Question

Dual nature of matter was proposed by de Broglie in 1923, it was experimentally verified by Davisson and Germer by the diffraction experiment. Wave character of matter has significance only for microscopic particles. De Broglie wavelength or the wavelength of a matter wave can be calculated using the following relation: $\lambda =\dfrac{h}{mv}$. Where, $m$ and $v$ are the mass and velocity of the particle. De Broglie hypothesis suggested that the electron waves were being diffracted by the target, much as X-rays are diffracted by planes of atoms in the crystals.
De Broglie equation is obtained by the combination of which of the following theories?
A. Planck’s quantum theory
B. Einstein’s theory of mass-energy equivalence
C. Theory of interference
D. Theory of diffraction
This question has multiple correct options

Answer 1

Hint: Think about all the formulae that the options mention and what quantities they can relate. Pay attention to whether they are formulae that define properties of matter or waves since the de Broglie hypothesis speaks about the dual nature of matter as particles and waves.

Complete step by step solution:
De Broglie formulated and derived his equation for the ‘De Broglie Wavelength’ using well established formulae that related the properties of particles and waves by a single entity. We know that both matter and waves have energy, de Broglie uses this same knowledge to tie together two hypothesis that were described for very different aspects.
We know that Planck's quantum theory states that the energy (E) of any wave is proportional to the frequency ($\nu $) of the wave. The constant of proportionality by which they are related is known as the Planck’s constant and is dented by h. This relation is defined by the formula:
\[E=h\nu \]
Now let us look at Einstein’s theory of mass-energy equivalence. This relation states that when particles travel at the speed of light, their energy is equal to the product of their mass and the square of their speed i.e. the speed of light. This relation implies that mass becomes energy when it travels at the speed of light. The relation is defined by the formula:
\[E=m{{c}^{2}}\]
We see that both these formulae define energy, but for particles and waves individually, since the de Broglie hypothesis suggested that matter has a dual nature of particles as well as waves, he equated both these relations and then proceeded to carry out a series of substitutions to get the de Broglie equation. The process was as follows:
\[m{{c}^{2}}=h\nu \]
Now, since all particles do not travel at the speed of light, the parameter ‘$c$’ was changed to velocity ‘$v$’. We all know the equation that relates wavelength and frequency $c=\lambda \nu $. Here too, the speed of light is replaced with the velocity of the wave at that point. So, now the equation is:
\[m{{v}^{2}}=\dfrac{hv}{\lambda }\]
Now, we will simplify and rearrange the equation for the wavelength.
\[\begin{align}
  & \lambda =\dfrac{hv}{m{{v}^{2}}} \\
 & \lambda =\dfrac{h}{mv} \\
\end{align}\]
This is de Broglie’s equation which is obtained by combining the equations described by Planck and Einstein.

Hence, the answer to this question is ‘A. Planck’s quantum theory’ and ‘B. Einstein’s theory of mass-energy equivalence’

Note: Do not get confused between the terms and notation for frequency and velocity. The notations should look different when written side by side; $v$ and $\nu $, here the former one is the velocity and the latter is the frequency. The letter $f$ can also be used to denote frequency.