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Physics 3 for Electrical Engineering Ben Gurion University of the Negev 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; פרקים בפיסיקה מודרנית, יחידה 2

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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.

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Meanwhile, a second revolution in modern physics, and in how we think about small energies, small distances, measurement and causality, was underway.

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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.

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Crucial experiments on the way to quantum theory: Blackbody spectrum ( )

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X-rays (1895) Crucial experiments on the way to quantum theory: Blackbody spectrum ( ) Spectroscopy ( )

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Radioactivity (1896) X-rays (1895) Crucial experiments on the way to quantum theory: Blackbody spectrum ( ) Photoelectric effect ( ) Spectroscopy ( )

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Discovery of the electron (1897) Radioactivity (1896) X-rays (1895) Crucial experiments on the way to quantum theory: Radium (1898) Blackbody spectrum ( ) Photoelectric effect ( ) Spectroscopy ( )

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Discovery of the electron (1897) γ-rays (1900) Specific heat anomalies ( ) Radioactivity (1896) X-rays (1895) Crucial experiments on the way to quantum theory: Radium (1898) Blackbody spectrum ( ) Photoelectric effect ( ) Spectroscopy ( )

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X-ray interference (1911) Rutherford scattering (1911) Superconductivity (1911) Discovery of the electron (1897) γ-rays (1900) Specific heat anomalies ( ) Radioactivity (1896) X-rays (1895) Crucial experiments on the way to quantum theory: Radium (1898) Blackbody spectrum ( ) Photoelectric effect ( ) Spectroscopy ( )

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X-ray interference (1911) Paschen- Back effect (1912) Rutherford scattering (1911) Superconductivity (1911) Discovery of the electron (1897) γ-rays (1900) Specific heat anomalies ( ) Radioactivity (1896) X-rays (1895) Crucial experiments on the way to quantum theory: Radium (1898) Blackbody spectrum ( ) Photoelectric effect ( ) Spectroscopy ( ) X-ray diffraction (1912)

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X-ray interference (1911) Paschen- Back effect (1912) Rutherford scattering (1911) Superconductivity (1911) Discovery of the electron (1897) γ-rays (1900) Specific heat anomalies ( ) Stern-Gerlach ( ) Radioactivity (1896) X-rays (1895) Crucial experiments on the way to quantum theory: Radium (1898) Blackbody spectrum ( ) Photoelectric effect ( ) Spectroscopy ( ) Franck-Hertz experiment (1914) X-ray diffraction (1912)

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X-ray interference (1911) Paschen- Back effect (1912) Rutherford scattering (1911) Superconductivity (1911) Discovery of the electron (1897) γ-rays (1900) Specific heat anomalies ( ) Stern-Gerlach ( ) Radioactivity (1896) X-rays (1895) Crucial experiments on the way to quantum theory: Radium (1898) Blackbody spectrum ( ) Photoelectric effect ( ) Spectroscopy ( ) electron diffraction (1927) Franck-Hertz experiment (1914) X-ray diffraction (1912) Compton effect (1923)

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X-ray interference (1911) Paschen- Back effect (1912) Rutherford scattering (1911) Discovery of the electron (1897) γ-rays (1900) Specific heat anomalies ( ) Stern-Gerlach ( ) Superconductivity (1911) Radioactivity (1896) X-rays (1895) Crucial experiments on the way to quantum theory: Radium (1898) Blackbody spectrum ( ) Photoelectric effect ( ) Spectroscopy ( ) electron diffraction (1927) Franck-Hertz experiment (1914) X-ray diffraction (1912) Compton effect (1923)

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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

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Receiver Spark Gap Transmitter

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Receiver Spark Gap Transmitter Hertz discovered that under ultraviolet radiation, sparks jump across wider gaps!

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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

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Ammeter Vacuum tube

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Ammeter Vacuum tube

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Ammeter Vacuum tube

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Ammeter Vacuum tube

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Ammeter Vacuum tube

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Ammeter Vacuum tube

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V … but the stopping potential V 0 does not depend on the light intensity. With an applied potential V, the saturation current is proportional to the light intensity …

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Einstein’s prediction (based on his “heuristic principle”): E max is the maximum energy of an ejected electron. V 0 is the stopping potential. h is Planck’s constant, h = × 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.

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V 0 = E max /e Measurements by Millikan (1914) showed that the coefficient of ν is indeed the h discovered by Planck. ν ν 0 = Φ/h

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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?

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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 S sphere = 4 (1 m) 2. The cross-section of an atom is S atom = (10 −10 m) 2. The atom absorbs (1 J/sec) (S atom /S sphere ). So the time required for 1 eV to build up at the atom is

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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!

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The Compton effect For almost two decades, no one believed in Einstein’s “quanta ” of light. Then came Compton’s experiment (1923):

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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 = pc 2 /E. Hence Thus p light = E light /c = hv/c. Since 0 = (E light ) 2 – (p light ) 2 c 2 = m 2 c 4, it follows that a “quantum of light” has zero mass.

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Consider light of frequency ν scattering from an electron at rest: Energy conservation: hν–hν′ = m e (γ–1)c 2, where. Forward momentum conservation: Transverse momentum conservation: e–e– ν′ ν θ φ

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}

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Compton ’ s data: θ λ′

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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.

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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.

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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.

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electrons Electron diffraction X-rays on zirconium oxide Electrons on gold

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Neutron diffraction Diffraction of X-rays on a single NaCl crystal Diffraction of neutrons on a single NaCl crystal

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Heiblum (1994): real experiment Bohr (1927): thought-experiment Electron interference

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λ=6 nm at T=300 K λ=600 nm at T=30 mK Electron interference

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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”.

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But already we can anticipate a strange, far-reaching and disturbing implication of the new mechanics: It limits what we can measure.

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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

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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.

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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?

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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.

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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.

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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. p = h/λ. θ

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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. p = h/λ. θ

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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. p = h/λ. Therefore (Δx)(Δp) ≥ h. θ

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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. ΔxΔx

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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. ΔxΔx

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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. ΔxΔx

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Example 1: Square barrier x L/2−L/2 k Δk ≥ 2/L Δx = L F(k) f(x)

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Example 2: Exponential decay f(x) ≈ x k F(k) ≈ Δk > 1/L Δx ≈ L F(k) ≈ f(x) ≈

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Example 3: Gaussian x k F(k) ≈ Δk > 1/L Δx ≈ L f(x) ≈

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

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We have already seen two proofs of Heisenberg ’ s uncertainty principle, and we will see at least one more proof.

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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.

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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.

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