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The historical development of the particle properties of waves, from newton's corpuscles to the modern concept of photons and quantum mechanics. It covers the wave-particle duality of light, planck's quantum theory of radiation, and the photoelectric effect. The document also explores the lorentz invariance of the phase of a photon and the particle properties of light.
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photons)
Relativisticmechanics,El.-Mag.(1905)^ Classicalphysics
Relativisticquantummechanics(1927-)^ Quantummechanics(1920’s-)
v / c
h / s S^ – the action=momentum
×distance, units g×
(^2) cm/s
Newton
( Opticks
, 1704): light as a stream of particles (corpuscles). Descartes
(1637),
Huygens
,^ Young
,^ Fresnel
(1821),
Maxwell
: by mid-
th^ century, the
wave nature of light was firmly established (
interference
and^ diffraction
, transverse nature
of e.-m. waves). Physics of the 19
th^ century: mostly investigation of light waves; physics of the 20
th^ century –
interaction of light with matter. One of the challenges – understanding the “blackbody spectrum” of thermal radiation (to beconsidered later in the course). Planck
(1900)
suggested
a^ solution
based
a^ revolutionary
new^
idea:^ emission
and
absorption of electromagnetic radiation by matter has quantum nature: the energy of aquantum of e.-m. radiation emitted or absorbed by a harmonic oscillator with the frequency
f
where^ h^ is the Planck’s constant
34
E^ ω=^ h^
where^
34
2 f ω π=
Also, in terms of theangular frequency
h =h 2 π
Wave equation in one dimensionfor any quantity
ψ:
2
(^22) 2 v^2 t^
x ψ
ψ ∂^
∂= ∂^
∂
(^ )
Solution:
a^ plane
wave
traveling
in^ the
negative (positive) direction
x^ with velocity
v :^
f^ x^ vt ψ = ±
x
t^ = 0^
t^ =^ t^0 vt^0
(^ ) Harmonic plane wave traveling in the positive direction
x : (^
)
sin^
sin^2
sin x^ t
A^ x
vt^
A^
A^ kx
t T
ψ
π
ω
⎡^ λ
⎤ ⎛^
⎞
=^
−^ =^
−^ =
−
⎜^
⎟ ⎢^
⎥ ⎝^
⎠ ⎣^
⎦
phase^
x^ vt ≡ −^
constant phase
/
x^ vt^
const^
v^ x^ t
→^ −^
=^
→^ =
v^ –^ the phase velocity
x λ ψ A^0 -A^0
2 2
f π ω^ π= =^ T
2 π k = λ
ω v = k
Electromagnetic waves: (transverse in free space)^2
2
(^22) 2
2
2 (^1) E E 0 0
Ec
t^
x^
x
∂^ ∂^ ε μ
∂ =^
= ∂^
∂^
∂ 2
2
(^22) 2
2
2 (^1) B B 0 0
Bc
t^
x^
x
∂^ ∂^ ε μ
∂ =^
= ∂^
∂^
∂
r , E^ x t^ (^ ) r , B x t ( )
angularfrequency
wavenumber
t T ψ A^0 -A^0^ (^
)^ (^ ) ,^
E^ x t^
cB x t =
According to the quantum theory of radiation, photons are
massless
bosons of spin
^1 (in units
ħ ). They move with the speed of light :
ph ph^
ph
“Light” – a shorthand notation for any e.-m. radiation (
ν^ from 0 to
∞).
2 2
2
2 0
ph^
ph^
ph E^
cp^
m^ c −^
=^
=
Quantum character of this equation is illustrated by the fact that theenergy is associated with the
frequency
of oscillations rather than
their^ amplitude
. Particle properties of light
Wave properties of light E^ r^^ ph^ , i p^ phc
Thus, if Planck’s idea
E=ħ^ ω^
is correct, than we must conclude that
p^ ph
r k = r^ h
The phase
is a^ Lorentz-invariant quantity
,
t^ the (scalar) product of two 4-vectors:
⎛^
⎞ ⎜^
⎟ ⎝^
r^ ⎠
k = =h^ λ
Historical Note
:^ The photoelectric effect was accidentally discovered by
Heinrich Hertz
in 1887 during the course of the experiment that discovered radio waves. Hertz died (atage 36) before the first Nobel Prize was awarded.Observation: when a negatively charged body was illuminated with UV light, its chargewas diminished. J.J. Thomson and P. Lenard
determined the ration
e/m^ for the particles emitted by the
body under illumination – the same as for electrons.The^ effect
remained
unexplained
until^
1905 when
Albert
Einstein
postulated
the
existence of quanta of light -- photons -- which, when absorbed by an electron near thesurface of a material, could give the electron enough energy to escape from the material. Robert Milliken
carried out a careful set of experiments, extending over ten years, that verified the predictions of Einstein’s photon theory of light.
Einstein was awarded the
1921 Nobel Prize in physics: "For his services to Theoretical Physics, and especially forhis discovery of the law of the photoelectric effect."
Milliken received the Prize in 1923
for his work on the elementary charge of electricity (the oil drop experiment) and on thephotoelectric effect.
Observations
:
1.^ For a given material of the cathode, the “stopping”voltage does not depend on the light intensity.2.^ The saturation current is proportional to the intensityof light at
f^ =const.
3.^ Material-specific “red boundary”
f exists: no^0
photocurrent at
f^ <^ f^0.
4.^ Practically instantaneous response – no delaybetween the light pulse and the photocurrent pulse(many applications are based on this property) V
retarding
I
intensity of lightincreases,
f^ =const^ V I^ intensity = const,
f^ increases
stoppingvoltage
light I
- e _+^ V^
f
stopping voltage
cesium
calcium
Parameters
: intensity (
S ) and frequency (
f ) of light,
applied voltage (
V ), measured photocurrent (
I )
“redboundary”
f^0
V^0 V ( f )^0
V ( f )^0 1 f > f 2 1
Absorption of a photon by an electron in metal(inelastic collision between these particles)
energy conservation
2
2 e^
ph^ e m c^
E^ m c
the rest RF of an electronafter the collision
(^0) Eph γ^ =^
before^
(^2) m c K + e after (^2) m c e hf
However, we’ve concluded that a
free^ electron
cannot
absorb a photon!
before^
after (^2) m c K +^ e
(^2) m c e
hf What’s wrong? The electron is not “free”, it is embedded in metal, and the chunk of metal is
the second body that participates in the collision 2
2
2
2
ph^ e
met^
e^
e^ met
met
E^ m c
c^ m c
M^ c^
ph^ e met p^ p
p = + r^ r
r^ met^
e M^
m >>> Thus, while the electron isstill
inside
metal
~ ph^
e^ met p^
p^ p <<
e met^ e
m met v^ v
2 met^ ~ 2
e^
e
met^
e^
e^
e
met^
met M^
m^
m
K^
v^
energy conservation E^ K = ph^ e
momentum conservation
(see Slide 6) The photon energy is absorbed by an electron (the energy absorbed by metal is negligibly small),but the momentum exchange between electron and metal is crucial for momentum conservation.
ph^ e met p^ p
p = + r^ r
r
In the experiment, the electron is observed
outside
the metal. It takes some energy to escape:
(consider an attraction between an electron and the positive
“image” charge induced on the metal surface)
metal
- q +q
The “escape” energy:
the work function W
(material-specific)
Thus, for the electron^ outside
metal^
K^ E^ e^ ph
K^ f^ h^ (^ ) e
f^ W = −
0 (^0) K hfe
“red” boundary of Ph. E.
(^ )^
(^
) 0 K^ f^ e
h^ f^
f =^
f^0
Planck’s constantmeasurements:
(^ )^
(^ ) 0 0
0 K^ f^ e
eV^ f h^ f^
f^ f
f =^
1.^ The work function of tungsten surface is 5.4eV. When the surface is illuminated by lightof wavelength 175nm, the maximum photoelectron energy is 1.7eV. Find Planck’s constantfrom these data.^ e
c K^ hf
(^
)^ (
)^
7
15
8 15
19
34
K^ W^ e
eV^
eV^
m
h^
eV^ s
c^
m^ s eV^ s^
J^ eV^
J^ s
−
−
−^
−^
−
2.^ The threshold wavelength for emission of electrons from a given metal surface is 380nm.(a) what will be the max kinetic energy of ejected electrons when
λ^ is changed to 240nm?
(b) what is the maximum electron speed?(c) the loss of electrons due to the photoelectric effect will cause an isolated sphere of thismetal to acquire a positive charge. Find the largest electric potential (in Volts) that couldbe achieved by this sphere for
λ^ = 240nm.
1
1 0
c e
c^ c
K^ h
h^
h^
hc^
eV
(a)^2
5 2 / 2^
e
e^ e
K e
K^ m v
v^
m^ s m
=^
(b)^
(c)
K^ f^ (^ ) e h f^
W metal^
E vacuum
electronenergy in the(repulsive)electric field
Ke^ 1. hf^ W
hf^ W
eV^
e^
e −
=^ +^
16
19
0.1^100 ph
nm^
f^
Hz^
keV
Production:
The upper “cut-off” of the spectrum correspondsto^ the^
full^ conversion
of^ the
electron
kinetic
energy into the photon energy. 2 (^ )
max
min
e^ e c hf^
h^
eV^ K
m c
34
8
11
min^
19
5
6.6^10
hc^
J^ s^
m^ s^
m
eV^
−
−
− ×^
o^
[^ ] [ ] 12.4min nmV kV
λ
γ-rays
UV X -rays 0.01 nm^^10
nm
