Question

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

Answer 1

Hint: 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
\[ \Rightarrow \lambda = \dfrac{{hc}}{{m{c^2}}} \\
\Rightarrow \lambda = \dfrac{h}{{mc}} \]
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.