Transcript Schrödinger and Matter Waves

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 (l)
Wave-Particle Duality
particle
wave function
Example: de Broglie wavelength of an electron

Mass = 9.11 x 10-31 kg
Speed = 106 m / sec
6.63 1034Joules  sec
10
l

7
.
28

10
m
31
6
(9.11 10 kg)(10 m/sec)

This wavelength is in the region of X-rays
Example: de Broglie wavelength of a ball

Mass = 1 kg
Speed = 1 m / sec
6.63 1034Joules  sec
l
 6.63 1034m
(1 kg)(1 m/sec)

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 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?
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
applet
Neils 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+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
Electron ‘Roller Coaster’
An incoming electron will oscillate between
the two outer negatively charged tubes
Schrödinger’s Equation

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
applet
Erwin Schrödinger (1887-1961)
Wave-like Behaviour of Matter
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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
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 mm,
an ordinary microscope
can only resolve objects
as small as this, such as
bacteria but not viruses
Scanning Electron Microscope (SEM)

To resolve even smaller objects, have to use electrons
with 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 ...