"Anyone who is not shocked by quantum theory has not understood it."—Niels Bohr

“I think I can safely say that nobody understands quantum mechanics."—Richard Feynman

Inside the Quantum World

Science's most precise and influential tool

Particles and waves, and their interactions and interrelationships

Atoms, molecules and material structure

Atomic, nuclear, particle, and condensed matter physics, chemistry, biology, information and

communication technology...

Quantum Theory or Science(“Quantum Mechanics” is Imprecise)

● Probably the most successful of all scientific theories (in terms of the accuracy and breadth of its predictions, and its impact)– Calculations of the fine structure constant, α,

within the context of quantum electrodynamics, agree with experiment to 0.32 parts per billion

– Applications: transistor, laser, semiconductors, integrated circuits, diagnostics, optical communications, clocks, computers....

Fluorescing Ions in a Linear Trap

Thirty-two ions, fluorescing under illumination by laser light in an electrodynamic trap

Annenberg Foundation 2013. All rights reserved.

Ultra-Cold Atoms Trapped in Standing Light Waves

Neutral rubidium atoms in an optical lattice trapAnnenberg Foundation 2013. All rights reserved.

Newton Wasn't Perfect

● Didn't know high speeds or really small objects● The Calculus is premised on continuous

distributions● On the microscopic level, nature is

– quantized: separated into discrete quantities

But he did claim that light was composed of particles...

Newton's Particle Theory of Light

● Little particles, so tiny they (intersecting beams, for example) don't scatter one another

● Obey the same laws of physics as all other objects

Foucault's Light Speed Experiment

Young's Double Slit Experiment

Young's Double Slit Experiment

Arago/Poisson Spot

Diffraction and Interference

Single Slit ~ 2 Wavelengths Wide© Eli Sidman, Technical Services Group, MIT.

© Eli Sidman, Technical Services Group, MIT.

Small Single Slit (top)Many Small Slits (bottom)

Photoelectric Effect

If Light Were a Wave...

● Electron ejection rate proportional to intensity● But at very low intensity, emission delayed● Maybe some rate dependence on frequency● Maximum kinetic energy of emitted electrons

probably related to intensity and maybe frequency

The Experiment

http://phet.colorado.edu/en/simulation/photoelectric

Homework

The Results

● Current always directly proportional to intensity—even down to the lowest intensities– “...no indication whatsoever of … a threshold...

intensity”

– Also, no measurable time delay even at lowest intensities

● The stopping potential unaffected by intensity, only by color (frequency, wavelength)– Maximum kinetic energy unaffected by intensity

The Interpretation (Einstein 1905)

● Electromagnetic radiation created by vibrating charged particles

● Particle energies are quantized● Emitted radiation therefore quantized● Quanton of radiation is called a photon.

What is a Wave?

● A wave is a periodic disturbance in a medium

● Essential properties:– Wavelength

– Frequency

– Velocity

– Amplitude

– Interference

© Adam Kleppner

© Annenberg Foundation 2013.

Additional Wave Characteristics

● Energy spreads out with wave fronts● Superposition

– Waves pass freely through one another

– At intersections, amplitudes add

(interference)

● In constrained systems, standing

waves© Eli Sidman, Technical Services Group, MIT.

© Annenberg Foundation 2013.

© Daniel Kleppner.

Wave Theory of Electromagnetic Radiation (Maxwell)

● Oscillations between electric and magnetic fields which spread according to a wave equation

● Can be reflected, refracted, polarized, and (most importantly) diffracted

● All travel at same speed, c

What is a Particle?

● A particle is an object of negligible size● Essential properties:

– Mass

– Momentum

– (Kinetic) Energy

“Particle” Theory of Electromagnetic Radiation (Einstein)

● Consists of bundles of energy called photons● Photon energy is proportional to the frequency

– The proportionality constant is called Planck's constant, h

● Being relativistic, photons carry moment = energy/c– Mechanical waves possess energy, but they carry

no momentum

Interference pattern built up photon by photon

Paradox

● Light exhibits all the behaviors predicted by Maxwell's wave theory of electromagnetic radiation (many of which particles do not exhibit)

● Einstein's particle-like energy bundles completely explain the photoelectric effect (wave theory doesn't)

[ The single-photon double-slit experiment is] a phenomenon which is impossible, absolutely

impossible to explain in any classical way, and which has in it the heart of quantum mechanics. In

reality, it contains the only mystery. We cannot make the mystery go away by explaining how it

works. — Richard Feynman

Single Photon Double Slit Experiment

Matter

● Ordinary matter consists of atoms● There are a finite number of different atoms

(elements)– Each type exhibits a unique pattern of radiated or

absorbed wavelengths

The Atom

Consists of a central nucleus surrounded by clouds of electrons

How the Cathode Ray Tube Works

Cathode Ray Tube

● Different gases produced different colored discharges

● Reduced pressure halts discharges and darkens tube except for a glow around the anode and florescence of the glass

Crookes's Tube

Cathode rays travel in straight lines

Originate at cathode and carry energy and momentum

Carry negative charge

J.J. Thomson Measures charge/mass

Thomson's Results

● Ratio the same regardless of accelerating voltage

● Ratio the same regardless of cathode material● Ratio 1836 times bigger than that of hydrogen

ions● The atom is not the smallest smallest

object

Note: Philipp Lenard detected cathode rays

passing undeflected through thin metal foil

and concluded that they must be waves;

charged objects would have to scatter (if it

couldn't be a particle it had to be a wave)

Discovering a New Object

● Something with a definite charge/mass ratio● Something found in cathode rays and

photoelectric ejection● Something ejected from elements under the

influence of X-rays● …

A particle called an electron: a small, light, negatively charged component of all atoms

Waves Through a Single Slit

Particles Through a Single Slit

Waves Through a Double Slit

Particles Through a Double Slit?

Single Electron Double Slit Experiment

700,000 Electrons Shot One at a Time

Louis de Broglie (1924)

“Following Einstein's introduction of photons in light waves, one knew that light contains

particles which are concentrations of energy incorporated into the wave, suggests that all

particles, like the electron, must be transported by a wave into which it is incorporated... My

essential idea was to extend to all particles the coexistence of waves and particles discovered

by Einstein in 1905 in the case of light and photons."

“Wave-Particle Duality”

● Identically prepared particles impact in an extended pattern– Different particles impact at different points

– Impact point of any particular particle is uncertain

● The overall pattern is reproducible and predictable

Wave-Particle Duality in the Double Slit Experiment

Erwin Schrödinger (1925)

● Take the wave nature of matter as fact and determine:– How do matter waves behave?

– What do we mean by a matter wave?

● Wave Mechanics– Wave equations for particles

psi

Werner Heisenberg (1926)

● Matrix Mechanics

= Wavefunction = Psi Field

● A particular solution to Schrödinger's equation● The magnitude squared is a probability

distribution● Predict the range for the result of a

measurement, not the exact value● Penetrates into classically forbidden regions:

tunneling– If the energy barrier is not too high, a particle can

pass from one classically allowed region to another through a region that is classically forbidden.

Single Electrons Through Two Slits

● Each electron passes through both slits, interfering with itself, before interacting with the screen at one point

Particle in a Box

● Confined system quantized energies● Smooth transition at boundaries integral

half-wavelength solutions● Energy increases as

Harmonic Oscillator

● Confined energies quantized● Energy proportional to frequency● Energy increases linearly as n

© Daniel Kleppner.

Psi Field (Wavefunction)

● Field: assign one or more values to every point in space(time)

● Square of psi field assigns a probability to every point in space(time)

Harmonic Oscillatorn=0

Harmonic Oscillatorn=10

Correspondence Principle

The transition between quantum and classical worlds should be smooth: in the limit of large

energy state quantum numbers, atomic systems should display classical-like behavior

Spectroscopy● Measurement of the absorption, scattering, or

emission of electromagnetic radiation by atoms or molecules

● Each element (type of atom), exhibits a unique pattern (radiation or absorption line spectrum) of individual wavelengths if sufficiently excited

● How is it that different atoms radiate and absorb characteristic spectra?

● What does this tell us about the structure of atoms?

Line Spectra

The Nuclear Atom (1905)

● Rutherford determines that most of the mass of an atom is located in a tiny volume—the nucleus—at the center of the atom

● Planetary model of the atom● But accelerating charges radiate: electrons

should spiral into nucleus, emitting ever higher frequency light—spectrum should be broad, not sharp

● Atoms would collapse in very little time

Bohr Model (1913)

● Hydrogen atoms exist in certain fixed energy states (stationary states), labeled by a quantum number

● “Jumping” between energy states involves absorbing or emitting radiation of specific frequencies

● Numerology got spectrum right

Limitations of Bohr Model

● Too many ad hoc assumptions● Too many unanswered questions● Didn't work for non-hydrogen-like atoms

● But it demonstrated that a new theory was necessary

● The theory would have to describe the microscopic and macroscopic worlds

Atoms and the Psi Field

● The Psi field should be the solution to the Schrödinger equation for the atom

● The Psi field should describe and predict the “orbits” of electrons

● The Schrödinger equation predicts that the “orbits” will be standing waves (in 3-d)

● Many different standing waves are possible: different quantum states

Hydrogen Atom

● Confined energies quantized● Psi field squared gives electron position

probability● Energy increases linearly as

Quantum Transitions

● Each quantum state is associated with a different frequency

● Frequency is proportional to energy● Each atomic quantum state is a different

energy state: atomic energy states are quantized

● The transition between energy states involves the absorption or emission of specific frequency photons

The ground state is the lowest energy state

because no more compact standing wave can

exist—no lower energy state to radiate to

Remarkably: quantum uncertainties prevent

atomic collapse

Complementary Values

● The Psi field capsulizes position

Complementary Values

● The Psi field capsulizes position– Localization: large magnitude in a specific region;

zero elsewhere

Particle in a Box

No localizedsingle state

Complementary Values

● The Psi field capsulizes position– Localization: large magnitude in a specific region;

zero elsewhere

– Superposition: overlapping waves create wave packets → greater localization

Complementary Values

● The Psi field capsulizes position– Localization: large magnitude in a specific region;

zero elsewhere

– Superposition: addition of odd waves in a box

Complementary Values

● The Psi field capsulizes position– Localization: large magnitude in a specific region;

zero elsewhere

– Superposition: atomic electrons● de Broglie waves

Complementary Values

● The Psi field capsulizes position– Localization: large magnitude in a specific region;

zero elsewhere

– Superposition: de Broglie wave packets

Complementary Values

● The Psi field capsulizes position– Localization: large magnitude in a specific region;

zero elsewhere

– Superposition

● Adding waves → additional momentum components

Complementary Values

● The Psi field capsulizes position– Localization: large magnitude in a specific region;

zero elsewhere

– Superposition

● Adding waves → additional momentum components

● Each momentum measurement yields one of the components

Complementary Values

● The Psi field capsulizes position– Localization: large magnitude in a specific region;

zero elsewhere

– Superposition

● Adding waves → additional momentum components

● Each momentum measurement yields one of the components

● Position and momentum complementary

(Heisenberg) Uncertainty Principle

● Reciprocal relation between the spreads in repeated position and momentum measurements

● Cannot simultaneously know both the position and the momentum with arbitrary precision

● Similarly for time and energy● Implications: zero-point energy, atomic size,

natural width of line spectra, virtual particles,...

Zero-Point Energy

● Sharpening Psi field around zero reduces average potential energy, but requires superposing short wavelength components

● Adding short wavelength components adds momentum / kinetic energy

● Ground state is optimum trade-off

Realm of Possibilities:

Classicalparticle

Quantumparticle

Area (uncertainty)cannot shrink

Proton 4000 timesmore predictable

than electron