Quantum Physics Tutorial 6 - Wave-Particle Duality

 Contents

De Broglie's Relationship

The French physicist and aristocrat, Louis de Broglie (1892 – 1987) [pronounced ‘de Broy’], reasoned that if waves have particle properties, it was reasonable to suppose that particles had wave properties.  He devised the relationship that states that particles have wave properties. Photo from Wikimedia

He combined the following equations:

• Energy of photons:                 E = hf

• Einstein’s mass equivalence:   E = mc2

Therefore:

hf = mc2.

Now

f = c/l

So

mc = h/l

The term mc is mass × speed, which gives us the value of momentum.  We give momentum the code p.

We can rewrite the equation as

l = h/p                        or                     l = h/mv

Therefore every particle with a momentum has an associated de Broglie wavelength, even something as absurd as a car travelling at 20 m s-1.

 What was de Broglie’s hypothesis? Diffraction is associated with waves.  Give two examples of diffraction with waves.

Electron Diffraction

Electrons can be shown to have wave properties by the simple use of an electron diffraction tube.  A slice of carbon is placed in a beam of electrons so that the electrons diffract, just like waves. We need to note a couple of points:

• l is the de Broglie wavelength;

• Strictly speaking we should count the mass and speed as relativistic.  As the speed of particles approaches the speed of light, the mass increases as kinetic energy is turned into mass.  We will not worry about this at this stage.

Diffraction is a wave property.  Interference patterns from electrons also can be observed.

An interference pattern of water waves looks like this: A similar pattern can be observed using electrons.  Images from light waves are very complicated interference patterns.  Images can be made from very complicated interference patterns using an electron beam in an electron microscope.

 What is the evidence that electrons behave like waves? What is the de Broglie wavelength of an electron travelling at 2.0 × 10 6 m s-1?  Give your answer to an appropriate number of significant figures.

The wave properties of electrons have led to the development of the electron microscope, which allows magnifications much bigger than was ever possible with the light microscope.  A good light microscope can magnify up to 1000 times.  The electron microscope can magnify up to about 1 million times, and can reveal the existence of individual atoms.  The electron beams are focused by magnets just like the lenses on a microscope. Do not confuse the de Broglie equation with the photon energy equation.  They look superficially similar, but they are not relevant to each other.

Conclusion

• Electrons, which are definitely particles, have wave-like properties.

• The wave behaviour of electrons explains how the electron microscope works.