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Lecture 14: Schr dinger and Matter Waves

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Lecture 14: Schr dinger and Matter Waves

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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 kg Speed = 106 m / sec This wavelength is in the region of X-rays

    7. Example: de Broglie wavelength of a ball Mass = 1 kg Speed = 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 a given 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 + V total 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 microscope can only resolve objects as small as this, such as bacteria but not viruses

    18. Scanning Electron Microscope (SEM) To resolve even smaller objects, have to use electrons with 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)

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