‘Modern’ Physics

At the beginning of the twentieth century, two new theoriesrevolutionized our understanding of the world and modified ‘old’physics that had existed for over 200 years:

Relativity: Describes macroscopic objects (planetary bodies,spaceships moving close to the speed of light…)

Quantum Mechanics: Describes microscopic objects (atoms,electrons, photons….)

Both theories explained phenomena that existing physics was unableto deal with.

Both theories challenged our fundamental intuition and perception ofthe world: space-time, mass-energy, locality, causality….

Quantum Mechanics

Blackbody radiationA black body is an object that absorbs all electromagnetic radiationthat falls on it. Since it does not reflect any radiation it appears black.

Black bodies are not only perfect absorbers but also perfect emitters ofradiation. They emit the maximum amount of energy possible at a giventemperature.

In reality no object is a perfect black body.Some examples of approximate black bodies: Graphite (absorbs over95% of incident radiation), Stars: absorb and emit at all wavelengthsA cavity with a small opening:

Blackbody radiationA black body emits radiation at all wavelengths.

The emitted power e(λ, T) depends onthe temperature (T) and wavelength(λ).

Measurement of emitted power

Blackbody radiationStefan-Boltzman law: The total power over all wavelengths (area undereach temperature curve) per unit area is:

etot= e(!,T )d! =

0

"

# $T 4

$ = 5.67 %10&8W m

2K4( )

For a non-ideal black-body:

etot= a!T

4, a < 1

! = 5.67 "10#8W m

2K4( )

This relationship was experimentally found by Stefan and theoreticallyconfirmed by Boltzman using Maxwell’s equations and thermodynamics

Blackbody radiationWien’s displacement law: The peak wavelength is inverselyproportional to temperature

!max

=a

T, a = 2.898 "10

#3m $K

Blackbody radiation

u !,T( ) =4

ce(!,T ) = E

aven(!) = kT

8"!4

#$%

&'(

Rayleigh-Jean’s law for the spectral energy density u:

Advantages:• Derived from Maxwell’s equations (not empirical)• Good agreement to experiments at long wavelengths (low frequencies)

Problems:• u goes to infinity at short wavelengths (high frequencies): U-Vcatastrophe

n(λ): number of oscillator modes with frequency c/λ in a cavityEave: average energy per mode

Blackbody radiation

u(!,T ) =Ac

3e"#c /!T

! 3= Af

3e"# f /T

Wien’s exponential law for the spectral energy density u:

Advantages:• Stefan’s law can be derived from this equation• Explains the peak wavelength• Good agreement to experiments at short wavelengths (highfrequencies)

Problems:• Derived empirically (no theoretical basis)• Does not agree with Rayleigh-Jeans theory at long wavelengths (lowfrequencies)

Blackbody radiation

u(!,T ) =8"h! 3

1

ehc !kT #1

$%&

'()=8" f 3

c3

1

ehf kT #1

$%&

'()

Planck’s law for the spectral energy density u:

Advantages:• Agrees with Stefan’s law, Wien’s displacement law• Agrees with Wien’s law at short wavelengths (high frequencies):

hf kT >> 1!1

ehf kT

"1# e

"hf kT$ u(%,T ) #

8& f3

c3e"hf kT

• Agrees with Rayleigh-Jean’s law at long wavelengths (low frequencies):

hf kT << 1!1

ehf kT

"1#

1

(1+ hf kT ) "1=kT

hf$ u(%,T ) #

8& f2

c3

kT

Blackbody radiationPlanck’s law relies on the quantization of energy:

Planck used the same approach as Rayleigh and Jeans:

Rayleigh and Jean’s considered a continuous distribution of energiesin a cavity to find Eave :

u !,T( ) = E

aven(!)

n(λ): number of oscillator modeswith frequency c/λ in a cavityEave: average energy per mode

E

ave= Ef (E)dE = EAe!E / kT dE = kT""

Planck considered a discrete distribution of energies in a cavity to findEave :

Eave

= En

fn(E

n) =

n

! En

Ae"E

n/ kT( ) =

n

! nhf Ae"E

n/ kT( ) =

n

!hf

ehf / kT"1

Photoelectric EffectThe photoelectric effect is the emission of electrons from a surfacewhen light is incident on it.

1. Electrons emitted with a range of velocities2. Current I (number of electrons) increases with light intensity3. Maximum kinetic energy of electrons does not depend on the

intensity of light

K

max=

1

2mu

max

2= eV

stop

Photoelectric EffectThe photoelectric effect is the emission of electrons from a surfacewhen light is incident on it.

4. There exists a threshold frequency below which no electrons areemitted. The threshold varies from metal to metal.

5. The maximum kinetic energy is proportional to the frequency6. The current appears without delay once the light is incident.

Photoelectric EffectClassical explanation of photoelectric effect:

• Electrons at the metal surface absorb energy K from the light wave.• The energy is uniformly distributed over the wave front.• The energy absorbed by an electron given an absorption cross

section A over a time t from light with intensity I is

• When the energy absorbed overcomes the work W required to ‘lift’the electrons from the surface, electrons are emitted with

• Kmax depends on the intensity (contradicts result 3)• Kmax does not depend on the frequency (contradicts results 4,5)• Time lag required to absorb enough energy to just overcome the

work function W:• (contradicts result 6)

E = cIAt

K

max= cIAt !W

aIAt =W ! t =W

aIA

Photoelectric EffectEinstein’s explanation of photoelectric effect:

• Light of frequency f consists of discrete bundles of energy calledphotons.

• The energy of each photon is

• Photons cannot be divided: each photon imparts all its energy to asingle electron.

• If the energy imparted by a photon is greater than the work function,then an electron is emitted with a maximum kinetic energy

E = hf =

hc

!

K

max= hf !W

Photoelectric EffectEinstein’s explanation of photoelectric effect:

E = hf

K

max= hf !W

• Kmax does not depend on the intensity (explains result 3)• Electrons are only emitted when a photon has enough energy to

overcome the work function W. Hence there is a thresholdfrequency which depends on the work function of the metal.(explains result 4)

• Kmax depends linearly on the frequency (explains result 5)• If a photon imparts its energy to a single electron it immediately can

escape: No time lag for ejecting electrons. (explains result 6)

K

max= 0 = hf

0!W " f

0=

W

h

Photoelectric EffectMillikan confirmed Einstein’s explanation by measuring a lineardependence of the stopping potential on frequency.

E = hf

V

stop=

Kmax

e=

h

ef ! f

0( )

Millikan’s experimental results

Note: Einstein won the Nobel prize in 1921 for the photoelectric effect,NOT relativity.

Photoelectric Effect

E =hc

!=6.63"10

#34 J s( ) 3.0 "108 m / s( )

388 "10#9

m"

1 eV

1.6 "10#19

J= 3.20 eV

When a metal is illuminated with light of wavelength 388nm, photoelectronsstart to be ejected. What is the work function of the metal.Will electrons be ejected if light with double the wavelength is incident?

The lowest photon energy that creates photoelectrons from the metal is

The work function of the metal is equal to the threshold energy 3.20 eV.

Photoelectric EffectThe maximum kinetic energy of photoelectrons is 2.8eV. When the wavelength

of light is increased by 50% this decreases to 1.1eV.(a) Find the work function(b) Find the initial wavelength

(a) The maximum kinetic energy of photoelectrons is Kmax = hf – E0. Substituting the given values,

2.8 eV = hc

!" E

0 1.1 eV =

hc

1.5!" E

0

Multiplying the second equation by 1.5 and subtracting the second equation from the first,

1.15 eV = 0.5 E0 # E0 = 2.3 eV

(b) Substituting E0 = 2.3 eV into the first equation,

2.8 eV =

4.14 $ 10"15

eV s( ) 3.0 $108

m /s( )!

" 2.3 eV # ! = 244 nm