Question

Which of the following is De-Broglie’s equation?
$\begin{align}
  & a)\lambda =\dfrac{m\upsilon }{h} \\
 & b)\lambda =hm\upsilon \\
 & c)\lambda =\dfrac{h\upsilon }{m} \\
 & d)\lambda =\dfrac{h}{m\upsilon } \\
\end{align}$

Answer 1

Hint: de Broglie equation is one of the equations that is commonly used to define the wave properties of matter. It basically describes the wave nature of the electron.

Step-by-Step Solution:
Let us first establish the particulars of the de Broglie equation and its significance in the study of wave properties of matter before answering this question.
In 1923, Louis de Broglie, a French physicist, proposed a hypothesis to explain the theory of the atomic structure. By using a series of substitution de Broglie hypothesizes particles to hold properties of waves. Within a few years, de Broglie's hypothesis was tested by scientists shooting electrons and rays of lights through slits. What scientists discovered was the electron stream acted the same way as light proving de Broglie correct.
He further proposed a relation between the velocity and momentum of a particle with the wavelength if the particle had to behave as a wave.
Let us now look at the derivation of De Broglie’s equation from Planck’s Wave equation derived from his quantum theory and Einstein’s mass-energy equation derived from his theory of special relativity.
Planck’s Wave Equation states that,
\[\begin{align}
  & \dfrac{hv}{\lambda }=m{{v}^{2}},\text{ for a particle moving with velocity v} \\
 & \Rightarrow \dfrac{h}{mv}=\lambda \\
\end{align}\]
Whereas Einstein’s mass-energy equation states,
$E=m{{c}^{2}}$
Equating these two equations, we get
\[\begin{align}
  & \dfrac{hv}{\lambda }=m{{v}^{2}},\text{ for a particle moving with velocity v} \\
 & \Rightarrow \dfrac{h}{mv}=\lambda \\
\end{align}\]
Therefore, as a result of this derivation, we can safely conclude that the answer to this question is d).

Note: Particle and wave nature of matter are actually ‘complementary’ to each other. It is not necessary for both to be present at the same time though. The significance of de Broglie relation is that it is more useful to microscopic, fundamental particles like electrons.