the fact that the most important developments inthis field were recently awarded the Nobel prize for lasercooling19–21 in 1997 and for the experimental realization ofBose–Einstein condensates with dilute atomic vapor22,23 in2001. In Bose–Einstein condensates all atoms have ex-tremely long de Broglie wavelengths and are coherent overmacroscopic distances up to a millimeter. However, similarto light quanta in a laser beam, the atoms in a Bose–Einsteincondensate are kept sufficiently apart to keep their interac-tion weak. Therefore, in spite of the large coherence length,the interfering object is still of small mass and complexity.Even experiments demonstrating interference between twoBose–Einstein condensates24 can be viewed as a double-slitexperiment with many individual atoms, as witnessed also bythe fact that to explain the fringe spacing the de Brogliewavelength corresponding to the individual atom rather thana wavelength using the total mass of the condensate is used.Different questions and new experimental challenges arise

if we study particles in the almost opposite parameter regimewhere the interaction among the particles is much stronger.Covalently bound atoms form a new entity, a molecule orcluster, and the de Broglie wavelength of this system is de-fined by the total mass of all the atoms and by the center-of-mass velocity of the bound system. In the following we shallfocus on these complex objects.The very first demonstration of molecule interference

dates back to the days of Estermann and Stern9 in 1930, who

demonstrated experimentally diffraction of H2 at a LiF crys-

tal surface. Further experiments with diatomic molecules hadto await progress and interest in atom optics. A Ramsey-Borde interferometer was already realized for the iodine

dimer in 199425 and was recently used26 for K2 . Similarly, a

Mach–Zehnder interferometer was demonstrated27 for Na2 .

The near-field analog to the Mach–Zehnder interferometer, aTalbot–Lau interferometer, was recently applied to experi-

ments with Li2 .28 Diffraction at nanofabricated gratings also

turned out to be the most effective way to prove the exis-tence of the weakly bound helium dimer16 and to measure itsbinding energy.29

Based on these historical achievements we ask how far wemight be able to extend such quantum experiments and forwhat kind of objects we might still be able to show thewave–particle duality. Recently, a new set of experimentsexceeding the mass and complexity of the previously usedobjects by about an order of magnitude has been developedin our laboratory. These experiments with the fullerene mol-

ecule C60 will be described in Sec. II.

II. THE C60 EXPERIMENT

The cage-like carbon molecules earned their names‘‘fullerenes’’ and ‘‘buckminster fullerenes’’ because of theirclose resemblance to geodesic structures that were first dis-cussed by Leonardo da Vinci30 and implemented in buildings

Fig. 1. The double-slit experiment is the prototype experiment demonstrat-

ing the wave–particle duality in quantum mechanics. !a" A wave impingingon a wall with one sufficiently small slit will spread out behind this obstacle.

An explanation based on Huygen’s principle tells us that each point in the

wave front can be imagined as being a source of a spherical wavelet. The

fields of many such sources interfere on the screen and form the single slit

pattern. !b" If we open a second slit, which sees the same wave as the firstone, the field amplitude at a sufficiently long distance from the slits drops to

zero at specific points: we observe destructive interference due to the over-

lap of wave troughs and hills. !c" Which pattern can we expect if we replacethe continuous source by one that emits quanta, that is, discrete packages of

energy and/or mass that are well localized in space and time in the source?

Can a single particle as massive as a buckyball acquire information of two

spatially separate locations?

320 320Am. J. Phys., Vol. 71, No. 4, April 2003 Nairz, Arndt, and Zeilinger

‘‘In reality, it contains the only mystery, the basic peculiarities of all of quantum mechanics.’’ R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals McGraw-Hill, New York, 1965.

diffraction experiment


diffraction of large molecules—buckyballs, etc. (2003)


heated molecules emit heat photons, in principle revealing which slit and destroying

interference


New York Times, May 4, 1935.

EPR, Bell, Quantum Entanglement

“hidden variables” behind QM correlations


1966


EPR, Bell, Quantum Entanglement ``spooky action at a distance”

entangled state spins are 100% anti-correlated

a

b

c

QM says can only measure spin up or down with some probability

with respect to any direction

e- is “spin 1/2”


number electron 1 electron 2

n1 +a+b+c -a-b-c

n2 +a+b-c -a-b+c

n3 +a-b+c -a+b-c

n4 +a-b-c -a+b+c

n5 -a+b+c +a-b-c

n6 -a+b-c +a-b+c

n7 -a-b+c +a+b-c

n8 -a-b-c +a+b+c

“hidden variable” list to get 100% anti-correlated outcomes

no selection of values for numbers are constant with QM theory and measurements!


Definitive exp. Nature V 526, October 29, 2015

measurements separated by distance greater than light travel time between measurements— no possible “communication” between electrons!

use entanglement to transmit “unbreakable” encryption

use entanglement to build “quantum computer” (Feynman, 1982)


``I think it is safe to say that no one understands quantum mechanics. Do not keep saying to yourself, if you can possibly avoid it, 'But how can it possibly be like that?' because you will go down the drain into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.”— Richard Feynman, 1964


Relativistic QM and anti-particles Dirac 1928

E=mc2 + QM ⇒

energy can be converted into matter (particle/anti-particle)

and matter (particle/anti-particle) can be converted into energy

every particle has an anti-particle: same mass, opposite charge

positron discovered in Cloud Chamber, Carl Anderson, 1932


BARYOGENESISEnergy of big bang converted into matter and anti-matter

Universe is observed to be virtually all matter (galaxies, stars, planets)

❓


QEDQuantum Electrodyamics

Shin'ichirō Tomonaga, Julian Schwinger, Richard Feynman

Freeman Dyson's paper showed they were different formulations of the same theory."The Radiation Theories of Tomonaga, Schwinger, and Feynman". Physical Review. 75 (3): 486–502, 1949

https://en.wikipedia.org/wiki/Shin%27ichir%C5%8D_Tomonaga

https://en.wikipedia.org/wiki/Julian_Schwinger

https://en.wikipedia.org/wiki/Richard_Feynman

https://en.wikipedia.org/wiki/Freeman_Dyson

https://en.wikipedia.org/wiki/Physical_Review


Feynman Diagrams

space

time

A graphic representation of a mathematical formula


Infinities!

These can be “renormalized” by redefining 3 constants


∞ is absorbed in def. of electron charge eobs; divergence is very, very slow. At Planck scale,

ebare = eobs/p1� C = 1.05eobs @⇤ = 1019GeV

C =↵

3⇡log

⇤2

m2+ constant


Lamb Shift (1947)


Cosmological constant

quantum field theoretical

= observed X 10120

QFT Vacuum energy is linearly divergent


https://physics.aps.org/articles/v11/103?utm_campaignArts & Culture: Poetry Takes on Quantum PhysicsOctober 11, 2018• Physics 11, 103

The poem “World Lines: A Quantum Supercomputer Poem” by Amy Catanzano can be read multiple ways. Whenever two lines meet at a “knot word” (shown in white), the reader can choose the upper or lower line to

continue.

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