Transcript L4 towards QM

Ben Gurion University of the Negev
www.bgu.ac.il/atomchip, www.bgu.ac.il/nanocenter
Physics 3 for Electrical Engineering
Lecturers: Daniel Rohrlich, Ron Folman
Teaching Assistants: Daniel Ariad, Barukh Dolgin
Week 4. Towards quantum mechanics – photoelectric effect •
Compton effect • electron and neutron diffraction • electron
interference • Heisenberg’s uncertainty principle • wave packets
Sources:
Tipler and Llewellyn, Chap. 3 Sects. 3-4 and Chap. 5 Sects. 5-7;
2 ‫ יחידה‬,‫פרקים בפיסיקה מודרנית‬
Einstein’s relativity theories (Special Relativity in 1905 and
General Relativity in 1915) were a revolution in modern physics,
and in how we think about space, time and motion at high speeds.
Einstein’s relativity theories (Special Relativity in 1905 and
General Relativity in 1915) were a revolution in modern physics,
and in how we think about space, time and motion at high speeds.
Meanwhile, a second revolution in modern physics, and in how
we think about small energies, small distances, measurement and
causality, was underway.
Einstein’s relativity theories (Special Relativity in 1905 and
General Relativity in 1915) were a revolution in modern physics,
and in how we think about space, time and motion at high speeds.
Meanwhile, a second revolution in modern physics, and in how
we think about small energies, small distances, measurement and
causality, was underway.
Crucial experiments on the way to quantum theory:
Blackbody
spectrum
(1859-1900)
Crucial experiments on the way to quantum theory:
Spectroscopy
(1885-1912)
Blackbody
spectrum
(1859-1900)
X-rays
(1895)
Crucial experiments on the way to quantum theory:
Spectroscopy
(1885-1912)
Blackbody
spectrum
(1859-1900)
Radioactivity
(1896)
Photoelectric
effect
(1887-1915)
X-rays
(1895)
Crucial experiments on the way to quantum theory:
Spectroscopy
(1885-1912)
Radioactivity
(1896)
Photoelectric
effect
(1887-1915)
Blackbody
spectrum
(1859-1900)
Radium
(1898)
Discovery of
the electron
(1897)
X-rays
(1895)
Crucial experiments on the way to quantum theory:
Spectroscopy
(1885-1912)
Radioactivity
(1896)
Photoelectric
effect
(1887-1915)
Blackbody
spectrum
(1859-1900)
Radium
(1898)
γ-rays
(1900)
Specific heat
anomalies
(1900-10)
Discovery of
the electron
(1897)
X-rays
(1895)
Crucial experiments on the way to quantum theory:
Spectroscopy
(1885-1912)
Superconductivity
(1911)
Blackbody
spectrum
(1859-1900)
γ-rays
(1900)
Specific heat
anomalies
(1900-10)
Radioactivity
(1896)
Photoelectric
effect
(1887-1915)
X-rays
Radium
(1895)
(1898) Rutherford
scattering
Discovery of
X-ray
(1911)
the electron
interference
(1897)
(1911)
Crucial experiments on the way to quantum theory:
PaschenBack effect
(1912)
Superconductivity
(1911)
Blackbody
spectrum
(1859-1900)
γ-rays
(1900)
Spectroscopy
(1885-1912)
Radioactivity
(1896)
Photoelectric
effect
(1887-1915)
X-rays
(1895)
X-ray diffraction Radium
(1898) Rutherford
(1912)
scattering
Specific heat Discovery of
X-ray
(1911)
anomalies
the electron
interference
(1900-10)
(1897)
(1911)
Crucial experiments on the way to quantum theory:
PaschenBack effect
(1912)
Superconductivity
(1911)
Blackbody
spectrum
(1859-1900)
γ-rays
(1900)
Franck-Hertz
experiment
(1914)
Spectroscopy
(1885-1912)
Radioactivity
(1896)
Photoelectric
effect
(1887-1915)
X-rays
(1895)
X-ray diffraction Radium
(1898) Rutherford
(1912)
scattering
Specific heat Discovery of
X-ray
(1911)
anomalies
the electron
Stern-Gerlach interference
(1900-10)
(1897)
(1911)
(1921-23)
Crucial experiments on the way to quantum theory:
Paschen- electron diffraction
Back effect
(1927)
(1912)
Superconductivity
(1911)
Blackbody
spectrum
(1859-1900)
Franck-Hertz
experiment
(1914)
Spectroscopy
(1885-1912)
Radioactivity
(1896)
Photoelectric
effect
(1887-1915)
Compton effect
X-rays
(1923)
(1895)
X-ray diffraction Radium
(1898) Rutherford
(1912)
γ-rays
scattering
Specific heat Discovery of
(1900)
X-ray
(1911)
anomalies
the electron
Stern-Gerlach interference
(1900-10)
(1897)
(1911)
(1921-23)
Crucial experiments on the way to quantum theory:
Paschen- electron diffraction
Back effect
(1927)
(1912)
Superconductivity
(1911)
Blackbody
spectrum
(1859-1900)
Franck-Hertz
experiment
(1914)
Spectroscopy
(1885-1912)
Radioactivity
(1896)
Photoelectric
effect
(1887-1915)
Compton effect
X-rays
(1923)
(1895)
X-ray diffraction Radium
(1898) Rutherford
(1912)
γ-rays
scattering
Specific heat Discovery of
(1900)
X-ray
(1911)
anomalies
the electron
Stern-Gerlach interference
(1900-10)
(1897)
(1911)
(1921-23)
The photoelectric effect
An irony in the history of physics:
Heinrich Hertz, who was the first (in 1886) to verify Maxwell’s
prediction of electromagnetic waves travelling at the speed of
light, was also the first to discover (in the course of the same
investigation) the photoelectric effect!
Receiver
Spark Gap
Transmitter
Receiver
Spark Gap
Transmitter
Receiver
Spark Gap
Transmitter
Hertz discovered that under ultraviolet
radiation, sparks jump across wider gaps!
Hallwachs (1888): Ultraviolet light on a neutral metal leaves it
positively charged.
Hertz died in 1894 at the age of 36, one year before the
establishment of the Nobel prize.
His assistant, P. Lenard, extended Hertz’s research on the
photoelectric effect and discovered (1902) that the energy of the
sparking electrons does not depend on the intensity of the
applied radiation; but the energy rises with the frequency of the
radiation.
photoelectric
Vacuum tube
Ammeter
Vacuum tube
Ammeter
Vacuum tube
Ammeter
Vacuum tube
Ammeter
Vacuum tube
Ammeter
Vacuum tube
Ammeter
With an
applied
potential V,
the saturation
current is
proportional
to the light
intensity…
V
…but the stopping potential V0 does
not depend on the light intensity.
Einstein’s prediction (based on his “heuristic principle”):
eV0  Emax  h  
• Emax is the maximum energy of an ejected electron.
• V0 is the stopping potential.
• h is Planck’s constant, h = 6.6260693 × 10−34 J · sec.
• ν is the frequency of the applied radiation.
• Φ is the “work function” – the work required to bring
an electron in a metal to the surface – a constant that
depends on the metal.
V0 = Emax/e
Measurements by Millikan (1914)
showed that the coefficient of ν is
indeed the h discovered by Planck.
ν0 = Φ/h
ν
Can we understand the physics?
Consider a light source, producing 1 J/sec = 1 W of power,
shining on metal at a distance of 1 meter.
If the metal has ionization energy (work function) Φ = 1 eV,
how long will it take to eject electrons from the metal?
Can we understand the physics?
Consider a light source, producing 1 J/sec = 1 W of power,
shining on metal at a distance of 1 meter.
A simple calculation: 1 J/sec of power is distributed (at 1 m)
over an area Ssphere = 4p(1 m)2. The cross-section of an atom is
Satom = p(10−10 m)2. The atom absorbs (1 J/sec) (Satom /Ssphere).
So the time required for 1 eV to build up at the atom is
(1 eV)(1.609  10 -19 J/eV)
 64 sec
(1 J/sec) Satom/Ssphere
Can we understand the physics?
Consider a light source, producing 1 J/sec = 1 W of power,
shining on metal at a distance of 1 meter.
In fact the light ejects electrons from the metal as soon as it
arrives!
The Compton effect
For almost two decades, no one believed in Einstein’s “quanta ”
of light. Then came Compton’s experiment (1923):
If the energy of a “light quantum” of frequency ν is hν, what is
its momentum?
Theorem: the velocity v of a particle of relativistic energy E and
momentum p is v = pc2/E. Hence
c  vlight  plightc 2 / Elight  plightc 2 / h
Thus plight = Elight/c = hv/c. Since 0 = (Elight)2 – (plight)2c2 = m2c4,
it follows that a “quantum of light” has zero mass.
Consider light of frequency ν scattering from an electron at rest:
ν′
θ
ν
φ
e–
Energy conservation: hν–hν′ = me(γ–1)c2, where   1 / 1   2 .
Forward momentum conservation:
h / c  (h ' / c) cos  (me c) cos
Transverse momentum conservation:
(h ' / c) sin   (me c) sin 
h / c  (h ' / c) cos 2  (me c)2 cos2 
(h ' / c) sin   (me c) sin 
2
2

2
2
2

}
h
2
2
2
2 2
2


(

'
)

2

'
cos


(
m


c
)

m
c
(

 1)
 
e
e
c
2 4
m
c
2
2
e
  ( ' )  2 ' cos  2 (  1)(  1)
h
(  ' ) 2  2 ' (1  cos )  (  ' )(  '2mec 2 / h)
 ' (1  cos )  (  ' )mec 2 / h
h
1  cos 
 ' 
mec
Compton’s data:
λ′
θ
Compton’s data finally convinced most physicists that
light of frequency ν indeed behaves like particles –
“quanta” or “photons” – with energy E = hν and
momentum p=E/c = hν/c or p= h/λ.
Compton’s data finally convinced most physicists that
light of frequency ν indeed behaves like particles –
“quanta” or “photons” – with energy E = hν and
momentum p=E/c = hν/c or p= h/λ.
Soon (1924) Louis de Broglie conjectured that, just as an
electromagnetic wave could behave like a particle, an
electron – indeed, any particle – of momentum p could
behave like a wave of wavelength p= h/λ.
Compton’s data finally convinced most physicists that
light of frequency ν indeed behaves like particles –
“quanta” or “photons” – with energy E = hν and
momentum p=E/c = hν/c or p= h/λ.
Soon (1924) Louis de Broglie conjectured that, just as an
electromagnetic wave could behave like a particle, an
electron – indeed, any particle – of momentum p could
behave like a wave of wavelength p= h/λ.
Confirmation of de Broglie’s conjecture came in 1927
with the experiments of C. Davisson and L. Germer, and
of G. P. Thompson, who showed that a beam of electrons
falling on a thin layer of metal or crystal produces
interference rings just like a beam of X-rays.
Electron diffraction
X-rays on zirconium oxide
electrons
Electrons on gold
Neutron diffraction
Diffraction of X-rays on
a single NaCl crystal
Diffraction of neutrons on
a single NaCl crystal
Electron interference
Bohr (1927):
thought-experiment
Heiblum (1994):
real experiment
Electron interference
λ=6 nm at T=300 K
λ=600 nm at T=30 mK
A world in which electromagnetic waves interact like particles,
and particles diffract and interfere like waves, is very different
from the world we know on a larger scale. It forces us to search
for a new mechanics – “quantum mechanics”.
A world in which electromagnetic waves interact like particles,
and particles diffract and interfere like waves, is very different
from the world we know on a larger scale. It forces us to search
for a new mechanics – “quantum mechanics”.
But already we can anticipate a strange, far-reaching and
disturbing implication of the new mechanics:
It limits what we can measure.
A world in which electromagnetic waves interact like particles,
and particles diffract and interfere like waves, is very different
from the world we know on a larger scale. It forces us to search
for a new mechanics – “quantum mechanics”.
But already we can anticipate a strange, far-reaching and
disturbing implication of the new mechanics:
It limits what we can measure.
Heisenberg (1926) stated this limit as an “uncertainty relation”:
(Δx) (Δp) ≥ h
Heisenberg’s uncertainty principle
Any optical device resolves objects
in its focal plane with a limited
precision Δx. According to
Rayleigh’s criterion, Δx is defined
by the first zeros of the image.
Heisenberg’s uncertainty principle
Any optical device resolves objects
in its focal plane with a limited
precision Δx. According to
Rayleigh’s criterion, Δx is defined
by the first zeros of the image.
By the way,
how did
Heisenberg
know about
Rayleigh’s
criterion?
Heisenberg’s uncertainty principle
1. If a lens with aperture θ focuses
light of wavelength λ, Rayleigh’s
criterion implies Δx ≈ λ/2sinθ.
p = h/λ.
Heisenberg’s uncertainty principle
1. If a lens with aperture θ focuses
light of wavelength λ, Rayleigh’s
criterion implies Δx ≈ λ/2sinθ.
2. A wave of wavelength λ has
momentum p = h/λ.
p = h/λ.
Heisenberg’s uncertainty principle
1. If a lens with aperture θ focuses
light of wavelength λ, Rayleigh’s
criterion implies Δx ≈ λ/2sinθ.
2. A wave of wavelength λ has
momentum p = h/λ.
3. From geometry we see here that
Δp ≥ 2p sinθ.
θ
p = h/λ.
Heisenberg’s uncertainty principle
1. If a lens with aperture θ focuses
light of wavelength λ, Rayleigh’s
criterion implies Δx ≈ λ/2sinθ.
2. A wave of wavelength λ has
momentum p = h/λ.
3. From geometry we see here that
Δp ≥ 2p sinθ.
θ
p = h/λ.
Heisenberg’s uncertainty principle
1. If a lens with aperture θ focuses
light of wavelength λ, Rayleigh’s
criterion implies Δx ≥ λ/2sinθ.
2. A wave of wavelength λ has
momentum p = h/λ.
3. From geometry we see here that
Δp ≥ 2p sinθ.
Therefore (Δx)(Δp) ≥ h.
θ
p = h/λ.
Another derivation of Heisenberg’s uncertainty principle:
1. We can produce a signal of length Δx by superposing waves of
various wave numbers k, where k = 2π/λ.
Δx
Another derivation of Heisenberg’s uncertainty principle:
1. We can produce a signal of length Δx by superposing waves of
various wave numbers k, where k = 2π/λ.
2. The Fourier transform of the signal will contain wave numbers
in a range Δk ≥ 2π/Δx.
Δx
Another derivation of Heisenberg’s uncertainty principle:
1. We can produce a signal of length Δx by superposing waves of
various wave numbers k, where k = 2π/λ.
2. The Fourier transform of the signal will contain wave numbers
in a range Δk ≥ 2π/Δx.
3. Therefore Δp = Δ(h/λ) = Δ(hk/2π) = h(Δk)/2π ≥ h/Δx and so
(Δx) (Δp) ≥ h.
Δx
Example 1: Square barrier
f(x)
−L/2
x
Δx = L
k
Δk ≥ 2/L
L/2
F(k)
1 1

L L
Example 2: Exponential decay
f(x) ≈ e
F(k) ≈
x / L
x
Δx ≈ L
k
Δk > 1/L
1
1  k 2 L2
Example 3: Gaussian
f(x) ≈ e
 x 2 / 2 L2
x
Δx ≈ L
k
Δk > 1/L
2 2

k
L /2
F(k) ≈ e
Wave packets
All these localized signals f(x) are examples of wave packets,
sums over waves of different wavelengths:
f ( x) 
f ( x) 
f ( x) 

2p

1

1
2p


eikx
sin kL
dk
k
eikx
dk
e

2p 
1
1  k 2 L2
ikx  k 2 L2 / 2
e
dk
We have already seen two proofs of Heisenberg’s uncertainty
principle, and we will see at least one more proof.
We have already seen two proofs of Heisenberg’s uncertainty
principle, and we will see at least one more proof.
Is the uncertainty principle a fundamental limit on what we can
measure? Or can we evade it? Einstein and Bohr debated this
question for years, and never agreed.
We have already seen two proofs of Heisenberg’s uncertainty
principle, and we will see at least one more proof.
Is the uncertainty principle a fundamental limit on what we can
measure? Or can we evade it? Einstein and Bohr debated this
question for years, and never agreed.
Today we are certain that uncertainty will not go away.
Quantum uncertainty is even the basis for new technologies such
as quantum cryptology.
It may be that the universe is not only stranger than we imagine,
but also stranger than we can imagine.