Lecture 14: Schrödinger and Matter Waves

Particle-like Behaviour of Light

Planck’s explanation of blackbody radiation

Einstein’s explanation of photoelectric effect

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)

de Broglie Wavelength

Relates a particle-like property (p) to a wave-like property ()

particle wave function

Wave-Particle Duality

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

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

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?

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)

Imagine a Roller Coaster ...

By conservation of energy, the car will climb up to exactly the same height it started

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

Electron ‘Roller Coaster’

An incoming electron will oscillate betweenthe two outer negatively charged tubes

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)

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

Electron Diffraction

X-rays electrons

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

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

Scanning Electron Microscope (SEM)

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

SEM Images

Guess the images ...

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)

Conventional Lithography

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…

Proton Beam Micromachining (NUS)

More details ...