Birth of Quantum Mechanics

PHYS 521

Necessity of QM “There is nothing new to be discovered in physics now.

All that remains is more and more precise measurement.”

--- Lord Kelvin, 1900

By the end of the nineteenth century a number of serious discrepancies had been found between experimental results and classical theory.

Blackbody radiation law Photo-electric effect Atom and atomic spectra

Blackbody radiation Exp. Measurements: the

radiation spectrum was well determined --- a continuous spectrum with a shape that dependent only on temperature

Blackbody radiation Exp. Measurements: the

radiation spectrum was well determined --- a continuous spectrum with a shape that dependent only on temperature

Theory: classical kinetic theory (Rayleigh and Jeans) predicts the energy radiated to increase as the square of the frequency. Completely wrong! Ultraviolet catastrophe!

Planck’s solution Planck’s assumption (1900): radiation of a given

frequency ν could only be emitted and absorbed in “quanta” of energy E=hν

h=6.6261E-34 J·s : Planck’s constant With this assumption, Planck came up with a formula that

fits well with the data. Planck called his theory “an act of desperation”. Planck neither envisaged a quantization of the radiation

field, nor did he quantize the energy of an individual material oscillator.

What Planck assumed is that the total energy of a large number oscillators is made up of finite energy elements hν.

Einstein’s interpretation of Planck’s formula

Einstein in 1906 interpreted Planck’s result as follows:

“Hence, we must view the following proposition as the basis underlying Planck’s theory of radiation: The energy of an elementary resonator can only assume values that are integral multiples of hν; by emission and absorption, the energy of a resonator changes by jumps of integral multiples of hν”

Photo-electric effect Experimental facts

Shining light on metal can liberate electrons from metal surface

Whether the metal emit electrons depends on the freq. of the light: only light with a freq. greater than a given threshold will produce electrons

Increasing the intensity of light increases the number of electrons, but not the energy of each electron

Energy of electron increases with the increase of light frequency.

Einstein on photo-electric effects

Light consists of a collection of “light quanta” of energy hν

The absorption of a single light quantum by an electron increases the electron energy by hν

Some of this energy must be expended to separate the electron from the metal (the work function, W), which explains the threshold behavior, and the rest goes to the kinetic energy of the electron.

Electron kinetic energy = hν – W Einstein won Nobel Prize in 1921

"for his services to Theoretical Physics, and especially for

his discovery of the law of the photoelectric effect"

Reactions to Einstein’s light quanta idea

For a long, long time, nobody else believed that. Planck and others in their recommendation of Einstein’s

membership in Prussian Academy (1913): “One can say that there is hardly one among the great

problems in which modern physics is so rich to which Einstein has not made a remarkable contribution. That he may sometimes have missed the target in his speculations, as, for example, in his hypothesis of light quanta, cannot really be held too much against him, for it is not possible to introduce really new ideas even in the most exact sciences without sometimes taking a risk”

Experimental confirmation Experimental confirmation came in 1915 by Millikan

Millikan didn’t like Einstein’s light quanta idea, which he saw as an attack on the wave theory of light.

Tried very hard (for 10 years) to disprove Einstein’s theoretical prediction.

For all his efforts, he confirmed Einstein’s theory and provided a very accurate measurement of Planck’s constant.

Millikan got Nobel prize in 1923. Still didn’t like Einstein’s light quanta idea, in a 1916 paper:

“This hypothesis may well be called reckless …” “Despite the apparently complete success of the Einstein

equation, the physical theory of which it was designed to be the symbolic expression is found so untenable …”

Compton scattering

Compton experiment 1923: Inelastic collision between photon and electron changes the wavelength of the photon.

Even Millikan was impressed: “It may be said that it is not merely the Einstein equation which is having extraordinary success at the moment, butthe Einstein conception as well.” But not without reservation: “But until it can account for the facts of Interference and the other effects, we must withhold our full assent.”(Nobel lecture, 1923)

Einstein on light quanta “All these fifty years of conscious brooding

have brought me no nearer to the answer to the question `What are light quanta?’ Nowadays every rascal thinks he knows, but he is mistaken.”

--- letter to Michel Besso,

1951

Problems with atom stability Rutherford’s experiment (1911): atom is

composed of electrons moving 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 electron will spiral into the nucleus, no stable atom should exist!

Problems with atomic spectrum Atomic radiation spectrum consists of discrete lines.

Bohr’s solution (1912) An atomic system can only

exist in a discrete set of stationary states, with discrete values of energy.

Change of the energy, including emission and absorption of light, must take place by a complete transition between two such stationary states.

Bohr’s solution (1912) An atomic system can only exist in a discrete set of stationary

states, with discrete values of energy. Change of the energy, including emission and absorption of

light, must take place by a complete transition between two such stationary states.

The radiation absorbed or emitted during a transition between two states of energies E1 and E2 has a frequency: hν=E1 - E2

Bohr’s formula explains some of the spectral lines in hydrogen atom (but not all), does not do well with other atoms.

A truly revolutionary idea, even Einstein was impressed: “… appeared to me like a miracle. This is highest form of

musicality in the sphere of thought.” (1951)

summary Energy quantization is necessary to explain

the blackbody radiation, the photo-electric effects, the stability of atoms and their spectra

Classical physics must be given up: physical properties that are quantized and not continuous are completely different from the ideas of continuous space and time in classical physics.

Later developments De Broglie: matter wave λ=h/p Exp. with electron diffraction (Davisson and Germer,

1927) Today: interferometers with neutrons, atoms and

molecules Born’s statistical interpretation of matter wave Matrix mechanics (Heisenberg, Born and Jordan) Wave mechanics, Schroedinger’s equation

(Schroedinger) Relativistic QM (Dirac) Exclusion principle (Pauli)

Birth of QM The necessity for quantum mechanics was thrust upon

us by a series of observations. The theory of QM developed over a period of 30 years,

culminating in 1925-27 with a set of postulates. QM cannot be deduced from pure mathematical or

logical reasoning. QM is not intuitive, because we don’t live in the world of

electrons and atoms. QM is based on observation. Like all science, it is subject

to change if inconsistencies with further observation are revealed.

Fundamental postulates of QM How is the physical state described? How are physical observables

represented? What are the results of measurement? How does the physical state evolve in

time?These postulates are fundamental, i.e., their explanation is beyond the scope of the theory. The theory is rather concerned with the consequences of thesepostulates.

Goal of PHYS521 and 522 We will study non-relativistic QM. Our goal is to understand the meaning of the postulates

the theory is based on, and how to operationally use the theory to calculate properties of systems.

The first semester will lay out the ground work and mathematical structure, while the second will deal more with computation of real problems.