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1. Lecture 14: Schrödinger and Matter Waves
2. Particle-like Behaviour of Light Planck’s explanation of blackbody radiation
Einstein’s explanation of photoelectric effect
3. de Broglie: Suggested the converse All matter, usually thought of as particles, should exhibit wave-like behaviour
Implies that electrons, neutrons, etc., are waves!
4. de Broglie Wavelength
5. Wave-Particle Duality
6. 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
7. 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
8. 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?
9. Copenhagen Interpretation:probability waves The quantity |y|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
10. Imagine a Roller Coaster ...
11. 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
12. Electron ‘Roller Coaster’
13.
Solve this equation to obtain y
Tells us how y evolves or behaves in a given potential
Analogue of Newton’s equation in classical mechanics Schrödinger’s Equation
14. 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
15. Electron Diffraction
16. Bragg Scattering
17. 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
18. Scanning Electron Microscope (SEM) To resolve even smaller objects, have to use electronswith wavelengths equivalent to X-rays
19. 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
20. Conventional Lithography
21. 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 …
22. Proton Beam Micromachining (NUS)