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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 and the Sept. 16: Lecture 2: Dilation and Lorentz Contraction Lecture 3: The and Paradoxes

Relativity Sept. 23: Lecture 4: and Gravitational Waves

Lecture 5: The Early Quantum Theory Sept. 30: Lecture 6: Feynman’s Double Slit Experiment

Oct. 7: Lecture 7: Wheeler’s Delayed Choice and Schrodinger’s Cat Quantum Mechanics Lecture 8: Quantum Reality and the EPR Paradox

Lecture 9: The Standard Model and Antimatter Oct. 14: Lecture 10: The Large Hadron Collider Model Standard

Lecture notes, written for this course, are available: www.nikhef.nl/~i93/Teaching/ Prerequisite for the course: High school level & mathematics. Relativity Theory 2 General Relativity All observers moving in inertial frames: • A free falling person is also inertial frame, • Have identical laws of physics, • Acceleration and gravitation are equivalent: • Observe the same speed of light: c. Inertial mass = gravitational mass

Consequences: Consequences: • Simultaneity is not the same for everyone, -time is curved: • Distances shrink, time slows down at • Light bends around a massive object, high speed, • Time slows down and space shrinks • do not add-up as expected. in gravitational fields, • Gravitational radiation exists. Relativity and Quantum Mechanics 3

Classical Mechanics Quantum Mechanics Smaller Sizes (ħ) Higher Speed

Newton Bohr Classical mechanics is not “wrong”. It is limited to macroscopic objects and moderate velocities. ( c )

Dirac Feynman Einstein Relativity Theory Quantum Field Theory 4

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).” - ”Einstein, stop telling God what to do!” - Niels Bohr Key Persons of Quantum Mechanics 5

Niels Bohr Erwin Schrodinger Werner Heisenberg

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: � Deterministic Universe 6 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

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

• Classical Mechanics leads to a deterministic universe. - From exact initial conditions can be predicted. • Quantum mechanics introduces a fundamental element of chance in the laws of nature: Planck’s constant: ℎ. - Quantum mechanics only makes statistical predictions. The Nature of Light 7

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

Newton

Huygens Thomas Young Waves & Interference : water, sound, light 8 Principle of a wave Water: Interference pattern: � = �⁄� � = 1⁄�

Sound: Active noise cancellation: Light: Thomas Young experiment:

light + light can give darkness! Interference with Water Waves 9 Interfering Waves 10

Double slit experiment: 11 Particle nature: Quantized Light (1858 – 1947) “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”.

h = 6.62 ×10-34 Js Nobel prize 1918

Classical theory: There are more short wavelength “oscillation modes” of atoms than large wavelength “oscillation modes”.

Quantum theory: Light of high frequency (small wavelength) requires more energy: E = h f (h = Planck’s constant) Photoelectric Effect 12 Photoelectric effect: light Light kicks out electron with E = h f (Independent on light intensity!) electrons Light consists of quanta. (Nobelprize 1921)

Wave: � = ℎ� = ℎ�⁄� à � = ℎ�⁄� Albert Einstein Momentum: � = �� = �� = �⁄� à � = �� It follows that: � = ℎ⁄� ℎ � − � = 1 − cos � �� Compton Scattering: “Playing billiards” with light quanta and electron electrons. light Light behaves as a particle with: λ = h / p (Nobelprize 1927) Arthur Compton Photoelectric Effect 13 Photoelectric effect: light Light kicks out electron with E = h f (Independent on light intensity!) electrons Light consists of quanta. (Nobelprize 1921)

Wave: � = ℎ� = ℎ�⁄� à � = ℎ�⁄� Albert Einstein Momentum: � = �� = �� = �⁄� à � = �� It follows that: � = ℎ⁄� ℎ � − � = 1 − cos � �� Compton Scattering: “Playing billiards” with light quanta and electron electrons. light Light behaves as a particle with: λ = h / p (Nobelprize 1927) Arthur Compton 14 Matter Waves 15 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. Original idea: a physical wave è Quantum mechanics: probability wave!

Particle wavelength: � = ℎ⁄� à � = ℎ⁄ �� Louis de Broglie

Wavelength visible light: graphene 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 Matter Waves 16 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. Original idea: a physical wave è Quantum mechanics: probability wave!

Particle wavelength: ELECTRON� = ℎ⁄� à � = ℎ⁄ �� Louis de Broglie

Wavelength visible light: graphene 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 17 The Quantum Atom of Niels Bohr 18 The classical Atom is unstable! Expect: t < 10-10 s Niels Bohr: Atom is only stable for specific orbits: “energy levels”.

An electron can jump from a high to Niels Bohr lower level by emitting a light quantum 1885 - 1962 with corresponding energy difference.

Balmer spectrum of wavelengths: Schrödinger: Bohr atom and de Broglie waves 19

If orbit length “fits”: 2π r = n λ with n = 1, 2, 3, … The wave positively interferes with itself! n = 1 è Stable orbits! Erwin Schrödinger

Periodic Table of the Elements de Broglie: λ = h / p L = r p Energy levels explained L = r h/ λ à atom explained L = r n h/ (2 π r) Outer shell electrons L = n h/(2π) = n ħ à “chemistry explained” (L = angular momentum) Not yet explained 20 !"#$%#&'()'*$!+,-#$./$0*(1*"$2"'345 !"#" $%%& .89#&3

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

Niels Bohr 1885 - 1962

Copenhagen Interpretation (Niels Bohr) One can observe wave characteristics or particle characteristics of quantum objects, 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). Heisenberg Uncertainty Relation 22

A measurement of a characteristic of quantum matter affects the object. Heisenberg’s “non-commuting” observables:

Energy and time: E t – t E = i ħ ΔE Δt ≥ ħ / 2

Position and momentum: x p – p x = i ħ Δx Δp ≥ ħ / 2 Werner Heisenberg Erwin Schrödinger Paul Adrian Maurice Dirac “Matrix mechanics” “Wave Mechanics” “q - numbers”

It is a fundamental aspect of nature. Not related to limited technology! Heisenberg Uncertainty Relation 23

A measurement of a characteristic of quantum matter affects the object. Heisenberg’s “non-commuting” observables:

Energy and time: E t – t E = i ħ ΔE Δt ≥ ħ / 2

Position and momentum: x p – p x = i ħ Δx Δp ≥ ħ / 2 Werner Heisenberg Erwin Schrödinger Paul Adrian Maurice Dirac “Matrix mechanics” “Wave Mechanics” “q - numbers”

It is a fundamental aspect of nature. Not related to limited technology! Waves and Uncertainty 24 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 Waves Waves

1 1

0.5 x and p are non-commuting observables0.5 also 0 E and t are non-commuting variables0

-0.5 -0.5

-1 -1 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25

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. Waves and Particles 25

“Particle:”

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

not localized well localized Measure precise frequency à no position information.

Dp Measure precise position à no momentum (frequency) information The uncertainty relation at work 26

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.

Δx Δp ~ ħ/2 x Image of a laser pointer after passing through a slit:

Δpx

Δx The wave function y 27

Position fairly known Momentum badly known The wave function y 27

Position fairly known Momentum badly known

Position badly known Momentum fairly known Imaginary Numbers 28 The Copenhagen Interpretation 29

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.

Prob(x,t) = |y(x,t)|2 = y y* Niels Bohr Max Born

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.”

The mathematics for the probability of the quantum wave-function is the same as the mathematics of the intensity of a classical wave function. The wave function y 30

Position fairly known Momentum badly known

Position badly known Momentum fairly known Next Lecture: double slit experiment 31

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.