Question

Write the expression for the de-Broglie wavelength of a particle.

Answer 1

We will first write what is the de-Broglie equation and then with Einstein's mass-energy and momentum of the photon formulas we will derive an expression for the de-Broglie wavelength equation.
Formulae Used: $\lambda = \dfrac{h}{{mv}}$, $E = hv$, $v = \dfrac{c}{\lambda }$

Complete step-by-step solution:
We know that the de-Broglie wavelength equation is $\lambda = \dfrac{h}{{mv}}$.
Now, we will derive an expression for de-Broglie’s wavelength equation.
Considering photon as an electromagnetic wave of frequency $v$, its energy from Planck’s quantum theory is given by
$E = hv................................\left( 1 \right)$
Where $h$ is Planck’s constant. Considering photon as a particle of mass $m$, the energy associated with it is given by Einstein’s mass-energy relationship as
$E = m{c^2}..........................\left( 2 \right)$
From equation (1) and (2), we get
$\Rightarrow hv = m{c^2}$
As we know that $v = \dfrac{c}{\lambda }$, therefore the above equation becomes as
$\Rightarrow \dfrac{{hc}}{\lambda } = m{c^2}$
Now, the expression for de-Broglie's wavelength equation is given by

\Rightarrow \lambda = \dfrac{h}{{mc}} $$ We know that the momentum of the photon is given by $$p = mc$$. Substituting this in the equation we have $$\therefore \lambda = \dfrac{h}{p}$$ Hence the expression for de-Broglie's wavelength equation is $$\lambda = \dfrac{h}{p}$$ where $$\lambda $$ is the wavelength of the radiation of frequency $$v$$ and $$p = mc$$, is the momentum of the photon. The above equation has been derived for a photon of radiation. According to de-Broglie’s hypothesis, it must be true for material particles like electrons, protons, neutrons, etc. Hence a particle of mass $$m$$ moving with velocity $$v$$ must be associated with a matter wave of wavelength $$\lambda $$ given by, $$ \Rightarrow \lambda = \dfrac{h}{p} \\\ \therefore \lambda = \dfrac{h}{{mv}} $$ Where $$p = mv$$, is the momentum of a particle. This is de-Broglie’s wavelength equation for material particles. **Note:** The waves associated with material particles in motion are called matter or de-Broglie waves and their wavelength is called the de-Broglie wavelength. These kinds of questions are simple but one should remember each and every step of the expression and some basic formulas.