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Quantum Ideas 2007
Axel Kuhn
Quantum Ideas
Syllabus:
The success of classical physics, measurements in classical physics. The nature of light,
the ultraviolet catastrophe, the photoelectric effect and the quantisation of radiation. Atomic
spectral lines and the discrete energy levels of electrons in atoms, the Frank-Hertz experiment and the Bohr model of an atom.
Magnetic dipoles in homogeneous and inhomogeneous magnetic fields and the SternGerlach experiment showing the quantisation of the magnetic moment. The Uncertainty
principle by considering a microscope and the momentum of photons, zero point energy,
stability and size of atoms. Measurements in quantum physics, the impossibility of measuring two orthogonal components of magnetic moments. The EPR paradox, entanglement,
hidden variables, non-locality and Aspect's experiment, quantum cryptography and the
BB84 protocol. Schrödinger's cat and the many-world interpretation of quantum mechanics. Interferometry with atoms and large molecules. Amplitudes, phases and wavefunctions. Interference of atomic beams, discussion of two-slit interference, Bragg diffraction of
atoms, quantum eraser experiments. A glimpse of quantum engineering and quantum
computing. Schrödinger's equation and boundary conditions. Solution for a particle in an
infinite potential well, to obtain discrete energy levels and wavefunctions.
Spectral lines
De Brogie
Wavelength
Interference of
massive particles
Uncertainty
Principle
Blackbody
Radiation
Photoelectric
Effect
Failure of
Classical Physics
Quantised
energy levels
Wave-Particle
Duality
Planck’s
Hypothesis
Quantisation
of radiation
Schrödinger’s
Equation
Quantum Computing
Quantum Cryptography
Modern Applications
Molecules, Solid state, etc.
Atom model (Bohr)
Structure of Matter
Quantum
Physics
Many-world interpretation
Break-down of the
Wavefunction
Role of the observer
Probabilities
Superposition
Entanglement
Paradoxa in early
Gedanken-experiments
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
Classical Physics:
• classical mechanics (Newton; F = m a)
• electricity and magnetism (Coulomb, Faraday, Maxwell)
• electromagnetic waves (rf ... light ... x-ray ... gamma)
• thermodynamics (energy conservation, equilibration, statistical mechanics)
• accurate measurement of all observables (position x(t)and momentum p(t) )
Quantum Physics:
• probabilistic - not deterministic (Einstein: „Good does not play dice“)
• probability wave function ψ(x,t) to describe a particle
• superposition and entanglement
• non-local behaviour („Spooky interaction at a distance“ that bothered Einstein)
• uncertainty principle: Δx Δp ≥ ħ/2 and ΔE Δt ≥ ħ/2
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
illumination with a mercury lamp, filtering a single spectral line.
Cathode metal with a binding energy (or work function) of Ebind = 2.02 eV
yellow, 578nm, 5.19E+14 Hz, Ekin = 0.13 eV
green, 546nm, 5.50E+14 Hz, Ekin = 0.27 eV
blue, 436nm, 6.88E+14 Hz, Ekin = 0.81 eV
violet, 405nm, 7.41E+14 Hz, Ekin = 1.02 eV
Planck‘s constant is obtained from the slope of the kinetic energy, Ekin(ν)
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Hints for quantisation:
a) threshold (minimum frequency required):
• resonance phenomenon
• quantised medium or light
b) linear in the intensity (for ν=const).
• electron number proportional to photon number
c) photo current insensitive to ν (provided hν > Ebind)
• no change of the electron current if photon flux constant
• albeit the intensity is increasing: Iphoto ν
d) no delay
direct evidence!
• it lasts seconds until a single atom accumulates enough energy,
so the radiation cannot be continuous
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Blackbody = Cavity?
- Multiple reflections -> absorption of incident light
- Thermal equilibrium -> Walls <-> Cavity modes
- Spectral energy density ρ(ν)dν R(ν)dν (Radiance through whole)
- Boundary conditions: Nodes on the walls
- Standing waves along x,y,z
Consider a 2D problem and decompose λ into
λx = λ / cos(α)
λy = λ / sin(α)
with nx λx = 2L etc... ==> nx = (2L/ λ) cos(α) and ny = (2L/ λ) sin(α)
square and add these conditions (generalise into 3D):
(2L/ λ)2 = nx2 + ny2 + nz2
Axel Kuhn
Quantum Ideas 2007
Number of Modes in the cavity with frequencies smaller than ν:
- sphere of radius R = √(nx2 + ny2 + nz2) = 2L/ λ = 2L/c × ν
- mode number N(ν) = 4/3 π R3 × 2/8 = ...
- same for N(ν+dν) = ...
Mode number in the interval ν...ν+dν
ΔN = N(ν+dν) - N(ν) = 8π ν2 L3 / c3 dν
Spectral density per unit volume
ρ(ν)dν = ΔΝ/L3 = 8π ν2 / c3 dν
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
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Quantum Ideas 2007
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Axel Kuhn
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Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
Blackbody Radiation
Average energy <E>per frequency mode, using the Boltzmann distribution P(E)
∞
E =
∑ E P(E)
n=0
∞
n
∑ P(E)
∞
=
hν ∑ ne
n=0
∞
∑e
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with
and
∴
− nhν kT
− nhν kT
A
= hν
B
n=0
(
)
A − A exp ( − hν kT ) = B exp ( − hν kT )
B − B exp − hν kT = 1
A
1
=
and
B exp hν
kT − 1
(
)
E =
hν
exp hν kT − 1
(
)
Quantum Ideas 2007
Axel Kuhn
Planck‘s law
Spectral energy density (energy per unit volume in the frequency range ν...ν+dν):
hν
1
8πν 2
8π hν 3
ρ(ν )dν = 3
=
c exp hν
c 3 exp hν
−
1
kT
kT − 1
(
)
total energy density:
8π 5 k 4
4
4
ρ = ∫ ρ(ν )dν =
×
T
=
σ
T
15(hc)3
(
)
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
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Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
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Quantum Ideas 2007
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Axel Kuhn
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Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn
Quantum Ideas 2007
Hanbury-Brown and Twiss:
Intensity correlation measurements
- dead time of the detectors
- beam splitter
- pair of photon counters
- cross correlation
Single-photon emitters:
- single atoms or ions
- crystal defects (Quantum dots)
Axel Kuhn
Quantum Ideas 2007
Axel Kuhn