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Introduction
De Broglie wavelength is used to determine the wavelength of matter and hence is used to explain the dual behaviour of matter. This can be explained with the help of examples.
De Broglie wavelength formula is demonstrated with a few examples below:
It is observed that when a person is riding a bicycle in a straight line, at a certain point, he will come back to his starting point. This is because while he moves forward, the bicycle keeps rotating forward, and on returning, it returns to its original position. It is observed that at that moment, the person seems to be standing still. However, he can’t stand still. He has some speed that would have taken him far away from his origin if there was no rotation of the bicycle. So, to understand this situation, let us consider the following example:
If we apply de Broglie relationship between momentum and wavelength, then we can get an equation:
λ=h/p=h/mv
Where m=Mass of an object, v=Velocity of the object and h= planck’s constant
Assumptions
De Broglie Relationship is derived by using the following assumptions:
- The particles of matter are indistinguishable from the waves.
- The particle’s velocity is very much less than the velocity of light.
- The wavelength of matter waves is very much greater than the size of the particle.
- There is no external force applied to the system, which means that there is no acceleration and hence, no change in momentum of the particle (p=mv).
- There is a constant force applied to the system, which means a change in momentum (p=mv+F/t) and hence changes in velocity of the particle.
About the relationship
As discussed above, the Bohr model of an atom is an example of a wave theory of the atom. It was not able to address many complexities related to the spectrum of various atoms and the splitting of spectral lines under magnetic and electric fields. To address the flaws in Bohr’s atomic model, efforts were undertaken to build a more comprehensive atomic model.
Altman et al., in their paper published in 1966, discussed and analysed the shortcomings of Bohr’s model in detail. They also emphasised that since Bohr’s nuclear models failed to explain many spectral lines, they needed to be replaced by models which could predict and explain a number of spectral lines to make them more useful.
A more generalised approach was proposed by Gerhard Herzberg, who introduced the concept of valence shell electron pair repulsion (VSEPR). While working on this approach, he considered two main factors while working on this approach – bond angles around single bonds and bond angles around double bonds as viewed from any one atom. He then proposed his molecular orbital theory as a replacement for the wave theory of an atom.
He suggested that bonding occurs when there are overlapping p-orbitals available. This overlapping results in the formation of a new set of orbitals known as bonding orbitals containing paired electrons which are held together.
Equation
We all know that matter can be either solid, liquid, or gaseous, depending upon the conditions under which it is kept. The Einstein equation was as follows-
E= mc2 ……….(i)
Where E is energy , m is mass and c is speed of light(3 x 108 ms-1)
The Plank’s equation tells that Energy of quantum (E) is directly proportional to frequency of radiation (n) .
E=hv = h c/ λ ……….(ii)
Where E = Energy, h = Plank’s constant. v = frequency= c/ λ, λ = wavelength and c = velocity of light.
That, the energy emitted from a moving object is proportional to the frequency of waves and inversely proportional to the wavelength. The emission of electromagnetic radiation (EMR) occurs due to the acceleration of charged particles in an atom. In simpler terms, when an atom is accelerated it radiates a photon which is a fundamental quantum particle that carries energy away in the form of EM waves.
Putting value of E=mc2 from equation (i) in equation (ii)
mc2 = hc/ λ
( c can be replaced by velocity v for a general particle )
mv2 = hv/ λ
λ = h/mv ……..(iii)
The equation (iii) is known as de broglie relationship or equation
Significance
Now, we will have a look at the significance of De Broglie’s relationship.
λ = h/p Where p is momentum =mv
This means that the more momentum a particle has, the shorter its wavelength will be. And when it comes to a cricket ball, which has a very large momentum, it would therefore have a very small wavelength.
But here we have a problem. If the cricket ball’s electrons have a very small wavelength, they will also have a very small energy and a very small mass. According to Einstein’s famous equation E=mc2 , if an object’s mass decreases, its energy must increase. And if the cricket ball had such energetic electrons inside of it, they should all have escaped long ago!
De Broglie was aware of this problem. To get round it he had to take another one of his great leaps into the unknown: he suggested that an electron could somehow be in two places at once! That might sound absurd, but de Broglie wasn’t just plucking it out of thin air. He argued that as well as being in one place at one time, an electron could also exist in another place at another time.
Conclusion
In this material, we discussed the concept of De Broglie’s relationship in detail. We also discussed how it has been derived and the number of changes that were made to it during the development of the same. Along with that, we also discussed the significance of de Broglie’s relationship.
