Download Week 10

Waves, Light & Quanta
Tim Freegarde
Web Gallery of Art; National Gallery, London
1
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
2
Quantum measurement
• allowed energies
energy
THE HYDROGEN ATOM
me4
1
E
2
2
240   n
n=
0
n=3
hcR
4
n=2
 hcR
n=1

QUANTUM MEASUREMENT
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
3
The experiment with the two holes
y
p0 

h


0

a
0
s
x

• fringe maxima when
a
• equivalent to change in illumination angle and hence
 by   p y p0
a sin   m0
 fringe spacing
  0 a
• smallest visible feature size 
 illumination wavelength   a
 illumination momentum p y  h a
a
4
Single slit diffraction

amplitude
y

a
x
intensity

s
5
Uncertainty
HEISENBERG’S UNCERTAINTY PRINCIPLE
• certain pairs of parameters may not simultaneously be exactly determined
• {position, momentum}
• {position, wavelength}
• {time, energy}
• {time, frequency}
• {orientation, angular momentum}
• {linear, circular} polarization
• {intensity, phase}
• {x, y}, {x, z}, {y, z} components of angular momentum
• conjugate parameters cannot be simultaneously definite
6
Uncertainty
BEATING OF TWO DIFFERENT FREQUENCIES
cos 1t  cos 2t  cos
1  2
  2
t cos 1
t
2
2
7
Bandwidth theorem
2
  0.10 Hz
1.5
1
1
0.9
0.8
0.5
0
-40
-30
-20
-10
0
10
20
30
time
40
amplitude
0.7
-0.5
0.6
0.5
0.4
0.3
0.2
-1
0.1
1.
4
1.
3
1.
2
1.
1
1
0.
9
0.
8
0.
7
0.
5
0.
6
0
-1.5
frequency
-2
6
1
4
0.9
0.8
2
0
-10
0
-2
10
20
30
time
40
0.5
0.4
0.3
0.2
0.1
1.
4
1.
3
1.
2
1.
1
1
0.
9
0.
8
0
-4
0.
7
-20
0.
6
-30
0.6
0.
5
-40
amplitude
0.7
frequency
-6
8
Bandwidth theorem
  0.20 Hz
2
1.5
1
1
0.9
0.8
0.5
0
-40
-30
-20
-10
0
10
20
30
time
40
amplitude
0.7
-0.5
0.6
0.5
0.4
0.3
0.2
-1
0.1
1.
8
1.
6
1.
6
1
1
1.
4
0.
8
0.
8
1.
4
0.
6
0.
6
1.
2
0.
4
0.
4
1.
2
0.
2
0
0.
2
0
-1.5
frequency
-2
6
4
1
0.9
0.8
2
0
-20
-10
0
-2
10
20
30
time
40
0.5
0.4
0.3
0.2
0.1
0
-4
1.
8
-30
0.6
0
-40
amplitude
0.7
frequency
-6
9
Bandwidth theorem
  0.05 Hz
2
1.5
1
1
0.9
0.8
0.5
0
-40
-30
-20
-10
0
10
20
30
time
40
-0.5
amplitude
0.7
0.6
0.5
0.4
0.3
0.2
-1
0.1
1.
2
1.
15
1.
1
1
1.
05
0.
95
0.
9
0.
85
0.
75
0.
8
0
-1.5
frequency
-2
6
4
1
0.9
0.8
2
0
-10
0
-2
10
20
30
time
40
0.5
0.4
0.3
0.2
0.1
1.
2
1.
15
1.
1
1
1.
05
0.
95
0.
9
0
-4
0.
85
-20
0.
8
-30
0.6
0.
75
-40
amplitude
0.7
frequency
-6
10
Terminology
UNCERTAINTY IN MEASUREMENT
• repeated experiment yields range of results
• expectation value = mean
• uncertainty = standard deviation
1
a   an
n n
1
2
2
a    a   an  a
n n


2
• before measurement, system was in a superposition
• probability of given result
a
given by
 a 
2
11
Uncertainty
HEISENBERG’S UNCERTAINTY PRINCIPLE
• certain pairs of parameters may not simultaneously be exactly determined
• {position, momentum}
• {position, wavelength}
• {time, energy}
• {time, frequency}
• {orientation, angular momentum}
• {linear, circular} polarization
• {intensity, phase}
• {x, y}, {x, z}, {y, z} components of angular momentum
QUANTUM MEASUREMENT
• measurement changes observed system so that parameter measured
is subsequently definite
• conjugate parameters cannot be simultaneously 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
12
The LASER
LIGHT AMPLIFICATION
by Stimulated Emission of Radiation
• Theodore Maiman, 16 May 1960
mirror
beam splitter
693.4 nm
ruby
flash tube
light amplifier
optical resonator
13
Absorption and emission of photons
ABSORPTION
energy
SPONTANEOUS
EMISSION
n=
0
2
1
hcR

4
STIMULATED
EMISSION
2
1
1
dN 2
  A21 N 2
dt
ABSORPTION
dN 1
  B12 N1 
dt
n=3
absorption
n=2
emission
1
 hcR
n=1
14
Absorption and emission of photons
ABSORPTION
SPONTANEOUS
EMISSION
2
1
dN 2
  A21 N 2
dt
ABSORPTION
dN 1
  B12 N1 
dt
dN 2
 B  N1N 2   AN 2
dt
N1  N 2  N
• Einstein A and B coefficients
STIMULATED
EMISSION
2
1
1
EINSTEIN EQUATIONS
1
• thermal equilibrium
 blackbody spectrum
• spontaneous emission
stimulated by vacuum field
• amplification of light if
atomic population is
inverted i.e. N 2  N1
15
mirror
beam splitter
energy
The ruby LASER
693.4 nm
ruby
flash tube
light amplifier
metastable
optical resonator
• Cr3+ ions in sapphire (Al2O3) absorb blue
and green from flash light
• internal transitions to metastable state
• spontaneous emission is amplified by
passage through ruby
absorption
emission
Cr3+
• repeatedly reflected/amplified near-axial
light builds up to form coherent laser beam
16
Laser beam characteristics
mirror
beam splitter
693.4 nm
ruby
flash tube
• as initial source recedes down unfolded cavity,
emission approaches that from distant point source
• divergence determined by diffraction by limiting aperture
• focusable
• constructive interference between reflections for certain wavelengths
• long pulse  continuous wave (c.w.)
• narrow linewidth for long pulses ( 
t  1 )
• monochromatic
• noise from spontaneous emission gives lower limit to linewidth
• nonlinear processes have various effects in detail
• Hecht section 13.1
17
The ruby LASER
mirror
beam splitter
693.4 nm
ruby
flash tube
light amplifier
optical resonator
• ray optics
• colour
• diffraction
• interference
• quantum physics
• refraction, polarization, …
18