A BRIEF STORY OF QUANTUM THEORY

PETER BALLO

Classical Mechanics

Electromagnetism and Optics Heat

END OF 19th CENTURY

State of Physics at 1895

•Lord Kelvin—arguably the greatest physicist of his day:• Strong opponent of existence of atoms•“There is nothing new to be discovered in physics now. All that remains is more and more precise measurement."

•Albert A Michleson—the first US Physics Nobel laureate•“The grand underlying principles have been firmly established...further truths of physics are to be looked for in the sixth place of decimals" (Science, 1892)

Hydrogen Spectrum: Balmer series

IT WORKS!

Balmer Formula:

• “X-rays” discovered in 1895 by Roentgen - World Wide sensation!

– Unknown ray produced from electric discharge that penetrates matter!

• J.J. Thomson discovers the electron in 1897.

• Henri Becquerel (1896) tries to produce X-rays from natural sources.

– Finds radiation (less penetrating than X-rays) given off from solid containing Uranium.

• Marie Curie (1897) discovers immense radiation energy

from element she named Radium.

– Surprising? Yes!

Black body radiation

Otto Lummer Ernst Pringsheim

Blackbody Radiation Spectra

• The true beginnings of the quantum theory lie in a strange place: the frequency spectrum emitted by a solid when it is heated (“blackbody” radiation).

• Experimental measurements: the frequency spectrum was well determined.. a continuous spectrum with a shape that depended only on the temperature (light bulb, … )

• Theoretical prediction: Classical kinetic theory predicts the energy radiated to increase as the square of the frequency (Completely Wrong! - “ultraviolet catastrophe”).

Planck’s Solution• Max Planck (1900): In order to describe the data Planck made the

bold assumption that light is emitted in packets or quanta, each with energy E = h f, where f is the frequency of the light.

– Some texts use the notation for frequency.• The factor h is now called

Planck’s constant, h = 6.626 (10-27) erg-sec.

o Data Theory

Atomic Models• Conclusion: Atoms contain electrons.

Questions: How are they arranged? Since atoms are neutral, where is the positive charge?

• Two models:–“Plum pudding”: Electrons are embedded in

continuum of positive electricity like plums in a pudding.

–“Planetary model”: Electrons orbit a small nucleus of positive charge like planets orbit the Sun.

Electrons

Positive Charge

Or

probeFor plum pudding: expect only small angle scattering.

probe

For planetary model: may see small angle or large angle scattering.

Rutherford saw large angles… planetary model!

The Problem of the atom

• Experiments supported the picture that an atom is composed of light electrons around a heavy nucleus

• Problem: if the electrons orbit the nucleus, classical physics predicts they should emit electromagnetic waves and loose energy. If this happens, the electrons will spiral into the nucleus!

• The atom would not be stable!

• What is the solution to this problem?

E=hf

Bohr’s Revolutionary Idea

• Can the new quantum theory explain the stability of the atom?

• If the energies can take on only certain discrete values, i.e., it is quantized, there would be a lowest energy orbit, and the electron is not allowed to fall to a lower energy!

• What is the role of Planck’s Constant h?

• The allowed orbits are labeled by the integers: n = 1, 2, 3, 4.

• The radii of these orbits can be determined from the quantization condition: radius = n2 a0 = n2 (h/2)2/ me2

• The energy can only have the values En= E1 /n2, E1 = - (1/2)(e2 / a0)/n2

• The spectra are the result of transitions between these orbits, with a single photon (f = E/h) carrying off the difference in energy E between the two orbits.

The Bohr Atom (NOT Correct in detail!)

12

34

The Schrodinger Equation• In 1926 Erwin Schrodinger proposed an equation which

describes completely the time evolution of the matter wave

where m = characteristic mass of “particle” V = potential energy function to describe the forces

Newton: SchrodingerGiven the force, find motion Given potential, find wave

F = ma = m (d2x/dt2) (- (h2 / 2m) 2 + V) i h (d /dt)solution: x = f(t) solution: = f(x,t)

Note: Schrodinger’s equation is more difficult to solve, but it is just as well-defined as Newton’s. If you know the forces acting, you can calculate the potential energy V and solve the Schrodinger equation to find .

Key Results of Schrodinger Eq.• The energy is quantized

–Only certain energies are allowed–Agrees with Bohr’s Idea in general–Predicts the spectral lines of Hydrogen exactly–Applies to many different problems - still one of the

key equations of physics!

• The wavefunction is spread out–Very different from Bohr’s idea–The electron wavefunction is not at a given radius

but is spread over a a range of radii.

Probability interpretation for • The location of an electron is not determined by . The probability of finding it is high where is large, and small where is small.

• Example: A hydrogen atom is one electron around a nucleus. Positions where one might find the electron doing repeated experiments:

Nucleus

Higher probabilityto find electronnear nucleus

Lower probabilityto find electronfar from nucleus

The Two-Slit Experiment

• We will first examine an experiment which Richard Feynman says contains “all of the mystery of quantum mechanics”.

• The general layout of the experiment consists of a source, two-slits, and a detector as shown below;

source detector

x

The idea is to investigate three different sources (a classical particle (bullets), a classical wave (water), and a quantum object (electron or

photon)). We will study the spatial distribution (x) of the objects which arrive at the detector after passing through the slits.

slits

Classical Particles

• Classical particles are emitted at the source and arrive at the detector only if they pass through one of the slits.

• Key features:

– particles arrive “in lumps”. ie the energy deposited at the detector is not continuous, but discrete. The number of particles arriving per second can be counted.

– The number which arrive per second at a particular point (x) with both slits open (N12) is just the sum of the number which arrive per second when only the top slit is opened (N1) and the number which arrive per second when only the bottom slit is opened (N2).

x

N

only bottom slit open

only top slit open

x

N

Both slits open

• Classical waves are emitted at the source and arrive at the detector only if they pass through the slits.

• Key features:

– detector measures the energy carried by the waves. eg for water waves, the energy at the detector is proportional to the square of the height of the wave there. The energy is measured continuously.

– The energy of the wave at a particular point (x) with both slits open (I12) is NOT just the sum of the energy of the wave when only the top slit is opened (I1) and the energy of the wave when only the bottom slit is opened (I2). An interference pattern is seen, formed by the superposition of the piece of the wave which passes through the top slit with the piece of the wave which passes through the bottom slit.

x

I

only bottom slit openonly top slit open

x

I

Both slits open

Timeline - Modern Physics

• “Modern Physics” was a sudden revolution starting around 1900, and ending ????

Einstein

200019501900

Michelson

PlanckThomson

RutherfordBohr

SpecialRelativity

GeneralRelativity

Quantum Mechanics

De BroglieSchrodingerHeisenberg

TransistorInvented

All the Quarksdiscovered

LaserInvented

Nuclear Energy Released

Expansion of Universe discovered

Neutron Stars discovered

Start ofQuantum Mechanics

Curie

“Thirty Years That Shook Physics”• 1887 Michelson-Morley exp. debunks “ether” • 1895 Rontgen discovers x rays• 1897 Becquerel discovers radioactivity• 1897 Thomson discovers the electron• 1900 Planck proposes energy quantization• 1905 Einstein proposes special relativity• 1915 Einstein proposes general relativity• 1911 Rutherford discovers the nucleus• 1911 Braggs and von Laue use x rays to determine crystal structures• 1911 Ones finds superconductors• 1913 Bohr uses QM to explain hydrogen spectrum• 1923 Compton demonstrates particle nature of light• 1923 de Broglie proposes matter waves• 1925 Davisson & Germer prove matter is wavelike• 1925 Heisenberg states uncertainty principle• 1926 Schrodinger develops wave equation• 1924-6 Boson and Fermion distributions developed

• 1949 Murphy's Law stated

• The two most important formulas in modern physics • E = mc2 (Einstein – special relativity - 1905)

E = h f (Planck – quantum mechanics - 1900)

• Planck initially called his theory “an act of desperation”.– “I knew that the problem is of fundamental significance for

physics; I knew the formula that reproduces the energy distribution in the normal spectrum; a theoretical interpretation had to be found , no matter how high.”

• Leads to the consequence that light comes only in certain packets or “quanta”

• A complete break with classical physics where all physical quantities are always continuous