Chapter 39: Introduction to Quantum Physics

Chapter 39: Introduction to Quantum Physics

You can't see atoms with light, but you can with electrons because ...

Blackbody Radiation and Planck's Hypothesis

Thermal vibrations causes charged particles to accelerate, emitting radiation. Blackbody radiation depend only on temperature and not on the material.

Blackbody Radiation: Classical Treatment

Consider a cubic cavity with length L on each side, inside which

electromagnetic radiation due to thermal vibration of the walls are resonating

in standing waves. Standing wave requires L to be exact multiples (n: integer)

of the half-wavelength.

L n / 2

Allowed frequency nc / (2L)

is (for each axis) :

For a 3D standing wave with nx, ny, and nz, its frequency is

2

c2 4L2

(nx2

n

2 y

nz2 )

c2 4L2

n2

2 d c2 2n dn

4L2

Each standing wave is an independent oscillator and, according to equipartition of energy principle, would have an average energy (per degree of freedom) of kT. We are therefore in position to estimate the energy spent in electromagnetic radiation due to thermal motion classically. We just need to find out how many oscillators are allowed in each frequency range!

Rayleigh-Jeans Law and the Ultraviolet Catastrophe

In the "phase space", each non-negative (nx,ny,nz) represents an allowed point, which contains two allowed states for the two possible polarizations of light. In a shell of volume in phase space with radius N and thickness dN, the energy is

dE kBT 4 n2dn (1 / 8) (2)

Energy density (energy/volume)

dE L3

8 kBT

d 4

Rayleigh-Jean Law

Intensity (power/area)

I T 2 ckBT

4

Actual Experimental Observations

1. The total power of the emitted radiation increases with temperature.

StefanBoltzmann

P AeT 4

2. The peak of the wavelength distribution shifts to shorter wavelengths as the temperature increases.

maxT 2.898103 m K

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