The Relativistic Quantum World A lecture series on Relativity Theory - - PowerPoint PPT Presentation

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A lecture series on Relativity Theory and Quantum Mechanics

The Relativistic Quantum World

University of Maastricht, Sept 16 – Oct 14, 2020

Marcel Merk

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The Relativistic Quantum World

Relativity Quantum Mechanics Standard Model

Lecture notes, written for this course, are available: www.nikhef.nl/~i93/Teaching/ Prerequisite for the course: High school level physics & mathematics.

Lecture 1: The Principle of Relativity and the Speed of Light Lecture 2: Time Dilation and Lorentz Contraction Lecture 3: The Lorentz Transformation and Paradoxes Lecture 4: General Relativity and Gravitational Waves Lecture 5: The Early Quantum Theory Lecture 6: Feynman’s Double Slit Experiment Lecture 7: Wheeler’s Delayed Choice and Schrodinger’s Cat Lecture 8: Quantum Reality and the EPR Paradox Lecture 9: The Standard Model and Antimatter Lecture 10: The Large Hadron Collider

• Sept. 16:
• Sept. 23:
• Sept. 30:
• Oct. 7:
• Oct. 14:

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Relativity Theory

Special Relativity

All observers moving in inertial frames:

• Have identical laws of physics,
• Observe the same speed of light: c.

Consequences:

• Simultaneity is not the same for everyone,
• Distances shrink, time slows down at

high speed,

• Velocities do not add-up as expected.

General Relativity

• A free falling person is also inertial frame,
• Acceleration and gravitation are equivalent:

Inertial mass = gravitational mass

Consequences:

Space-time is curved:

• Light bends around a massive object,
• Time slows down and space shrinks

in gravitational fields,

• Gravitational radiation exists.

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Relativity and Quantum Mechanics

Classical Mechanics Quantum Mechanics

Smaller Sizes (ħ) Higher Speed (c)

Relativity Theory Quantum Field Theory

Newton Bohr Einstein Feynman Classical mechanics is not “wrong”. It is limited to macroscopic objects and moderate velocities. Dirac 3

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Lecture 5 The Early Quantum Theory

“If Quantum Mechanics hasn’t profoundly shocked you, you haven’t understood it yet.”

• Niels Bohr

“Gott würfelt nicht (God does not play dice).”

• Albert Einstein

”Einstein, stop telling God what to do!”

• Niels Bohr

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Key Persons of Quantum Mechanics

Niels Bohr Erwin Schrodinger Werner Heisenberg Paul Dirac

Niels Bohr: Nestor of the ”Copenhagen Interpretation” Erwin Schrödinger: Inventor of the quantum mechanical wave equation Werner Heisenberg: Inventor of the uncertainty relation and “matrix mechanics” Paul Dirac: Inventor of relativistic wave equation: Antimatter! Max Born: Inventor of the probability interpretation of the wave function We will focus of the Copenhagen Interpretation and work with the concept of Schrödinger’s wave-function: 𝜔

5 Max Born

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Deterministic Universe

Mechanics Laws of Newton:

• 1. The law of inertia: a body in rest moves with a constant

speed

• 2. The law of force and acceleration: F= m a
• 3. The law: Action = - Reaction
• Classical Mechanics leads to a deterministic universe.
• From exact initial conditions future can be predicted.
• Quantum mechanics introduces a fundamental element
• f chance in the laws of nature: Planck’s constant: ℎ.
• Quantum mechanics only makes statistical predictions.

Isaac Newton (1642 – 1727) “Principia” (1687)

6

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The Nature of Light

Isaac Newton (1642 – 1727): Light is a stream of particles. Christiaan Huygens (1629 – 1695): Light consists of waves. Thomas Young (1773 – 1829): Interference observed: Light is waves!

Isaac Newton Christiaan Huygens Thomas Young

7 Newton Huygens

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Waves & Interference : water, sound, light

Sound: Active noise cancellation:

Light: Thomas Young experiment:

Water: Interference pattern: Principle of a wave

light + light can give darkness!

𝜇 = 𝑤 𝑔 ⁄ 𝑔 = 1 𝑈 ⁄

8

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Interference with Water Waves

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Interfering Waves

Double slit experiment:

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Particle nature: Quantized Light

“UV catastrophe” in Black Body radiation spectrum:

If you heat a body it emits radiation. Classical thermodynamics predicts the amount of light at very short wavelength to be infinite! Planck invented an ad-hoc solution: For some reason material emitted light in “packages”.

Max Planck (1858 – 1947)

Classical theory:

There are more short wavelength “oscillation modes”

• f atoms than large wavelength “oscillation modes”.

Nobel prize 1918

Quantum theory:

Light of high frequency (small wavelength) requires more energy: E = h f (h = Planck’s constant)

h = 6.62 ×10-34 Js

11

Paul Ehrenfest

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Photoelectric Effect

Photoelectric effect:

Light kicks out electron with E = h f (Independent on light intensity!) Light consists of quanta. (Nobelprize 1921)

Compton Scattering:

“Playing billiards” with light quanta and electrons. Light behaves as a particle with: λ = h / p (Nobelprize 1927)

light electrons light electron Albert Einstein Arthur Compton

𝜇* − 𝜇 = ℎ 𝑛-𝑑 1 − cos 𝜄

Wave: 𝐹 = ℎ𝑔 = ℎ𝑑 𝜇 ⁄ à 𝜇 = ℎ𝑑 𝐹 ⁄ Momentum: 𝑞 = 𝑛𝑤 = 𝑛𝑑 = 𝐹 𝑑 ⁄ à 𝐹 = 𝑞𝑑 It follows that: 𝜇 = ℎ 𝑞 ⁄ 12

SLIDE 14

Photoelectric Effect

Photoelectric effect:

Light kicks out electron with E = h f (Independent on light intensity!) Light consists of quanta. (Nobelprize 1921)

Compton Scattering:

“Playing billiards” with light quanta and electrons. Light behaves as a particle with: λ = h / p (Nobelprize 1927)

light light electron Albert Einstein Arthur Compton

𝜇* − 𝜇 = ℎ 𝑛-𝑑 1 − cos 𝜄

Wave: 𝐹 = ℎ𝑔 = ℎ𝑑 𝜇 ⁄ à 𝜇 = ℎ𝑑 𝐹 ⁄ Momentum: 𝑞 = 𝑛𝑤 = 𝑛𝑑 = 𝐹 𝑑 ⁄ à 𝐹 = 𝑞𝑑 It follows that: 𝜇 = ℎ 𝑞 ⁄ 13 electrons

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Matter Waves

Louis de Broglie - PhD Thesis(!) 1924 (Nobel prize 1929):

If light are particles incorporated in a wave, it suggests that particles (electrons) “are carried” by waves.

Louis de Broglie

Particle wavelength: 𝜇 = ℎ 𝑞 ⁄ à 𝜇 = ℎ 𝑛𝑤 ⁄

Original idea: a physical wave è Quantum mechanics: probability wave!

Wavelength visible light: 400 – 700 nm Use h= 6.62 × 10-34 Js to calculate:

• Wavelength electron with v = 0.1 c:

0.024 nm

• Wavelength of a fly (m = 0.01 gram,

v = 10 m/s): 0.0000000000000000000062 nm graphene

15

SLIDE 17

Matter Waves

Louis de Broglie - PhD Thesis(!) 1924 (Nobel prize 1929):

If light are particles incorporated in a wave, it suggests that particles (electrons) “are carried” by waves.

Louis de Broglie

Particle wavelength: 𝜇 = ℎ 𝑞 ⁄ à 𝜇 = ℎ 𝑛𝑤 ⁄

Original idea: a physical wave è Quantum mechanics: probability wave!

Wavelength visible light: 400 – 700 nm Use h= 6.62 × 10-34 Js to calculate:

• Wavelength electron with v = 0.1 c:

0.024 nm

• Wavelength of a fly (m = 0.01 gram,

v = 10 m/s): 0.0000000000000000000062 nm graphene

ELECTRON

16

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17

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The Quantum Atom of Niels Bohr

The classical Atom is unstable!

Expect: t < 10-10 s Niels Bohr: Atom is only stable for specific

• rbits: “energy levels”.

Niels Bohr 1885 - 1962

An electron can jump from a high to lower level by emitting a light quantum with corresponding energy difference.

Balmer spectrum of wavelengths: 18

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Schrödinger: Bohr atom and de Broglie waves

L = r p L = r h/ λ L = r n h/ (2 π r) L = n h/(2π) = n ħ de Broglie: λ = h / p

n = 1

Erwin Schrödinger

Energy levels explained à atom explained Outer shell electrons à “chemistry explained” (L = angular momentum) If orbit length “fits”: 2π r = n λ with n = 1, 2, 3, … The wave positively interferes with itself! è Stable orbits!

Periodic Table of the Elements

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Copenhagen Interpretation (Niels Bohr)

One can observe wave characteristics or particle characteristics of quantum

• bjects, never both at the same time.

Particle and Wave aspects of a physical object are complementary. Similarly one can never determine from a quantum object at the same time: energy and time, position and momentum (and more).

Subatomic matter is not just waves and it is not just particles. It is nothing we know from macroscopic world.

Complementarity

Niels Bohr 1885 - 1962 21

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Heisenberg Uncertainty Relation

Werner Heisenberg “Matrix mechanics” Erwin Schrödinger “Wave Mechanics” Paul Adrian Maurice Dirac “q - numbers”

A measurement of a characteristic of quantum matter affects the object. Heisenberg’s “non-commuting” observables: It is a fundamental aspect of nature. Not related to limited technology! Position and momentum: x p – p x = i ħ Δx Δp ≥ ħ / 2 Energy and time: E t – t E = i ħ ΔE Δt ≥ ħ / 2

22

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Heisenberg Uncertainty Relation

Werner Heisenberg “Matrix mechanics” Erwin Schrödinger “Wave Mechanics” Paul Adrian Maurice Dirac “q - numbers”

A measurement of a characteristic of quantum matter affects the object. Heisenberg’s “non-commuting” observables: Position and momentum: x p – p x = i ħ Δx Δp ≥ ħ / 2 Energy and time: E t – t E = i ħ ΔE Δt ≥ ħ / 2 It is a fundamental aspect of nature. Not related to limited technology!

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Waves and Uncertainty

• 25
• 20 -15 -10
• 5

5 10 15 20 25

• 1
• 0.5

0.5 1

Waves

• 25
• 20 -15 -10
• 5

5 10 15 20 25

• 1
• 0.5

0.5 1

Waves

Use the “wave-mechanics” picture of Schrödinger A wave has an exactly defined frequency. A particle has an exactly defined position. Two waves: p1 = hf1/c , p2 = hf2/c Wave Packet: sum of black and blue wave

The more waves are added, the more the wave packet looks like a particle, or, If we try to determine the position x, we destroy the momentum p and vice versa. x and p are non-commuting observables also E and t are non-commuting variables

24

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Dp

Waves and Particles

Measure precise frequency à no position information. Measure precise position à no momentum (frequency) information “Particle:”

not localized well localized

Pure waves of different frequency, i.e. different momentum p = hf/c Wave package, i.e. “particle” 25

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The uncertainty relation at work

Shine a beam of light through a narrow slit which has a opening size Δx. The light comes out over an undefined angle that corresponds to Δpx. Δpx Δx

Δx Δpx ~ ħ/2

26 Image of a laser pointer after passing through a slit:

SLIDE 28

The wave function y

Momentum badly known Position fairly known

27

SLIDE 29

The wave function y

Momentum badly known Position fairly known Position badly known Momentum fairly known

27

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Imaginary Numbers

28

SLIDE 31

The Copenhagen Interpretation

Niels Bohr Max Born

Prob(x,t) = |y(x,t)|2 = y y*

The wave function y is not a real object. The only physical meaning is that its square gives the probability to find a particle at a position x and time t. The mathematics for the probability of the quantum wave-function is the same as the mathematics of the intensity of a classical wave function. Quantum mechanics allows only to calculate probabilities for possible outcomes of an experiment and is non-deterministic, contrary to classical theory. Einstein: “Gott würfelt nicht.”

29

SLIDE 32

The wave function y

Momentum badly known Position fairly known Position badly known Momentum fairly known

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SLIDE 33

Next Lecture: double slit experiment

Richard Feynman

The core of quantum mechanics illustrated by Feynman. Einstein and Schrödinger did not like it. Wheeler later took it to the extreme. Even today people are debating its interpretation.

31