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A lecture series on Relativity T heory and Quantum M echanics

The Relativistic Quantum World. A lecture series on Relativity T heory and Quantum M echanics. Marcel Merk. University of Maastricht, Sept 24 – Oct 15, 2014. The Relativistic Quantum World. Sept 24: Lecture 1: The Principle of Relativity and the Speed of Light

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A lecture series on Relativity T heory and Quantum M echanics

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  1. The Relativistic Quantum World A lecture series on Relativity Theory and Quantum Mechanics Marcel Merk University of Maastricht, Sept 24 – Oct 15, 2014

  2. The Relativistic Quantum World Sept 24: Lecture 1: The Principle of Relativity and the Speed of Light Lecture 2: Time Dilation and Lorentz Contraction Oct 1: Lecture 3: The Lorentz Transformation Relativity Lecture 4: The Early Quantum Theory Oct 8: Lecture 5: The Double Slit Experiment Lecture 6: Quantum Reality Quantum Mechanics Oct 15: Lecture 7: The Standard Model Lecture 8: The Large Hadron Collider Standard Model Lecture notes, written for this course, are available: www.nikhef.nl/~i93/Teaching/ Literature used: see lecture notes. Prerequisite for the course: High school level mathematics.

  3. Relativity and Quantum Mechanics ? ? lightspeed Special Relativity- theory Quantum- Field theory c Quantum- mechanics Classical- mechanics Speed Size Smallest ; elementary particles Human size Planck constant ħ

  4. Relativity Classical Transformation: Velocities: • Equivalence of inertial frames • Light-speed c is constant! Relativistic Transformation: fraction of light-speed Relativistic factor Velocities:

  5. From another perspective particle particle How does a photon see the universe? For a photon time does not exist!

  6. Quantum Mechanics Light is a stream of particles Yes, because it interferes Similar to sound light consists of waves Light is emitted in quanta Thomas Young Isaac Newton Christiaan Huygens Particles have a wave nature: l= h/p Max Planck Particles are probability waves The nature of light is quanta Yes, because photons collide! Louis de Broglie Albert Einstein Niels Bohr Arthur Compton

  7. Uncertainty relation You can not precisely determine position and momentum at the same time: Erwin Schrödinger Werner Heisenberg p = h/λ = hf/c A particle does not have well defined position and momentum at the same time.

  8. The wave function y Position fairly known Momentum badly known

  9. The wave function y Position fairly known Momentum badly known Position badly known Momentum fairly known

  10. Lecture 5 The 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

  11. Richard Feynman (1918 – 1988) . Nobelprize 1965: Quantum Electrodynamics (Path Integral formulation of quantum mechanics) • Mostly known from: • Feynman diagrams • Challenger investigation • Popular books Challenger disaster Feynman diagram

  12. The Double Slit Experiment Case 1: An Experiment with Bullets

  13. Case 1: Experiment with Bullets 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”. P1 is the probability curve when only slit 1 is open P2 is the probability curve when only slit 2 is open What is the probability curve when both slit 1 and slit 2 are open?

  14. Case 1: Experiment with Bullets 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 P1 is the probability curve when only slit 1 is open P2 is the probability curve when only slit 2 is open We can just add up the probabilities. When both slits are open: P12 = P1 + P2

  15. The Double Slit Experiment Case 2: An Experiment with Waves

  16. Waves & Interference : water, sound, light Water: Interference pattern: Waves: Interference principle: Light: Thomas Young experiment: Sound: Active noise cancellation: light + light can give darkness!

  17. Case 2: Experiment with Waves We replace the gun by a wave generator. Let’s 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”. I1 = |h1|2 I2 = |h2|2 I12 = ?? The intensity of a wave is the square of the amplitude…

  18. Intermezzo: Wave Oscillation & Intensity Energy in the oscillation (up-down) movement of the molecules: Ekin = ½ m v2and v is proportional to the amplitude or height: v ≈ h So that the intensity of the wave is: I ≈ h2 h h v Formula for the resulting oscillation of a water molecule somewhere in the wave: R(t) = hcos (2pft + f) f = frequency f= phase and the Intensity: I = h2

  19. Case 2: Experiment with Waves When both slits are open there are two contributions to the wave the oscillation at the detector: R(t) = R1(t) + R2(t) R1 (t) = h1cos (2pf t + f1) R2 (t) = h2cos (2pf t + f2) f1 and f2 depend on distance to 1 and 2 I1 = |h1|2 I2 = |h2|2 I12 = ?? First combine: R(t) = R1(t) + R2(t) Afterwards look at the amplitude and intensity of the resulting wave!

  20. Mathematics for the die-hards Interference!

  21. Interference of Waves cosDf= 1 h1 cosDf= -1 h2 Interfering waves: I12 = |R1 + R2|2 = h12 + h22 + 2h1h2 cos(Df) Regions of constructive interference: I12 = 2 × ( I1 + I2 ) Regions of destructive interference: I12 = 0

  22. Case 2: Experiment with Waves When both slits are open there are two contributions to the wave the oscillation at the detector: R(t) = R1(t) + R2(t) First combine: R(t) = R1(t) + R2(t) Afterwards look at the amplitude and intensity of the resulting wave!

  23. Case 2: Experiment with Waves When both slits are open there are two contributions to the wave the oscillation at the detector: R(t) = R1(t) + R2(t) Contrary to “bullets” we can not just add up Intensities. Interference pattern:I12 = |R1 + R2|2 = h12 + h22 + 2h1h2 cos(Df) Regions where waves are amplified and regions where waves are cancelled.

  24. Double Slit Experiment with Light (Young)

  25. The Double Slit Experiment Case 3: An Experiment with Electrons

  26. Case 3: Experiment with Electrons From the detector counts deduce again the probabilities P1 and P2 To avoid confusion use single electrons: one by one! Observation: Electrons come in “lumps”, like bullets |y1|2 |y2|2 What do we expect when both slits are open?

  27. Case 3: Experiment with Electrons |y1|2 |y2|2

  28. Case 3: Experiment with Electrons |y1|2 |y2|2

  29. Case 3: Experiment with Electrons |y1|2 |y2|2

  30. Case 3: Experiment with Electrons |y1|2 |y2|2

  31. Case3: Experiment with Electrons |y1|2 |y2|2

  32. Case 3: Experiment with Electrons 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. |y1|2 |y1 + y2|2 |y2|2 Add the wave amplitudes: y12 = y1 + y2 The probability is the square of the sum: P12 = |y12|2 = |y1 + y2|2 = |y1|2 + |y2|2 + 2y1y2*

  33. Case 3: Experiment with Electrons Perhaps the electrons interfere with each other. Reduce the intensity, shoot electrons one by one: same result. |y1|2 |y1 + y2|2 |y2|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.

  34. The Double Slit Experiment Case 4: A Different Experiment with Electrons

  35. Case 4: Watch the Electrons Let us try to out-smart the electron: just watch through which slit it goes! D1 D2 D1 and D2 are two “microscopes” looking at the slits 1 and 2, respectively.

  36. Case 4: Watch the Electrons When we watch through which slit the electrons go, we kill the interference! Now the electron behaves just like a classical particle (“bullet”). D1 D2 If you watch half the time; you only get the interference for the cases you did not watch. It requires an observation to let the quantum wave function “collapse” into reality. As long as no measurement is made the wave function keeps “all options open”.

  37. Wave Particle Duality Next lecture we will try to out-smart nature one step further… … and face the consequences.

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