The Relativistic Quantum World

A lecture series on Relativity Theory and Quantum Mechanics

The Relativistic Quantum World

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

Key Persons of Quantum Mechanics

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

Waves & Interference : water, sound, light

Sound:

Active noise cancellation:

Light: Thomas Young experiment:

Interfering Waves

Double slit experiment:

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”

of 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

The Quantum Atom of Niels Bohr

The classical Atom is unstable!

Expect:

t

< 10

-10

_s

Niels Bohr: Atom is only stable for specific

orbits: “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:

Not yet explained

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Heisenberg Uncertainty Relation

Werner Heisenberg

“Matrix mechanics”

Erwin Schrödinger

“Wave Mechanics”

Paul Adrian Maurice Dirac

“q - numbers”

Heisenberg Uncertainty Relation

Werner Heisenberg

“Matrix mechanics”

Erwin Schrödinger

“Wave Mechanics”

Paul Adrian Maurice Dirac

“q - numbers”

Waves and Uncertainty

-25 -20 -15 -10

-5

0

5

10

15

20

25

-1

-0.5

0

0.5

1

Waves

-25 -20 -15 -10

-5

0

5

10

15

20

25

-1

-0.5

0

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:

p

₁

= hf

₁

/c ,

p

₂

= hf

₂

/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

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”

The wave function y

Momentum badly known

Position fairly known

The wave function y

Momentum badly known

Position fairly known

Position badly known

Momentum fairly known

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.”

The wave function y

Momentum badly known

Position fairly known

Position badly known

Momentum fairly known