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A lecture series on Relativity T heory and Quantum M echanics

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The Relativistic Quantum World

A lecture series on Relativity Theory and Quantum Mechanics

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

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.

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

Relativity

Classical Transformation:

Velocities:

- Equivalence of inertial frames
- Light-speed c is constant!

Relativistic Transformation:

fraction of light-speed

Relativistic factor

Velocities:

From another perspective

particle

particle

How does a photon see the universe?

For a photon time does not exist!

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

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.

The wave function y

Position fairly known

Momentum badly known

Position badly known

Momentum fairly known

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

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

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?

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

Waves & Interference : water, sound, light

Water: Interference pattern:

Waves: Interference principle:

Light: Thomas Young experiment:

Sound: Active noise cancellation:

light + light can give darkness!

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…

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

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!

Mathematics for the die-hards

Interference!

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

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!

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.

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?

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*

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.

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.

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”.

Wave Particle Duality

Next lecture we will try to out-smart nature one step further…

… and face the consequences.

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