1

Waves, Light & QuantaTim Freegarde

Web Gallery of Art; National Gallery, London

22

Quantum mechanics

1. particles behave like waves, and vice-versa

2. energies and momenta can be quantized, ie measurements yield particular results

3. all information about a particle is contained within a complex wavefunction, which determines the probabilities of experimental outcomes

4. deterministic evolution of the wavefunction is determined by a differential (e.g. Schrödinger) wave equation

5. 80 years of experiments have found no inconsistency with quantum theory

6. explanation of the ‘quantum measurement problem’ – the collapse of the wavefunction upon measurement – remains an unsolved problem• non-deterministic

process• Heisenberg’s uncertainty principle

33

Quantum measurement

• allowed energies 22

0

4 1

42 n

meE

en

erg y

0

hcR

4

hcR

n = 1

n = 3

n =

n = 2

1. measured energy must be one of allowed values

2. …but until measurement, any energy possible

3. after measurement, subsequent measurements will give same value

THE HYDROGEN ATOM

QUANTUM MEASUREMENT

44

The experiment with the two holes

x

0

s

a

y

• fringe maxima when

0sin ma a0 fringe spacing

• smallest visible feature size

a illumination wavelength

ahpy illumination momentum• equivalent to change in illumination angle and

hence by 0ppy

00

hp

a a

55

Single slit diffraction

x

y

s

a

amplitude

intensity

66

Uncertainty

HEISENBERG’S UNCERTAINTY PRINCIPLE

• certain pairs of parameters may not simultaneously be exactly determined• {position, momentum}

• {time, energy}

• {orientation, angular momentum}• {intensity, phase}

• {x, y}, {x, z}, {y, z} components of angular momentum

• {position, wavelength}

• {time, frequency}

• {linear, circular} polarization

• conjugate parameters cannot be simultaneously definite

77

Uncertainty

BEATING OF TWO DIFFERENT FREQUENCIES

tttt2

cos2

coscoscos 212121

88

Bandwidth theorem

Hz10.0

99

Bandwidth theorem

Hz20.0

1010

Bandwidth theorem

Hz05.0

1111

Terminology

UNCERTAINTY IN MEASUREMENT

• repeated experiment yields range of results

• expectation value = mean n

nan

a1

n

na aan

a222 1• uncertainty = standard

deviation

• before measurement, system was in a superposition

• probability of given result given bya 2a

1212

Uncertainty

• conjugate parameters cannot be simultaneously definite

QUANTUM MEASUREMENT

• measurement changes observed system so that parameter measured is subsequently definite

• process measure A, measure B not the same as measure B, measure A• measure A, measure B are not commutative / do not commute

• commutator [measure A, measure B] 0

HEISENBERG’S UNCERTAINTY PRINCIPLE

• certain pairs of parameters may not simultaneously be exactly determined• {position, momentum}

• {time, energy}

• {orientation, angular momentum}• {intensity, phase}

• {x, y}, {x, z}, {y, z} components of angular momentum

• {position, wavelength}

• {time, frequency}

• {linear, circular} polarization

1313

The LASER

by Stimulated Emission of Radiation

LIGHT AMPLIFICATION

• Theodore Maiman, 16 May 1960

beam splitter

flash tube

mirror

ruby

693.4 nm

light amplifier

optical resonator

1414

Absorption and emission of photons

ABSORPTION

en

erg y

0

hcR

4

hcR

n = 1

n = 3

n =

n = 2

absorption emission

EMISSIONSPONTANEOUS

ABSORPTIONEMISSION

STIMULATED

1121

d

dNB

t

N

1

2

1

2

1

1

2212

d

dNA

t

N

1515

Absorption and emission of photons

ABSORPTIONEMISSION

SPONTANEOUS

ABSORPTIONEMISSION

STIMULATED

1121

d

dNB

t

N

1

2

1

2

1

1

2212

d

dNA

t

N

2212

d

dANNNB

t

N

EINSTEIN EQUATIONS

NNN 21

• spontaneous emission stimulated by vacuum field

• amplification of light if atomic population is inverted i.e. 12 NN

• thermal equilibrium blackbody

spectrum

• Einstein A and B coefficients

1616

The ruby LASER

beam splitter

flash tube

mirror

ruby

693.4 nm

light amplifier

optical resonator

en

erg y

absorption emission

Cr3+

• Cr3+ ions in sapphire (Al2O3) absorb blue and green from flash light

• internal transitions to metastable state

metastable

• spontaneous emission is amplified by passage through ruby

• repeatedly reflected/amplified near-axial light builds up to form coherent laser beam

1717

Laser beam characteristics

beam splitter

flash tube

mirror

ruby

693.4 nm

• as initial source recedes down unfolded cavity, emission approaches that from distant point source

• long pulse continuous wave (c.w.)

• narrow linewidth for long pulses ( )

1 t• noise from spontaneous emission gives lower limit to

linewidth• nonlinear processes have various effects in detail

• divergence determined by diffraction by limiting aperture

• Hecht section 13.1

• focusable

• monochromatic

• constructive interference between reflections for certain wavelengths

1818

The ruby LASER

beam splitter

flash tube

mirror

ruby

693.4 nm

light amplifier

optical resonator

• ray optics

• colour

• diffraction

• interference

• quantum physics

• refraction, polarization, …