Relativistic Doppler Effect

Statement of the problem: A source of light is moving at constant speed utoward a stationary observer (Stanley). The source emits EM waves with f0 = 1/T0 in its rest frame. What is the frequency measured by Stanley?

Relativistic Doppler EffectFrom Stanley point of view, ( )cT uT c u T

Then, since f = c, we have ( )

cfc u T

As we have seen, time intervals are measured differently by different observers.

T0 is measured in the rest frame of the source so that it is the propermeasurement and T is not and in fact, T is dilated, i.e.,

0 00 2 2 2 21

T cTT Tu c c u

This gives,2

00

2 2 21 1 fT

c u c uT c c

(dist. between wave crests)

0 0cT

0 0f c

Note:In the S (rest fr),

00

1fT

Relativistic Doppler EffectNow substitute expressions for 1/T into the boxed equation,

cf 2 2

( )c u

c u c

0

0( )( )

( )

f

c u c uf f

c u

0c uf fc u

(Doppler Shift for an approaching source)

For source receding, the only difference is the sign of the relative speed u -u,

0c uf fc u

(Doppler Shift for a receding source)

higher freq blue shifted

lower freq red shifted

Example: Red Shifted Red Dwarf Star

From Wein’s Displacement Law: A red dwarf star at 3000K has a peak in its emittance at 1000nm but observing from Earth, the red dwarf appears red (650 nm). Estimate its receding speed with respect to Earth. (see Q38.19)

0

0

f c uf c u

2200

20

10.406

1c u u c cc u

(Dopper Shift for Receding Object)

General Relativity and GravityEquivalence of gravity and acceleration.

General Relativity and Space-Time

Massive objects curve space-time

Object or light paths will be deflected along the curved surface. Verified by Arthur Eddington during a solar eclipse in 1919.

Gravity Lens

Distant galaxy lensed by Cluster Abell 2218J-P Kneib and R. Ellis (Caltech) 2004

Chapter 38: Light Waves Behaving as Particles

Photoelectric Effects X-ray Production Compton Scattering & Pair

Production Heisenberg Uncertainty

Principle Wave/Particle Duality

Quantum Nature of Light

By the end of the 19th century, most physicists (Maxwell, Hertz,

and others) have firmly established that electromagnetic waves are

waves which exhibit interference and diffraction (Ch. 35-36).

Two seemingly paradoxical nature of light (EM waves):

But newer experiments on the emissions and absorptions of EM

waves have shown behaviors which CANNOT be explained with light

being as waves… It requires a radical new thinking of light as

quantized packets of energy called photons (as particles).

Quantum Nature of Light

X Rays Production, Compton Scattering, & Pair Production:X-rays were discovered in 1895 in high-voltage electric discharge tubes but no one understood the process in their production and what determine their wavelengths. In particular, when x-rays collide with matter, the scatter rays can have a different (longer) wavelengths.

Photoelectric Effect: When light struck a metal surface, some electrons near the surface will be emitted. The absorption and emission process can only be explained by assuming light is quantized into packets of energy.

Absorption of Light as Particles:The Photoelectric EffectAn experimental demonstration of the particle nature of light.

Observation: Light causes the cathode to emit electrons (photoelectrons), which are pushed toward the anode by the electric-field force.

Electrons on the metal surface (cathode) are normally bounded to the positive ions on the surface.If an electron absorb enough energy from the incident radiations to overcome the potential-energy, it can be ejected.The minimum amount of energy an individual ehas to gain to escape is call the work function

Rationale:

and i is called the photocurrent

The Photoelectric Effect

The maximum KE (Kmax) of these electrons can be calculated by measuring this ,

Above a certain potential strength , NO e- can reach the anode!

max 0K eV

The minimum needed to stop all e-getting across to the anode is called the stopping potential V0 and

0(reversed)ACV V

0V

0V

0V

The Photoelectric Effect

Unexpected Results:•No electrons will be ejected if f < f0(threshold frequency) independent of light intensity

•Even at very low intensity with f > f0, emission is immediate

•V0 is independent of intensity

Classical Expectation: • Energy of EM wave depends on intensity emission will monotonically depends on intensity

• Intensity of light not dependent on f• For low intensity light, emission is

expected to be delayed

Photoelectric Effect

The dependence of V0 (Kmaxof the ejected electrons) on fis also another unexpected (unexplainable by classical physics) result. Energy of light was not expected (classically) to depend on its frequency f.

Einstein’s Photon ExplanationIn 1905, Einstein published his theory on photoelectric effect which resulted in his Nobel prize in 1921.

Built upon Max Planck’s hypothesis of quantized light (photon). [later]

hcE hf

(energy of a photon)

34where 6.626 10h J s is a universal constant called Planck’s Constant.

Note the smallness of this number.

Einstein’s Photon Explanation

e-

hf

Kmax

Incident light as a collection of photons (particles)

•Each photon has energy hf• Intensity of light ~ # of photons

The interaction is an all-or-none process. Electrons bounded to the surface of the metal can absorb a single photon at a time or none at all. If hf is large enough to overcome , an electron will be ejected with kinetic energy Kmax.

By energy conservation, we have: maxK hf depends on the metal surface

0eV hf

Einstein’s Photon Explanation

• Since Kmax has to be positive, if hf < , no electrons will gain enough kinetic energy to leave. Therefore, there is a threshold frequency and is given by,

Unexpected Results Explained:

• Since intensity I is proportional to the # of photons and Kmax (or V0) depends only on the energy of the individual photon hf.

0hf

• Kmax linear dependence on f: It is explicit with the above equation.

Increasing intensity will only increase the # of electrons being ejected and it will increase the photocurrent being observed but it will not affect the stopping potential V0.

Notes

1eV = energy required to move one unit of charge across an electric potential of 1 V.

19 191eV 1.602 10 1 1.602 10C V J

- Convenient Energy Units:

- Energy and Momentum of a Photon:

E hfc

hPc

(momentum of a photon)

34 1519

15 8 6

16.626 10 4.136 101.602 10

4.136 10 3.00 10 / 1.241 10 1241

eVh J s eV sJ

hc eV s m s eV m eV nm

- The duality of light (wave & particle) applied to the entire EM spectrum !

Emission of Light as Particles:X-Ray Production

X-rays are produced when rapidly moving electrons that have been accelerated through a largepotential difference (103 to 106 V) strike a metal target.

X-rays emission is the inverse of the photoelectric effect.

Photoelectric: hf K of eX-Ray prod: K of e hf

Two Processes1. Independent of target material: bremsstrahlung (braking radiation) gives maximum f (energy) or minimum

maxmin

AChceV hf

max KE of accelerated e

2. Dependent of target material: electrons with sufficient KE can excite atoms in the target material. When they decay back to their ground state, characteristic spectrum of X-Rays will be emitted. (The energy levels involved in these transition are typically separated by hundreds or thousands of eVs, rather than a few eVs as is typical for visible photons.)

Both of these cannot be explained by classical physics !

deceleration of high energy e- should produce EM waves of all f.

Medical Applications of X-Rays

High-energy photons (such as X-Rays) can penetrate denser materials such as bones which low-energy photons (such as visible light) can not. Then, by measuring the degree of penetration, one can map out different biology structures in your body.

High-energy photons can also damage biological tissues by breaking molecular bonds and creating highly reactive free radical such as H or OH .

Compton ScatteringIn 1923, Arthur H. Compton provided an additional direct conformation on the quantum nature of x-rays.

- X-rays of well defined are made to fall on a graphite target- For various scattering angle , intensities of scattered x-rays are measured

as a function of the wavelength.

NOTE: Energy of photon >> binding energy of e-

in graphite.

Compton ScatteringExperimental Observation:

The scattered x-rays have intensity peaks at twowavelengths: 0 and ’.

0'

is called the Compton Shift.

Compton ScatteringBut, most importantly,

is found to be a function of the observation angle !

Classical prediction:

Electrons in graphite absorbs x-rays and reemit them back.

- If the electron is stationary, then the reemitted ’ = 0.

- But electrons are moving before & after scattering, so that ’ will be Dopper’s shifted depending on e-’s velocity.

Compton ScatteringClassical prediction:

Since diff. electrons will have diff. velocity, the intensity profile for the scattered x-rays is expected to be a single peak with a spread around 0.

But, the actual experiment gives two peaks with a Compton Shift which depends on . Classical physics can’t explain this !

Compton ScatteringQuantum Explanation:Compton and his co-workers showed that if we take the x-rays as particles with

/E hfp hf c

and treat the scattering process as a “billiard-like” collision,

then, the observation of ‘s dependence on can be explained !