Week 6: Quantum Uncertainty (20 C) Flashcards
What are the origins of Quantum Physics?
Answering the problem of Black Body Radiation
• An ideal black body should absorb all light
o Was modeled as a metallic sphere, with a small pinhole in its surface
o Light could enter the pinhole, but once inside, would be unable to escape
Light would just reflect around, trapped within the sphere
• If a black body absorbs all light, how can it emit an EM spectrum?
o Still has a temperature, small amounts might leak out of the pinhole
• It was thought that thermal radiation could connect Thermodynamics and Electrodynamics
o Has radiation because of temperature (heat, and thus thermodynamics), but concerns radiation emission (electrodynamics)
o In the same vein as Maxell & Boltzmann trying to reduce thermodynamics to mechanical principles, the coalescence of the branches of physics was a common goal
• By the late 19th century, many were searching for a theory to explain black body thermal radiation
Rayleigh-Jeans?
Early theory by both English physicists to explain Black Body problem:
• Consider an ideal black box with a pin hole
• All the trapped light bouncing within will excite electrons on the surface of the sphere
• These electrons will oscillate
o And therefore, they will emit too!
• But a sphere has finite size – so only certain frequencies will be able to resonate within it
o These are the frequencies which will be observed
• R-J derives intensities from their frequency
o Longer wavelengths rapidly become improbable – as they are unable to resonate within the black box model
o But intensities are predicted to rise towards infinity as wavelength approaches zero
Releasing an infinite amount of energy and violates the conservation of energy
• Around this time, German researchers performed experiments to measure intensities
o R-J accurately predicted the fall off at higher wavelengths
o But at lower wavelengths, observed intensity falling back to zero
the ULTRAVIOLET CATASTROPHE
What was the problem with Rayleigh-Jeans? How was it answered?
UV catastrophe - RJ predicted intensity rising to infinite levels with decreasing wavelengths; experimentally false.
- Max Planck (1900) – “Quantization of Energy for Oscillating Electrons”
• More of a mathematical trick than a theory, but it could remove the infinite intensities
• Formulated by working backwards from R-J to see how he could force it to match experimental results
• Idea: Assume oscillators can’t occupy arbitrary energies
• Instead, only discrete energy levels could be occupied
• Energy levels are multiples of hν - Albert Einstein (1905) – Light Quanta
• Re-examines the black body formulation
• Instead of quantizing Energy levels (a la Planck), proposes quantizing the light
• Light has discrete momentum and Energy levels
• Treat the light like particles of an ideal gas
• Transforming thermodynamic results into radiation, his obtained new formulations which matched experimental results
• This was an even more dramatic proposal than Planck’s
Neils Bohr’s atomic model? Problems and answers?
A Danish physicist.
• After bombarding materials (e.g. gold foil) with alpha particle radiation, and observing the scattering pattern, noticed that most passed through but there were certain concentrated deflections
• Theorized “planetary model” – nucleus core with orbiting electrons
• This model is not stable
o Unlike planetary physics which can have stable orbits, these electrons have electrostatic repulsion between each other
o Further, particles require a constant centripetal acceleration to maintain orbit
Emitting electromagnetic radiation should drain these particles of their energy over time
So eventually they will have inadequate energy to maintain orbit
Rutherford helped resolve this:
o Postulated that electrons can only be in one of a finite set of orbits
These orbits are stable
o Electron energy is therefore quantum
• With Rutherford’s explanation, Bohr’s model could explain Hydrogen’s radiation spectrum
o When an electron drops to a lower orbit, it loses an amount of energy associated with the energy difference between those two orbitals
This Energy is emitted as EM radiation, dictated by E = hν
Was very consistent with the hydrogen spectrum, and was used for spectroscopy from 1913 – 1925
• This Bohr model had limitations though:
o Failed for any element but Hydrogen
o Required you to impose quantum conditions a priori
o While it was a theory consistent with experimental results and an important step, Bohr lacked a real proof for his theory
What was the new Quantum Physics of the 1920’s?
Call for a new-new quantum physics, moving past Bohr model and light quanta.
Erwin Schrödinger’s Wave Mechanics
1. treat orbiting electrons not as particles, but wave like entities with a distribution
Idea stems from de Broglie’s matter waves
2. Develops “wave equation” describing electron distribution in space
Discrete solutions of this equation describe the possible states
Unlike Bohr, he doesn’t a priori assume Quantum states – instead it presents itself as an outcome of the theory
Then came Matrix mechanics (Heisenberg, Bohr, Jordans)
Matrix Mechanics
Matrix Mechanics from the “Göttingen Three” (Heisenberg, Bohr, and Jordans)
1. Abandon the Electron as a macroscopic entity
Electron is just a mathematical representation of observable properties
• E.g. spectral lines
2. Understand electron not with physical properties, but instead as a shorthand notation to describe transitions between quantum states
Only have position/momentum for the transitioning between states
• E.g. transition from state n to m described with Xn,m, ρ¬n,m
These transitions are conveniently represented with matrices – hence the theory
3. They formulated matrices for the position and momentum of transitions
Can be reduced to be equivalent to wave mechanics
Though outcome is the same, conceptually they are very different
What were the hallmarks of the Copenhagen interpretation?
o Uncertainty Principle & Measurement Problem
-electron comes back, with position, momentum, trajectory
-but properties are intrinsically uncertain - ONTOLOGICAL UNCERTAINTY
o Complementarity
o Probabilistic Interpretation of the Wave Function
What was the Uncertainty Principle? Why was it a new direction for uncertainty?
Completely abolishing the physical meaning of electrons (as with matrix mechanics) was too radical
1926-1927, Heisenberg acknowledges electron has position, momentum, trajectory
But unlike with Maxwell’s microscopic particles, these have uncertainty:
ΔXΔP≥h/2π
Implications of Heisenberg’s interpretation:
• Measurement/observation shapes the object to be measured
• You will always affect the object through measurement
o This is not a case where better measuring tools can be employed – it is fundamental to its nature
o This was at odds with the classical understanding of the world
What was complementarity?
• Bohr wondered – how should we characterize light?
o Einstein had proposed quantized light, which had experimental support
o There was evidence for light to be a particle and for it to be a wave
• Bohr’s answer: It is neither – it is a hybrid of both!
o Electrons and light have elements of both waves and particles
o By measuring its wave properties, we have forced it to behave as a wave
Reverse case is the same for particle properties
o Measurement of the object collapses it to be either a particle or a wave state
This explains the experimental results showing it as both
What was the probabilistic interpretation of the wave function?
• Schrödinger had suggested the discrete solutions of the wave equation (Ψ) captured different states
• Max Born took it one level further (1926)
o Ψ represents a particle’s probabilistic density
o Ψ itself is not a probability density, |Ψ| is
o But Ψ gives a complex number!
Result: Particles can interfere with each other
• Like the interference observed in the double slit experiment
What was EPR?
Einstein-Podolsky-Rosen Paradox - developed by Einstein with his new Princeton colleagues
• Prepare a pair of “entangled” quantum states, and let them separate
• Due to properties of entanglement, measuring particle 1’s momentum informs you of particle 2’s momentum
o And similarly, measuring one particle’s position gave you the other’s
• This means that the other particle, which is in another location, can be determined immediately without disturbing it
o i.e. you can the other particles “elements of reality”
Heisenberg’s uncertainty principle does not explain how this is possible
o Thus Quantum Mechanics is an incomplete theory
-Probably were some “hidden variables” needed to explain this
