De Broglie Wavelength

The Bohr model of the atom ( as seen in the previous section ) did have a major issue - that the orbiting electrons did not emit radiation. Bohr simply ignored this issue and stated that they didn't, but an explanation was required.

It was Louis De Broglie that first came up with an explanation, based upon the idea that like light, Electrons could act as both particles and waves. It was in the form of a standing wave that the electron took within the atom, with no loss of energy. 

The diagram below shows a visual representation of De Broglie's theory :-

Non Examinable Derivation - De Broglie Wavelength

For a Photon, as it has no rest mass, it can be shown that the Energy can be given as :-

E  =  p x C

where :-

E  =  Energy of the Photon ( J ) 

p  =  Momentum of the Photon ( kg ms-1

C  =  Speed of Light ( ms-1

But from Higher Physics, we know that the Energy of a Photon is related to its Frequency. By combining these two formulae together, the following can be derived :-

E  =  p x v     and     E  =  h x f 

p x v  =  h x f 

p  =  hf /

p  =  h / λ

λ  =  h /

Where :-

h  =  6.63 x10-34 ( kg ms-1 )

p  =  momentum of the particle ( kg ms-1

λ  =  De Broglie Wavelength of the particle ( m ) 

Example 1 -

An electron microscope uses electrons travelling at 1.8 x107 ms-1 to image a sample. What is the De Broglie Wavelength of the Electrons ? 

h  =  6.63 x10-34 

m  = 9.11 x10-31

v  =  1.8 x107 

λ =  ? 

λ  = ( 6.63 x10-34 ) / ( 9.11 x10-31 x 1.8 x107 )

λ  = 0.04 x10-9

λ  =0.04 nm 

Note - In the above example, it can bee seen why scientists use Electrons to image extremely small objects. In order to image an object, the object must be larger than the wavelength of the light used. In the case of visible light microscopes, this gives a maximum resolution of ~ 400 nm. The above Electron Microscope can therefore resolve objects x10,000 smaller.

Double Slit experiment with Electrons

In the Higher Physics course, the interference pattern of light through two slits was discussed in great detail. A summary of this is given below :-

The pattern above always looks the same for a double slit, with a central bright maximum and a series of bright and dark bands, moving out symmetrically. 

Central Bright Maximum - the waves from each source travel the same distance (Path difference = 0) to reach this point. As the waves are coherent, this means that if they travel the same distance, then the waves will be in phase when they meet. This causes constructive interference and a central bright maximum.

Note - The Central Maximum is referred to as the 0th Order Maximum 

To reach other maxima or minima, waves from the 2 sources have to travel different

distances - The difference between these 2 distances is known as the path difference.

First ( Bright ) Maximum - As this is a bright band, the waves must meet in phase. This means the distance from the Slit S2 to the screen is greater that the distance from Slit S1 to the screen by 1 whole wavelength. This applies for each maxima out from the centre, giving the general rule:-

Path difference = ml

where m = number of maxima out from centre

First ( Dark ) Minimum - As this is a dark band, the waves must meet out of phase. This means that a peak meets a trough, which means the waves are a half wavelength out of step. This applies to each minima out from the centre, giving the general rule:-

Path difference = ( m + ½ ) l

where m = number of maxima out from centre

Double Slit experiment - Quantum Theory 

Just as light can exhibit both wave and particle behaviour, De Broglie postulated that Electrons, under the right conditions can act as a wave. This can be seen by passing a beam of electrons through the same double slit set up above, and will again generate a interference pattern, proving that the electrons are behaving as wave. 

Click the image below to open an applet showing the generation of an electron interference pattern :-

Click to Open

Single Photon - Electron Interference Patterns

The above applet shows the interference patterns for both Photons and Electrons on a large scale of many many particles. However, recently it has been possible to complete the above Young's Slits experiment for single Photons or Electrons. 

When single Photons or Electrons are incident on the slit, the usual interference pattern is gradually built up. The question this raises is that how does a single Photon have an awareness of the other slit? 

In order to understand what is happening here, it is possible to set up a set of detectors at the slits in order to find out which slit the Photon ( or Electron ) passes through. 

The image below shows the outcome of the system, with the detectors turned off  :-

As can be seen, the interference pattern is seen on the screen. If, however, one of the detectors is turned on, the following occurs :-

As can be seen, the interference pattern is lost and a pattern of particles passing through either slit is seen. 

As soon as the Photon ( or Electron ) is observed as a particle, all wave properties are lost. 

It appears from the results that the Photon really is "aware" of the other slit, and when unobserved, a single particle can simultaneously pass through both slits. 

This is an example of how Quantum Theory is very non-intuitive. However, this effect is very much in agreement with Quantum Theory. Quantum Theory states that we cannot measure wave and particle properties at the same time.

The video below shows a lecture by Miles Padgett explaining to university level the implications of the Young's slit experiment.