lecture 14: schrödinger and matter waves. particle-like behaviour of light n planck’s explanation...
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Lecture 14: Schrödinger and Matter Waves
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Particle-like Behaviour of Light
Planck’s explanation of blackbody radiation
Einstein’s 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 ()
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particle wave function
Wave-Particle Duality
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Example: de Broglie wavelength of an electron
Mass = 9.11 x 10-31 kgSpeed = 106 m / sec
m10287m/sec) kg)(10 10(9.11secJoules10636 10
631
34
.
.
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
m10636m/sec) kg)(1 (1
secJoules10636 3434
..
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 (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 Interpretation:probability waves
The quantity 2 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
appletNeils 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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Solve this equation to obtain
Tells us how evolves or behaves in a given potential
Analogue of Newton’s equation in classical mechanics
Schrödinger’s Equation
appletErwin 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 -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 m,
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
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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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