Lecture 14: Schrdinger and Matter Waves

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Lecture 14 Schrödinger and Matter Waves
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Particle-like Behaviour of Light

• Plancks explanation of blackbody radiation
• Einsteins explanation of photoelectric effect

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de Broglie Suggested the converse

• All matter, usually thought of as particles,
  should exhibit wave-like behaviour
• Implies that electrons, neutrons, etc., are waves!

Prince Louis de Broglie (1892-1987)
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de Broglie Wavelength
Relates a particle-like property (p) to a
wave-like property (l)
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Wave-Particle Duality
particle
wave function
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Example de Broglie wavelength of an electron

• Mass 9.11 x 10-31 kgSpeed 106 m / sec
• This wavelength is in the region of X-rays

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Example de Broglie wavelength of a ball

• Mass 1 kgSpeed 1 m / sec
• This is extremely small! Thus, it is very
  difficult to observe the wave-like behaviour of
  ordinary objects

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Wave Function

• Completely describes all the properties of
  agiven particle
• Called y y (x,t) is a complex function of
  position x and time t
• What is the meaning of this wave function?

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Copenhagen Interpretationprobability waves

• The quantity y2 is interpreted as the
  probability that the particle can be found at a
  particular point x and a particular time t
• The act of measurement collapses the wave
  function and turns it into a particle

applet
Neils Bohr (1885-1962)
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Imagine a Roller Coaster ...
By conservation of energy, the car will climb up
to exactly the same height it started
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Conservation of Energy

• E K Vtotal energy kinetic energy
  potential energy
• In classical mechanics, K 1/2 mv2 p2/2m
• V depends on the system
• e.g., gravitational potential energy, electric
  potential energy

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Electron Roller Coaster
An incoming electron will oscillate betweenthe
two outer negatively charged tubes
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Schrödingers Equation

• Solve this equation to obtain y
• Tells us how y evolves or behaves in a given
  potential
• Analogue of Newtons equation in classical
  mechanics

applet
Erwin Schrödinger (1887-1961)
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Wave-like Behaviour of Matter

• Evidence
• electron diffraction
• electron interference (double-slit experiment)
• Also possible with more massive particles, such
  as neutrons and a-particles
• Applications
• Bragg scattering
• Electron microscopes
• Electron- and proton-beam lithography

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Electron Diffraction
X-rays
electrons
The diffraction patterns are similar because
electrons have similar wavelengths to X-rays
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Bragg Scattering
Bragg scattering is used to determine the
structure of the atoms in a crystal from the
spacing between the spots on a diffraction
pattern (above)
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Resolving Power of Microscopes

• To see or resolve an object, we need to use light
  of wavelength no larger than the object itself
• Since the wavelength of light is about 0.4 to 0.7
  mm,an ordinary microscopecan only resolve
  objectsas small as this, such asbacteria but
  not viruses

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Scanning Electron Microscope (SEM)

• To resolve even smaller objects, have to use
  electronswith wavelengths equivalent to X-rays

Virtual SEM
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Particle Accelerator

• Extreme case of an electron microscope, where
  electrons are accelerated to very near c
• Used to resolve extremely small distances e.g.,
  inner structure of protons and neutrons

Stanford Linear Accelerator (SLAC)
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Conventional Lithography
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Limits of Conventional Lithography

• The conventional method of photolithography hits
  its limit around 200 nm (UV region)
• It is possible to use X-rays but is difficult to
  focus
• Use electron or proton beams instead

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Proton Beam Micromachining (NUS)
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