1. Quantum physics (quantum theory, quantum mechanics)
Part 1:

2. Outline Introduction Problems of classical physics
emission and absorption spectra
Black-body Radiation
experimental observations
Wien’s displacement law
Stefan – Boltzmann law
Rayleigh - Jeans
Wien’s radiation law
Planck’s radiation law
photoelectric effect
observation
studies
Einstein’s explanation
Summary

3. Question: What do these have in common?
lasers
solar cells
transistors
computer chips
CCDs in digital cameras
Ipods
superconductors
Answer:
They are all based on the quantum physics discovered in the 20th century.

4. Why Quantum Physics? “Classical Physics”:
developed in 15th to 20th century;
provides very successful description of “every day, ordinary objects”
motion of trains, cars, bullets,….
orbit of moon, planets
how an engine works,..
subfields: mechanics, thermodynamics, electrodynamics,
Quantum Physics:
developed early 20th century, in response to shortcomings of classical physics in describing certain phenomena (blackbody radiation, photoelectric effect, emission and absorption spectra…)
describes “small” objects (e.g. atoms and their constituents)

5. “Classical” vs “modern” physics

6. Quantum Physics QP is “weird and counterintuitive” But:
“Those who are not shocked when they first come across quantum theory cannot possibly have understood it” (Niels Bohr)
“Nobody feels perfectly comfortable with it “ (Murray Gell-Mann)
“I can safely say that nobody understands quantum mechanics” (Richard Feynman)
But:
QM is the most successful theory ever developed by humanity
underlies our understanding of atoms, molecules, condensed matter, nuclei, elementary particles
Crucial ingredient in understanding of stars, …

7. Features of QP key insights:
Quantum physics is basically the recognition that there is less difference between waves and particles than was thought before
key insights:
light can behave like a particle
particles (e.g. electrons) are indistinguishable
particles can behave like waves (or wave packets)
waves gain or lose energy only in "quantized amounts“
detection (measurement) of a particle  wave will change suddenly into a new wave
quantum mechanical interference – amplitudes add
QP is intrinsically probabilistic
what you can measure is what you can know

8. emission spectra continuous spectrum line spectrum
solid, liquid, or dense gas emits continuous spectrum of electromagnetic radiation (“thermal radiation”);
total intensity and frequency dependence of intensity change with temperature (Kirchhoff, Bunsen, Wien, Stefan, Boltzmann, Planck)
line spectrum
rarefied gas which is “excited” by heating, or by passing discharge through it, emits radiation consisting of discrete wavelengths (“line spectrum”)
wavelengths of spectral lines are characteristic of atoms

10. Emission spectra:

11. Absorption spectra first seen by Fraunhofer in light from Sun;
spectra of light from stars are absorption spectra (light emitted by hotter parts of star further inside passes through colder “atmosphere” of star)
dark lines in absorption spectra match bright lines in discrete emission spectra
Helium discovered by studying Sun's spectrum
light from continuous-spectrum source passes through colder rarefied gas before reaching observer;

12. Fraunhofer spectra

13. Spectroscopic studies

14. Thermal radiation
thermal radiation = e.m. radiation emitted by a body by virtue of its temperature
spectrum is continuous, comprising all wavelengths
thermal radiation formed inside body by random thermal motions of its atoms and molecules, repeatedly absorbed and re-emitted on its way to surface  original character of radiation obliterated  spectrum of radiation depends only on temperature, not on identity of object
amount of radiation actually emitted or absorbed depends on nature of surface
good absorbers are also good emitters (Kirchhoff)

15. warm bodies emit radiation

16. Black-body radiation perfect absorber must also be perfect emitter
ideal body which absorbs all e.m. radiation that strikes it, any wavelength, any intensity
such a body would appear black  “black body”
must also be perfect emitter
able to emit radiation of any wavelength at any intensity -- “black-body radiation”
“Hollow cavity” (“Hohlraum”) kept at constant T
hollow cavity with small hole in wall is good approximation to black body (Kirchhoff 1859)
thermal equilibrium inside, radiation can escape through hole, looks like black-body radiation
Kirchhoff’s challenge to theorists: calculate spectrum of hollow cavity radiation (black body radiation)

17. Studies of radiation from hollow cavity
hollow cavity behaves like black body
In 2nd half of 19th century, behavior of radiation within a heated cavity studied by many physicists, both theoretically and experimentally
Experimental findings:
spectral density ρ(f,T) (= energy per unit volume per unit frequency) of the heated cavity depends on the frequency f of the emitted light and the temperature T of the cavity and nothing else.

18. A small hole in the wall of the cavity approximating an ideal blackbody. Electromagnetic radiation (for example, light) entering the hole has little chance of leaving before it is completely adsorbed within the cavity.

19. Blackbody radiation – Stefan-Boltzmann
“Global description”, i.e without frequency dependence:
Descriptions successful, i.e. in agreement with observations
Joseph Stefan (1879) first measured temperature dependence of the total amount of radiation emitted by blackbody at all wavelengths and found it varied with absolute temperature
Ludwig Boltzmann: theoretical explanation
total power output of black body: Stefan-Boltzmann law:
For an “ideal” radiator (“black body”), total emitted power (per unit emitting area), P/A P/A = σ·T4 σ = · 10-8 W m-2 K-4 (Josef Stefan, Ludwig Boltzmann 1879, 1884)
Note that the power per unit area radiated by blackbody depends only on the temperature, and not of other characteristic of the object, such as its color or the material, of which it is composed.
An object at room temperature (300 K) will double the rate at which it radiates energy as a result of temperature increase of only 570.

20. Black body radiation – Wien’s law
Wien’s displacement law (1893)
peak vs temperature:
max ·T = C, C= · mK
inverse relationship between the wavelength max of the peak of the emission of a black body and its temperature when expressed as a function of wavelength
hyperphysics.phy-astr.gsu.edu/hbase/quantum/wien2.html
Example: The wavelength at the peak of the spectral distribution for a blackbody at 4300 K is 674 nm (red). At what temperature would the peak be 420 nm (violet)?
Solution: From Wien’s law, we have
λ1T1 = λ2T2
(674 x 10-9m)(4300 K) = (420 x 10-9m)(T2)
T2=6900 K

21. Wien’s displacement law
is used to
determine the surface temperatures of stars by analyzing their radiation.
also used to map out the variation in temperature over different regions of the surface of an object. Map = “thermograph”
Example: thermograph can be used to detect cancer because cancerous tissue results in increased circulation which produce a slight increase in skin temperature.

22. Radiation from the Sun
The radiation emitted by the surface of the sun has maximum power at wavelength of about 500 nm. Assuming the sun to be a blackbody emitter,
what is it surface temperature?
Calculate λmax for a blackbody at room temperature, T=300 K.

23. Attempts to get radiation spectrum
Wilhelm Wien (1896) r(f, T) = af3 exp(-bf/T), (a and b constants).
OK for high frequency but fails for low frequencies
Rayleigh-Jeans Law (1900)
r(f,T) = af2T (a = constant)
(constant found to be = 8pk/c3 by James Jeans, in 1906)
OK for low frequencies, but “ultra – violet catastrophe” at high frequencies, i.e. intensity grows  f2   for f  (corresponding to limit of wavelength  0)

24. Ultraviolet catastrophe
(Rayleigh-Jeans)

25. Raleigh – Jeans formula
Power radiated from cavity (out of the hole) is proportional to the total energy density U (energy per unit volume) of the radiation in the cavity.
Spectral distribution of power emitted, ρ()d is proportional to the spectrum of the energy density u()d in the cavity,, proportionality factor = c/4 (c=speed of light)
ρ() = radiation power per unit wavelength interval
u() is calculated from classical electrodynamics and thermodynamics

26. Energy density in hollow cavity
Energy density of radiation in the cavity modelled by electromagnetic oscillators in the cavity;
Need to find the number of modes of oscillation of the electromagnetic field in the cavity with wavelength λ in the interval dλ and multiply it by average energy per mode.
number of modes of oscillation per unit volume, n(λ), is independent of the shape of cavity and is given by:

27. Raleigh-Jeans equation
Equipartition principle: energy per degree of freedom =1/2 kT
oscillator has 2 degrees of freedom
Mechanical: kinetic and potential energy
Electromagnetic: electric and magnetic energy
Energy per oscillation mode = kT
Therefore

28. expt data for
T=1600
Comparison of the Rayleigh-Jeans Law with experimental data at T=1600 K. The u(λ) axis is linear.

30. Frequencies in cavity radiation
cavity radiation = system of standing waves produced by interference of e.m. waves reflected between cavity walls
many more “modes” per wavelength band  at high frequencies (short wavelengths) than at low frequencies
for cavity of volume V, n = (8πV/4)  or n = (8πV/c3) f2  f
if energy continuous, get equipartition, <E> = kT  all modes have same energy  spectral density grows beyond bounds as f
If energy related to frequency and not continuous (E = nhf), the “Boltzmann factor” e-E/kT leads to a suppression of high frequencies

31. Planck’s quantum hypothesis
Max Planck (Oct 1900) found formula that reproduced the experimental results
derivation from classical thermodynamics, but required assumption that oscillator energies can only take specific values E = 0, hf, 2hf, 3hf, …
for a multi-state system of particles in thermal equilibrium, probability for particle to be in state with energy E, P(E) = (1/Z)e-E/kT (“Boltzmann factor” )
<Eosc> is the average
energy of a cavity
“oscillator”

32. Planck’s calculation
Classically, the electromagnetic waves in the cavity are produced by accelerated electric charges in the walls vibrating like simple harmonic oscillators
Classical thermodynamics: average energy for simple harmonic oscillator from Maxwell-Boltzmann distribution function: f(E) = Ae-E/kT
where A is a constant (normalization)and f(E) is the fraction of oscillators with energy E.

33. Average energy of oscillator
Maxwell-Boltzmann distribution: A is normalization factor so that
Average energy
Planck: to derive function which agrees with data, need to assume that energy of oscillators is a discrete variable
M.B. factor

34. Planck’s calculation Normalization condition Average energy
Using geometric series formula and

35. Get average energy
and finally
Multiplying with number of oscillators, get Planck formula

36. Planck’s radiation formula
Multiplying average oscillator energy by the number of oscillators per unit volume in the interval dλ given by n(λ)=8πcλ-4 (the number of modes of oscillation per unit volume), finally obtain the energy distribution function for the radiation in cavity:
And for blackbody radiation:

37. Black-body radiation spectrum
Measurements of Lummer and Pringsheim (1900)
calculation

38. Consequences of Planck’s hypothesis
oscillator energies E = nhf, n = 0,1,…;
h = Js = eV·s now called Planck’s constant
 oscillator’s energy can only change by discrete amounts, absorb or emit energy in small packets – “quanta”; Equantum = hf
average energy of oscillator <Eosc> = hf/(ex – 1) with x = hf/kT; for low frequencies get classical result <Eosc> = kT, k = 1.38 · J·K-1

39. Problems estimate Sun’s surface temperature assuming:
Earth and Sun are black bodies
Stefan-Boltzmann law
Earth in energetic equilibrium (i.e. rad. power absorbed = rad. power emitted) , mean temperature T = 290K
Sun’s angular size Sun = 32’
show that for small frequencies, Planck’s average oscillator energy yields classical equipartition result <Eosc> = kT
show that for standing waves on a string, number of waves in band between  and + is n = (2L/2) 

40. Homework problems, cont’d
Thermal Radiation from the Human Body. The temperature of the skin is approximately 35°C. What is the wavelength at which the peak occurs in the radiation emitted from the skin?
The Quantized Oscillator. A 2-kg mass is attached to a massless spring of force constant k=25N/m. The spring is stretched 0.4m from its equilibrium position and released. (a) Find the total energy and frequency of oscillation according to classical calculations. (b) Assume that the energy is quantized and find the quantum number, n, for the system. (c) How much energy would be carried away in one-quantum change?
The Energy of a “Yellow” Photon. What is the energy carried by a quantum of light whose frequency equals 6 x 1014 Hz yellow light? What is the wavelength of this light?
Bonus problem: fill in the gaps in the derivation of Planck’s formula

41. Summary classical physics explanation of black-body radiation failed
Planck’s ad-hoc assumption of “energy quanta” of energy Equantum = h, modifying Wien’s radiation law, leads to a radiation spectrum which agrees with experiment.
old generally accepted principle of “natura non facit saltus” violated
Opens path to further developments