Lecture 14: Schrödinger and Matter Waves

Particle-like Behaviour of Light n Planck’s explanation of blackbody radiation n Einstein’s explanation of photoelectric effect

de Broglie: Suggested the converse n All matter, usually thought of as particles, should exhibit wave-like behaviour n Implies that electrons, neutrons, etc., are waves! Prince Louis de Broglie ( )

de Broglie Wavelength Relates a particle-like property (p) to a wave-like property ()

particlewave function Wave-Particle Duality

Example: de Broglie wavelength of an electron n Mass = 9.11 x kg Speed = 10 6 m / sec n This wavelength is in the region of X-rays

Example: de Broglie wavelength of a ball n Mass = 1 kg Speed = 1 m / sec n This is extremely small! Thus, it is very difficult to observe the wave-like behaviour of ordinary objects

Wave Function n Completely describes all the properties of a given particle Called (x,t); is a complex function of position x and time t n What is the meaning of this wave function?

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 n The act of measurement ‘collapses’ the wave function and turns it into a particle applet Neils Bohr ( )

Imagine a Roller Coaster... By conservation of energy, the car will climb up to exactly the same height it started

Conservation of Energy n E = K + V total energy = kinetic energy + potential energy n In classical mechanics, K = 1/2 mv 2 = p 2 /2m n V depends on the system –e.g., gravitational potential energy, electric potential energy

Electron ‘Roller Coaster’ An incoming electron will oscillate between the two outer negatively charged tubes

Solve this equation to obtain  Tells us how  evolves or behaves in a given potential n Analogue of Newton’s equation in classical mechanics Schrödinger’s Equation applet Erwin Schrödinger ( )

Wave-like Behaviour of Matter n Evidence: –electron diffraction –electron interference (double-slit experiment) Also possible with more massive particles, such as neutrons and -particles n Applications: –Bragg scattering –Electron microscopes –Electron- and proton-beam lithography

Electron Diffraction X-rayselectrons The diffraction patterns are similar because electrons have similar wavelengths to X-rays

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)

Resolving Power of Microscopes n 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 microscope can only resolve objects as small as this, such as bacteria but not viruses

Scanning Electron Microscope (SEM) n To resolve even smaller objects, have to use electrons with wavelengths equivalent to X-rays

SEM Images Guess the images...

Particle Accelerator n Extreme case of an electron microscope, where electrons are accelerated to very near c n Used to resolve extremely small distances: e.g., inner structure of protons and neutrons Stanford Linear Accelerator (SLAC)

Conventional Lithography

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

Proton Beam Micromachining (NUS) More details...