‘Modern’ Physics
At the beginning of the twentieth century, two new theories
revolutionized 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 unable
to deal with.
Both theories challenged our fundamental intuition and perception of
the world: space-time, mass-energy, locality, causality….
Quantum Mechanics
Blackbody radiation
A black body is an object that absorbs all electromagnetic radiation
that falls on it. Since it does not reflect any radiation it appears black.
Black bodies are not only perfect absorbers but also perfect emitters of
radiation. They emit the maximum amount of energy possible at a given
temperature.
In reality no object is a perfect black body.
Some examples of approximate black bodies: Graphite (absorbs over
95% of incident radiation), Stars: absorb and emit at all wavelengths
A cavity with a small opening:
Blackbody radiation
A black body emits radiation at all wavelengths.
Measurement of emitted power
The emitted power e(λ, T) depends on
the temperature (T) and wavelength
(λ).
Blackbody radiation
Stefan-Boltzman law: The total power over all wavelengths (area under
each temperature curve) per unit area is:
"
etot = # e(!,T )d ! =$ T 4
0
$ = 5.67 % 10 &8 W ( m 2 K 4 )
For a non-ideal black-body:
etot = a! T 4 , a < 1
! = 5.67 " 10 #8 W ( m 2 K 4 )
This relationship was experimentally found by Stefan and theoretically
confirmed by Boltzman using Maxwell’s equations and thermodynamics
Blackbody radiation
Wien’s displacement law: The peak wavelength is inversely
proportional to temperature
!max
a
= , a = 2.898 " 10 #3 m $ K
T
Blackbody radiation
Rayleigh-Jean’s law for the spectral energy density u:
# 8" &
4
u ! ,T = e( ! ,T ) = Eave n( ! ) = kT % 4 (
c
$! '
(
)
n(λ): number of oscillator modes with frequency c/λ in a cavity
Eave: average energy per mode
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-V
catastrophe
Blackbody radiation
Wien’s exponential law for the spectral energy density u:
Ac 3e" # c / !T
3 " # f /T
u(!,T ) =
=
Af
e
3
!
Advantages:
• Stefan’s law can be derived from this equation
• Explains the peak wavelength
• Good agreement to experiments at short wavelengths (high
frequencies)
Problems:
• Derived empirically (no theoretical basis)
• Does not agree with Rayleigh-Jeans theory at long wavelengths (low
frequencies)
Blackbody radiation
Planck’s law for the spectral energy density u:
3
8" h $
1
1
' 8" f $
'
u(!,T ) = 3 & hc ! kT
=
)
&
)
! %e
# 1(
c 3 % ehf kT # 1 (
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
8& f 3 " hf
$ u(%,T ) # 3 e
c
kT
• Agrees with Rayleigh-Jean’s law at long wavelengths (low frequencies):
hf kT << 1 !
1
ehf
kT
1
kT
8& f 2
#
=
$ u(%,T ) # 3 kT
" 1 (1 + hf kT ) " 1 hf
c
Blackbody radiation
Planck’s law relies on the quantization of energy:
Planck used the same approach as Rayleigh and Jeans:
(
)
u ! ,T = Eave n( ! )
n(λ): number of oscillator modeswith frequency c/λ in a cavity
Eave: average energy per mode
Rayleigh and Jean’s considered a continuous distribution of energies
in a cavity to find Eave :
Eave = " Ef (E)dE = " EAe! E / kT dE = kT
Planck considered a discrete distribution of energies in a cavity to find
Eave :
(
Eave = ! En f n (En ) = ! En Ae
n
n
" En / kT
) =! nhf ( Ae
n
" En / kT
)=e
hf
hf / kT
"1
Photoelectric Effect
The photoelectric effect is the emission of electrons from a surface
when light is incident on it.
1. Electrons emitted with a range of velocities
2. Current I (number of electrons) increases with light intensity
3. Maximum kinetic energy of electrons does not depend on the
intensity of light
K max
1 2
= mumax = eVstop
2
Photoelectric Effect
The photoelectric effect is the emission of electrons from a surface
when light is incident on it.
4. There exists a threshold frequency below which no electrons are
emitted. The threshold varies from metal to metal.
5. The maximum kinetic energy is proportional to the frequency
6. The current appears without delay once the light is incident.
Photoelectric Effect
Classical 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
E = cIAt
•
When the energy absorbed overcomes the work W required to ‘lift’
the electrons from the surface, electrons are emitted with
K max = cIAt ! W
•
•
•
•
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:
W
(contradicts result 6)
aIAt = W ! t =
aIA
Photoelectric Effect
Einstein’s explanation of photoelectric effect:
•
•
Light of frequency f consists of discrete bundles of energy called
photons.
The energy of each photon is
hc
E = hf =
!
•
•
Photons cannot be divided: each photon imparts all its energy to a
single electron.
If the energy imparted by a photon is greater than the work function,
then an electron is emitted with a maximum kinetic energy
K max = hf ! W
Photoelectric Effect
Einstein’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 threshold
frequency which depends on the work function of the metal.
(explains result 4)
K max
•
•
W
= 0 = hf0 ! W " f0 =
h
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)
Photoelectric Effect
Millikan confirmed Einstein’s explanation by measuring a linear
dependence of the stopping potential on frequency.
E = hf
Vstop =
K max
e
=
h
f ! f0
e
(
)
Millikan’s experimental results
Note: Einstein won the Nobel prize in 1921 for the photoelectric effect,
NOT relativity.
Photoelectric Effect
When a metal is illuminated with light of wavelength 388nm, photoelectrons
start 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
(
)(
)
6.63 " 10 #34 J s 3.0 " 10 8 m / s
hc
1 eV
E=
=
"
= 3.20 eV
#9
#19
!
388 " 10 m
1.6 " 10 J
The work function of the metal is equal to the threshold energy 3.20 eV.
Photoelectric Effect
The 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
!
" E0
1.1 eV =
hc
1.5 !
" E0
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 $ 10 m /s
!
8
) " 2.3 eV # ! = 244 nm