The Relativistic Quantum World A lecture series on Relativity Theory and Quantum Mechanics Marcel Merk University of Maastricht, Sept 16 – Oct 14, 2020
The Relativistic Quantum World 1 Lecture 1: The Principle of Relativity and the Speed of Light Sept. 16: Relativity Lecture 2: Time Dilation and Lorentz Contraction Lecture 3: The Lorentz Transformation and Paradoxes Sept. 23: Lecture 4: General Relativity and Gravitational Waves Lecture 5: The Early Quantum Theory Mechanics Sept. 30: Quantum Lecture 6: Feynman’s Double Slit Experiment Lecture 7: Wheeler’s Delayed Choice and Schrodinger’s Cat Oct. 7: Lecture 8: Quantum Reality and the EPR Paradox Standard Model Lecture 9: The Standard Model and Antimatter Oct. 14: Lecture 10: The Large Hadron Collider Lecture notes, written for this course, are available: www.nikhef.nl/~i93/Teaching/ Prerequisite for the course: High school level physics & mathematics.
Quantum Mechanics 2 No, similar to Yes, because Light is a stream sound light consists it interferes of particles of waves Light is emitted in quanta Isaac Newton Thomas Young Christiaan Huygens Max Planck Particles have a The nature of Particles are Yes, because wave nature : light is quanta probability waves photons collide! l = h/p Albert Einstein Louis de Broglie Arthur Compton Niels Bohr
Quantum Mechanics 2 No, similar to Yes, because Light is a stream sound light consists it interferes of particles of waves Light is emitted in quanta “Particle” and “Wave” are Isaac Newton Thomas Young Christiaan Huygens complementary aspects. Max Planck Particles have a The nature of Particles are Yes, because wave nature : light is quanta probability waves photons collide! l = h/p Albert Einstein Louis de Broglie Arthur Compton Niels Bohr
Uncertainty Relation 3 Δ𝑦 Δ𝑞 ≥ ℏ It is not possible to determine position and momentum at the same time: 2 D p Erwin Schrödinger Werner Heisenberg 𝑞 = ℎ 𝜇 = ℎ𝑔 𝑑 A particle does not have well defined position and momentum at the same time.
The wave function y 4 Position fairly known Momentum badly known
The wave function y 4 Position fairly known Momentum badly known Position badly known Momentum fairly known
5 Lecture 6 Feynman’s Double Slit Experiment “It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment it’s wrong.” - Richard Feynman
Richard Feynman (1918 – 1988) . 6 Nobelprize 1965: Quantum Electrodynamics (Path Integral formulation of quantum mechanics) Mostly known from: ・ Feynman diagrams ・ Challenger investigation ・ Popular books Challenger disaster Feynman diagram
Richard Feynman (1918 – 1988) . 7 Nobelprize 1965: Quantum Electrodynamics (Path Integral formulation of quantum mechanics) Challenger disaster Feynman diagram
Richard Feynman and the double slit experiment 8 The double slit experiment demonstrates the fundamental aspect of the quantum world.
The Double Slit Experiment 9 Case 1: An Experiment with Bullets
Case 1: Experiment with Bullets 10 A gun fires bullets in random direction. Slits 1 and 2 are openings through which bullets can pass. A moveable detector “collects” bullets and counts them. Observation: Bullets come in “lumps”. P 1 is the probability curve when only slit 1 is open P 2 is the probability curve when only slit 2 is open What is the probability curve when both slit 1 and slit 2 are open?
Case 1: Experiment with Bullets 11 A gun fires bullets in random direction. Slits 1 and 2 are openings through which bullets can pass. A moveable detector “collects” bullets and counts them. P 1 is the probability curve when only slit 1 is open P 2 is the probability curve when only slit 2 is open We can just add up the probabilities. When both slits are open: P 12 = P 1 + P 2
The Double Slit Experiment 12 Case 2: An Experiment with Waves
Waves & Interference : water, sound, light 13 Water: Interference pattern: Waves: Interference principle: Light: Thomas Young experiment: Sound: Active noise cancellation: light + light can give darkness!
Case 2: Experiment with Waves 14 We replace the gun by a wave generator: think of water waves. Slits 1 and 2 act as new wave sources. The detector measures now the intensity (energy) in the wave. Observation: Waves do not Come in “lumps”. I 1 = |h 1 | 2 I 2 = |h 2 | 2 I 12 = ?? The intensity of a wave is the square of the amplitude…
Intermezzo: Wave Oscillation & Intensity 15 Energy in the oscillation (up-down) movement of the molecules: ⁄ 𝑛𝑤 : and 𝑤 is proportional to the amplitude or height: 𝑤 ≈ ℎ 𝐹 678 = 9 : So that the intensity of the wave is: 𝐽 ≈ ℎ : h h v Formula for the resulting oscillation of a water molecule somewhere in the wave: 𝑆 𝑢 = ℎ cos 2𝜌𝑔𝑢 + 𝜚 𝑔 = frequency 𝜚 = phase and the Intensity: 𝐽 = ℎ :
Case 2: Experiment with Waves 16 When both slits are open there are two contributions to the wave the oscillation at the detector: 𝑆 𝑢 = 𝑆 9 𝑢 + 𝑆 : 𝑢 𝑆 9 𝑢 = ℎ 9 cos 2𝜌𝑔 𝑢 + 𝜚 9 𝑆 : 𝑢 = ℎ : cos 2𝜌𝑔 𝑢 + 𝜚 : 𝜚 9 and 𝜚 : depend on distance to 1 and 2 𝐽 9 = ℎ 9 : 𝐽 : = ℎ : : 𝐽 9: = ? ? First combine: 𝑆 𝑢 = 𝑆 9 𝑢 + 𝑆 : 𝑢 Afterwards look at the amplitude and intensity of the resulting wave!
Mathematics for the die-hards 17 𝑺 𝟐𝟑 𝒖 = 𝒊 𝟐 𝐝𝐩𝐭 𝟑𝝆𝒈𝒖 + 𝝔 𝟐 + 𝒊 𝟑 𝐝𝐩𝐭 𝟑𝝆𝒈 𝒖 + 𝝔 𝟑 Assume equal size waves: 𝒊 𝟐 = 𝒊 𝟑 = 𝒊 First find amplitude of sum wave 𝑺 𝟐𝟑 𝒖 . From math textbook: cos 𝐵 + cos 𝐶 = 2 cos 1 cos 1 2 𝐵 − 𝐶 2 𝐵 + 𝐶 𝟐 Use this to find: 𝑺 𝟐𝟑 𝒖 = 𝒊′ 𝐝𝐩𝐭 𝟑𝝆𝒈𝒖 + 𝟑 𝝔 𝟐 + 𝝔 𝟑 With 𝒊 L = 𝟑𝒊 𝐝𝐩𝐭 𝟐 𝟑 𝝔 𝟐 − 𝝔 𝟑 𝑱 𝟐𝟑 = 𝒊 L𝟑 = 𝟓𝒊 𝟑 𝐝𝐩𝐭 𝟑 𝟐 Resulting wave has the intensity: 𝟑 𝝔 𝟐 − 𝝔 𝟑 Use math textbook: cos : 𝐵 = 9 : + 9 𝑱 𝟐𝟑 = 𝟑𝒊 𝟑 + 𝟑𝒊 𝟑 𝐝𝐩𝐭 𝝔 𝟐 − 𝝔 𝟑 : cos 2𝐵 , so: Interference!
Interference of Waves 18 cos Df = 1 ℎ 9 cos Df = -1 ℎ : Interfering waves: 𝐽 9: = 𝑆 9 + 𝑆 : : = ℎ 9 : + ℎ : : + 2ℎ 9 ℎ : cos Δ𝜚 Regions of constructive interference: 𝐽 9: = 2× 𝐽 9 + 𝐽 : Regions of destructive interference: 𝐽 9: = 0
Case 2: Experiment with Waves 19 When both slits are open there are two contributions to the wave the oscillation at the detector: 𝑆 𝑢 = 𝑆 9 𝑢 + 𝑆 : 𝑢 First combine: 𝑆 𝑢 = 𝑆 9 𝑢 + 𝑆 : (𝑢) Afterwards look at the amplitude and intensity of the resulting wave!
Case 2: Experiment with Waves 20 When both slits are open there are two contributions to the wave the oscillation at the detector: 𝑆 𝑢 = 𝑆 9 𝑢 + 𝑆 : 𝑢 Contrary to “bullets” we can not just add up Intensities. Interference pattern: 𝐽 9: = 𝑆 9 + 𝑆 : : = ℎ 9 : + ℎ : : + 2ℎ 9 ℎ : cos Δ𝜚 Regions where waves are amplified and regions where waves are cancelled .
Double Slit Experiment with Light (Young) 21
The Double Slit Experiment 22 Case 3: An Experiment with Electrons
Case 3: Experiment with Electrons 23 From the detector counts deduce again the probabilities P 1 and P 2 To avoid confusion use single electrons: one by one! Observation: Electrons come in “lumps”, like bullets | y 1 | 2 | y 2 | 2 What do we expect when both slits are open?
Case 3: Experiment with Electrons 24 | y 1 | 2 | y 2 | 2
Case 3: Experiment with Electrons 24 | y 1 | 2 | y 2 | 2
Case 3: Experiment with Electrons 24 | y 1 | 2 | y 2 | 2
Case 3: Experiment with Electrons 24 | y 1 | 2 | y 2 | 2
Case 3: Experiment with Electrons 24 | y 1 | 2 | y 2 | 2
Case 3: Experiment with Electrons 29 | y 1 | 2 | y 2 | 2
Case 3: Experiment with Electrons 30 An Interference pattern! The electron wave function behaves exactly like classical waves. De Broglie waves Just like “waves” we can not just add up Intensities. | y 1 | 2 | y 1 + y 2 | 2 | y 2 | 2 The probability is the square of the sum: Add the wave amplitudes: P 12 = | y 12 | 2 = | y 1 + y 2 | 2 = | y 1 | 2 + | y 2 | 2 + 2 y 1 y 2 y 12 = y 1 + y 2 *
Case 3: Experiment with Electrons 31 Perhaps the electrons interfere with each other. Reduce the intensity, shoot electrons one by one: same result. | y 1 | 2 | y 1 + y 2 | 2 | y 2 | 2 P.S.: Classically, light behaves light waves. However, if you shoot light, photon per photon, it “comes in lumps”, just like electrons. Quantum Mechanics: for photons it is the same story as for electrons.
Case 3: Experiment with Electrons 31 Perhaps the electrons interfere with each other. Reduce the intensity, shoot electrons one by one: same result. Although the electron is detected as a “lump” on the screen, apparently it has gone through both slits! | y 1 | 2 | y 1 + y 2 | 2 | y 2 | 2 P.S.: Classically, light behaves light waves. However, if you shoot light, photon per photon, it “comes in lumps”, just like electrons. Quantum Mechanics: for photons it is the same story as for electrons.
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