Quantum Mechanics, part 2: Quantum Mechanics, part 2 Particles as Waves??
Summary of Conclusions for em radiation: Summary of Conclusions for em radiation The model for electromagnetic radiation is incomplete unless
em radiation is explained by wave properties for some experiments
em radiation is explained by particle properties for some experiments
Wave and particle characteristics complement each other to provide a complete picture of em radiation
1923 Doctoral dissertation of Count Louis de Broglie: 1923 Doctoral dissertation of Count Louis de Broglie particles may have a periodicity (wave characteristics)
the wavelength of a particle is inversely proportional to its momentum The photon Particles with Mass
Young Double Slit done by Taylor in 1909 and Davidson and Germer 1927 : Young Double Slit done by Taylor in 1909 and Davidson and Germer 1927 We get interference
for electron waves!
Davisson Germer Experiment (1927): Davisson Germer Experiment (1927)
Experimental Evidence and Complementarity (1927-28): Experimental Evidence and Complementarity (1927-28) Davisson - Germer experiment
54 eV electrons incident on a nickel crystal
G.P. Thompson
electron diffraction by gold foil
Principle of Complementarity (Niels Bohr)
The wave and particle models of matter or radiation complement each other
Neither the wave model or the particle model is a complete picture by itself
Diffraction from an Aluminum Foil(Multi crystalline structure) : Diffraction from an Aluminum Foil (Multi crystalline structure) X Ray Diffraction
(l = 0.071 nm) Electron Diffraction
(E = 600 eV)
A NEW Description: A NEW Description
We cannot predict the location ( for an event occurring) for a single photon or electron when diffracted; we can only predict the probability that the event will occur at that location
Classically the intensity of an em wave is proportional to amplitude squared
Intensity is proportional to the number of photons
It follows that the probability of locating a photon is proportional to E2 or the square of the amplitude
We apply the same idea to particles – probability is proportional to the amplitude (Y) of the wave property squared
Young’s Double Slit: Young’s Double Slit Intensity behavior
Intensity by Phasor Method: Intensity by Phasor Method
Young’s Double Slit: Young’s Double Slit Intensity behavior algebraic approach
Double Slit Diffraction of Electrons: Double Slit Diffraction of Electrons Probability of locating an electron
Double Slit Diffraction of Electrons: Double Slit Diffraction of Electrons Probability distribution is diffraction pattern
Double Slit Diffraction of Electrons: Double Slit Diffraction of Electrons Conclusion: electron (or photon) must be present at both slits simultaneously
The Schrödinger Wave Equation use to describe WAVE: The Schrödinger Wave Equation use to describe WAVE Recall Classical Electro-mag Wave Equation RECALL the total energy of a
Massive particle Maxwell’s differential equation for light propagation as a wave Schrödinger's differential equation for light propagation as a wave
The General Solution without boundary conditions fixed : The General Solution without boundary conditions fixed and NOTE: and where y* is the complex conjugate of y
The Free Particle: The Free Particle U = 0
Particle travels in +x direction General solution Solution with boundary condition of wave traveling in positive x direction
Probability to find a particle from the wave properties of an “particle”: Probability to find a particle from the wave properties of an 'particle' Intensity=probability to locate a particle
Sum of two waves that has some localization Sum of two waves that has no localization
The Uncertainty Principle: The Uncertainty Principle Momentum well known
x is completely unknown Position well known
The Uncertainty Principle: The Uncertainty Principle Werner Heisenberg (1927)
Derived conclusion that
'it is fundamentally impossible to make simultaneous measurements of a particle’s position and speed with infinite accuracy'
or
'it is physically impossible to simultaneously measure the exact position and exact momentum of a particle'
Radioactivity Barrier “Tunneling”: Radioactivity Barrier 'Tunneling' How do get out of a nucleus?
classically a particle has to scale the hill….with an electrical potential barrier and electron has to climb an electrical potential hill
'Can’t get there from here'
Tunneling through a potential barrier(principle of Scanning Tunneling Electron Microscope): Tunneling through a potential barrier (principle of Scanning Tunneling Electron Microscope) Solution (transmission coefficient)
Nobel Prize 1973 : Nobel Prize 1973 Leo Esaki, Ivar Giaver and Brian Josephson
Josephson junction: a rapid quantum switching device based on tunneling
Example 38-7…
Introduction to Wave Mechanics: Introduction to Wave Mechanics The wave function
Interpretation - Probability function and density
Normalization
Probability of locating a particle
Expectation value