Physics 3 for Electrical Engineering Ben Gurion University of the Negev

Physics 3 for Electrical EngineeringPhysics 3 for Electrical Engineering

Ben Gurion University of the Negevwww.bgu.ac.il/atomchip, www.bgu.ac.il/nanocenter

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; 2פרקים בפיסיקה מודרנית, יחידה

http://www.bgu.ac.il/atomchip

http://www.bgu.ac.il/nanocenter

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.


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(

X-rays (1895)


Blackbody spectrum

)1859-1900(

Spectroscopy(1885-1912)

Radioactivity (1896)

X-rays (1895)


Blackbody spectrum

)1859-1900(Photoelectric

effect)1887-1915(


Discovery of the electron

(1897)


X-rays (1895)


Radium (1898)

Blackbody spectrum


effect)1887-1915(



(1897)

γ-rays (1900)

Specific heat anomalies(1900-10)


X-rays (1895)


Radium (1898)

Blackbody spectrum


effect)1887-1915(


X-ray interferenc

e(1911)

Rutherford scattering

(1911)

Superconductivity(1911)


(1897)

γ-rays (1900)



X-rays (1895)


Radium (1898)

Blackbody spectrum


effect)1887-1915(


X-ray interferenc

e(1911)

Paschen-Back effect

(1912)


(1911)



(1897)

γ-rays (1900)



X-rays (1895)


Radium (1898)

Blackbody spectrum


effect)1887-1915(


X-ray diffraction(1912)

X-ray interferenc

e(1911)

Paschen-Back effect

(1912)


(1911)



(1897)

γ-rays (1900)

Specific heat anomalies(1900-10) Stern-Gerlach

(1921-23)


X-rays (1895)


Radium (1898)

Blackbody spectrum


effect)1887-1915(


Franck-Hertz experiment

(1914)


X-ray interferenc

e(1911)

Paschen-Back effect

(1912)


(1911)



(1897)

γ-rays (1900)


(1921-23)


X-rays (1895)


Radium (1898)

Blackbody spectrum


effect)1887-1915(


electron diffraction(1927)


(1914)


Compton effect(1923)

X-ray interferenc

e(1911)

Paschen-Back effect

(1912)


(1911)Discovery of the electron

(1897)

γ-rays (1900)


(1921-23)



X-rays (1895)


Radium (1898)

Blackbody spectrum


effect)1887-1915(


electron diffraction(1927)


(1914)


Compton effect(1923)

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


Receiver


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

Ammeter

Vacuum tube

Ammeter

Vacuum tube

Ammeter

Vacuum tube

Ammeter

Vacuum tube

Ammeter

Vacuum tube

Ammeter

Vacuum tube

V

…but the stopping potential V0 does not depend on the light intensity.

With an applied potential V, the saturation current is proportional to the light intensity…

Einstein’s prediction (based on his “heuristic principle”):

• 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.

hEeV max0

V0 =

Em

ax/e

Measurements by Millikan (1914) showed that the coefficient of ν is indeed the h discovered by Planck.

ν

ν0 = Φ/h

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?



A simple calculation: 1 J/sec of power is distributed (at 1 m) over an area Ssphere = 4(1 m)2. The cross-section of an atom is Satom = (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

sec 64/SS J/sec) (1

J/eV) 10 V)(1.609 1(

sphereatom

-19

e

http://images.google.co.il/imgres?imgurl=http://i97.photobucket.com/albums/l227/emilycool_2006/LightBulb.jpg&imgrefurl=http://www.habidoo.com/blog/&usg=__orl5WcwOkmW865B5J3i_uqAaYDs=&h=631&w=800&sz=217&hl=en&start=17&tbnid=g3L7J2aE04bMVM:&tbnh=113&tbnw=143&prev=/images%3Fq%3Dlight%2Bbulb%26gbv%3D2%26hl%3Den



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

hcpEcpvc // 2lightlight

2lightlight

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:

Energy conservation: hν–hν′ = me(γ–1)c2, where .

Forward momentum conservation:

Transverse momentum conservation:

e–

ν′

νθ

φ

cos) (cos)/'(/ cmchch e

sin) (sin)/'( cmch e

21/1

222 cos) (cos)/'(/ cmchch e

2222 sin) (sin)/'( cmch e

)1( ) ( cos'2)'( 2222222

cmcm

c

hee

}

)1)(1(cos'2)'(2

4222

h

cme

)/2')('()cos1('2)'( 22 hcme

hcme /)'()cos1(' 2

cos1' cm

h

e

Compton’s data:

θ

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.


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.

electrons

Electron diffraction

X-rays on zirconium oxide Electrons on gold

Neutron diffraction

Diffraction of X-rays on a single NaCl crystal

Diffraction of neutrons on a single NaCl crystal

Heiblum (1994):real experiment

Bohr (1927):thought-experiment

Electron interference

λ=6 nm at T=300 Kλ=600 nm at T=30 mK

Electron interference

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.


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.


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?


1. If a lens with aperture θ focuses light of wavelength λ, Rayleigh’s criterion implies Δx ≈ λ/2sinθ.

p = h/λ.



2. A wave of wavelength λ has momentum p = h/λ.

p = h/λ.




3. From geometry we see here that Δp ≥ 2p sinθ.

p = h/λ.

θ





p = h/λ.

θ


1. If a lens with aperture θ focuses light of wavelength λ, Rayleigh’s criterion implies Δx ≥ λ/2sinθ.



p = h/λ.

Therefore (Δx)(Δp) ≥ h.

θ

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



2. The Fourier transform of the signal will contain wave numbers in a range Δk ≥ 2π/Δx.

Δx



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

x L/2−L/2

k

L

1

L

1

Δk ≥ 2/L

Δx = L

F(k)

f(x)

Example 2: Exponential decay

f(x) ≈

x

k

F(k) ≈

Δk > 1/L

Δx ≈ L

F(k) ≈

f(x) ≈ Lxe /

221

1

Lk

Example 3: Gaussian

x

k

F(k) ≈

Δk > 1/L

Δx ≈ L

2/22Lke

f(x) ≈ 22 2/ Lxe

Wave packets

All these localized signals f(x) are examples of wave packets, sums over waves of different wavelengths:

dk

k

kLex ikx sin

2

1)(f

2212

1)(f

Lk

dkex ikx

dkeex Lkikx 2/

22

2

1)(f

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.


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.

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Physics 3 for Electrical Engineering Ben Gurion University of the Negev