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Minkowski Space

- Consider a 4 dimensional vector space

Example

Example

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Time Dilation Again

Four velocity

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The relativistic Addition of velocities

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Relativistic addition of velocities

The momentum Energy 4 vector

- As we have seen the classical momentum is not

relativistically invariant

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Doppler again

y

x

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Relativistic Center of Mass System

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- The energy available for inelastic processes is

End of Special Relativistic Section

The wanning of the classical world view

Classical Physics

- The physical universe is deterministic, given

enough information one can predict exactly the

evolution of the system - Light consists of electromagnetic waves while

ordinary matter consists of discrete particles - Physical quantities like postion momentum,

angular momentum and energy are continuous

variables - Newtonian Mechanics and Electromagnetism depend

typically on second order differential equations

Thermal Radiation

- We see objects by scattering electromagnetic

radiation from them - When we heat an object it can also emit radiation

Observations

- As the Temperature of a body is increased the

intensity of the thermal radiation rises - The higher the temperature the shorter the wave

length of the most intense emitted radiation - A body becomes red hot and then white hot

- Stefan showed that the total power emitted per

unit area,R, called the total emissive power or

total emittance is given by the empirical

formula

Constant independent of surface

Emissivity,characterisic of surface, 1?e

Temperature on absolute scale

- If a body is in thermal equilibrium with its

surroundings, it must absorb and admit the same

amount of radiant energy(otherwise temperature

would rise) - A blackbody is a perfect absorber so if it is

emitting thermal radiation we must have - e1

- Early attempts to study these observations

quantatively ran into difficulties because it was

found that the thermal radiation emitted from a

given body at a given temperature depended on

the material from it was made, the roughness of

the surface etc.

Cavity Radiator

- To avoid these problems the idea of a cavity

radiator was introduced. - Idea form a cavity in a material with its walls

held at a constant temperature - A small hole is created which allows radiation to

escape - The radiation emerging from this hole does not

depend on the nature of the cavity or the

material just on the temperature

- "Blackbody radiation" refers to an object or

system which absorbs all radiation incident upon

it and re-radiates energy which is characteristic

of this radiating system only, not dependent upon

the type of radiation which is incident upon it.

The radiated energy can be considered to be

produced by standing wave or resonant modes of

the cavity which is radiating.

Cavity blackbody radiation

- The radiation emitted from a cavity through a

small hole ("cavity radiation") is very close to

the theoretical blackbody curve for the same

temperature. In the cavity, the radiation is in

equilibrium with the material - most of the

radiation stays inside the cavity, being

continually emitted and re-absorbed by the walls.

Radiation emitted from the outer surface of a

material will not necessarily be fully

thermalized - some frequencies corresponding to

certain transitions of the material, will be

emitted preferentially. So, the blackbody curve

is not material-specific, but the actual emission

from an object will be. Cavity radiation will

depend less on the material, and the smaller the

hole, the closer it will correspond to the

theoretical blackbody curve.

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Intensity versus wavelength for different

temperatures

Rayleigh-Jeans distribution

- The radiation detected outside the hole will be a

sample of the radiation inside the box, so some

analysis is required to understand whats

happening inside the box. - The box is filled with electromagnetic standing

waves. If the walls are metal, the radiation

bounces around inside the box with the electric

field stopping at each wall, creating a node at

each wall. - The number of standing waves with wavelengths

between ?? and ?d? is N(??) d? (8? V / ?4) d?

where V is the volume of the box. - This can be proven by regular analysis of

standing waves and expanding it to three

dimensions. - Each individual wave contributes an energy kT to

the radiation in the box. From classical

thermodynamics, we know that the radiation in the

box is in thermal equilibrium with the walls at

temperature T. Radiation is absorbed and quickly

reemitted by the walls, which creates

oscillations in the frequency of the radiation.

The mean thermal kinetic energy of an oscillating

atom is 0.5kT. Since these are simple harmonic

oscillators, the mean kinetic energy is equal to

the mean potential energy, so the total energy is

kT. - The radiance is related to the energy density

(energy per unit volume) u(?) in the relationship

Some ideas from Thermodynamics

- Consider a collection of electromagnetic waves

inside a blackbody cavity of temperature T. - The energy density is just the average energy of

the waves multiplied by their number density

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- k is the Boltzmann constant

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From all this we get

known as the Rayleigh-Jeans formula)

Ultra violet Catastrophe

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Ultra violet Catastrophe

- Plank rederived the formula and avoided the

catastrophe by assuming that the oscillators

could only take energies which were integer

multiples of some energy

- He further showed that

frequency

constant

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