De-Broglie Wavelength Formula: Equations, Solved Examples

De-Broglie Wavelength Formula – Einstein proposed that any electromagnetic radiation, including light which was, till then, considered an electromagnetic wave, in fact, showed particle-like nature. He coined the word “photon” for the quanta or particle of light. Soon, scientists began to wonder if other particles could also have a dual wave-particle nature.

In 1924, French scientist Louis de Broglie derived an equation, known as the De Broglie Wavelength Formula, that described the wave nature of any particle. Thus, establishing the wave-particle duality for the matter. Microscopic particle-like electrons also proved to possess this dual nature property. Let us learn about the equation proposed by de-Broglie in detail in this article.

De Broglie’s Hypothesis and Equation

Louis-de-Broglie explained the concept of de-Broglie waves in the year 1923. In his thesis, he suggested that any moving particle, whether microscopic or macroscopic, will be related to a wave character. This was later experimented with and proved by Davisson and Germer within the year 1927. The waves associated with matter were called ‘Matter Waves’. These waves explain the character of the wave associated with the particle. We know that electromagnetic radiation exhibit the dual nature of a particle (having a momentum) and wave (expressed in frequency, and wavelength).

He further proposed a relation between the speed and momentum with the wavelength if the particle had to behave as a wave. Since, at non-relativistic speeds, the momentum of a particle will be adequate to its mass \(\text {m}\), multiplied by its velocity \(\text {v}\). Thus, according to de-Broglie, the wavelength \(\left( \lambda \right)\) of any moving object is given by:

\(\lambda = \frac{{\rm{h}}}{{\rm{p}}}\)

Where \(\text {h}\) is Planck’s constant and \(\text {p}\) is the momentum of the particle.

Derivation of De Broglie’s Wavelength

From Einstein’s relation of mass-energy equivalence, we know that,

\({\text{E}} = {\text{m}}{{\text{c}}^2} \cdots (1)\)

Where,

\(\text {E}=\) energy of the particle

\(\text {m}=\) mass of the particle

\(\text {c}=\) speed of light

According to Planck’s theory, every quantum of a wave has a discrete amount of energy associated with it, and he gave the equation:

\({\text{E}} = {\text{hf}} \cdots (2)\)

Where,

\(\text {E}:\) energy of the particle

\(\text {h}=6.62607 \times 10^{-34} \mathrm{Js}:\) Planck’s constant

\(\text {f}=\) frequency

De-Broglie’s hypothesis suggested that particles and waves behave as similar entities. Thus, he equated the energy relation for both particle and wave; equating equations \((1)\) and \((2)\), we get:

\(\text {mc}^{2}=\text {hf}\)

Since the particles generally do not travel at the speed of light, De Broglie substituted the speed of light \(\text {c}\), with the velocity of a real particle \(\text {v}\), and obtained:

\({\text{m}}{{\text{v}}^2} = {\text{hf}} \cdots (3)\)

If \(\lambda \) be the wavelength of the wave, then the frequency will be: \({\rm{f}} = \frac{{\rm{v}}}{\lambda }\)

 Substituting this in equation \((3)\), we get:

\({\rm{m}}{{\rm{v}}^2} = \frac{{{\rm{hv}}}}{\lambda }\)

\(\lambda = \frac{{\rm{h}}}{{{\rm{mv}}}}\)

or, \(\lambda = \frac{{\rm{h}}}{{\rm{p}}}\,\,\, \cdots (4)\)

Where \(\text {p}\) is the momentum of the particle.

De Broglie Wavelength and Kinetic Energy

The kinetic energy of an object of mass \(\text {m}\) moving with velocity \(\text {v}\) is given as:

\({\text{K}} = \frac{1}{2}{\text{m}}{{\text{v}}^2}\)

or, \({\text{K}} = \frac{1}{2}{\text{mv}} \cdot {\text{v}}\)

\({\text{m}} \cdot {\text{K}} = \frac{1}{2}{({\text{mv}})^2}\)

Since, \(\text {p}=\text {mv}\), Thus:

\({\text{m}}.{\text{K}} = \frac{1}{2}{({\text{p}})^2}\)

From equation \((4),\,{\rm{p}} = \frac{{\rm{h}}}{\lambda }\)

\( \Rightarrow {\rm{m}} \cdot {\rm{K}} = \frac{1}{2}{\left( {\frac{{\rm{h}}}{\lambda }} \right)^2}\)

\({\lambda ^2} = \frac{{{{\rm{h}}^2}}}{{2\,{\rm{mK}}}}\)

\(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{mK}}} }}\)

De-Broglie Wavelength and Potential

When a charged particle, having a charge \(\text {q}\) is accelerated through an external potential difference \(\text {V}\), the energy of the particle can be given as:

\({\text{E}} = {\text{qV}} \cdots ({\text{i}})\)

According to Planck’s equation,

\(\text {E}=\text {hf}\)

Since, \({\rm{f}} = \frac{{\rm{v}}}{\lambda }\)

Therefore, \({\rm{E}} = {\rm{h}}\frac{{\rm{v}}}{\lambda }\,\,\, \ldots ({\rm{ii}})\)

Equating the equations \(\left({\text{i}} \right)\) and \(\left({\text{ii}} \right)\),

\({\rm{qV}} = {\rm{h}}\frac{{\rm{v}}}{\lambda }\)

or, \(\lambda = \frac{{{\rm{hv}}}}{{{\rm{qV}}}}\)

Thermal De Broglie Wavelength

There exists a relation between the De-Broglie equation and the temperature of the given gas molecules, and the thermal de Broglie wavelength gives it \(\left( {{\lambda _{{\rm{Th}}}}} \right).\) The Thermal de Broglie equation represents the average value of the de Broglie wavelength of the gas particles at the specified temperature in an ideal gas.

The expression gives the thermal de Broglie wavelength at temperature \(\text {T}\):

\({\lambda _{{\rm{Th}}}} = \lambda  = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{m}}{{\rm{k}}_{\rm{B}}}{\rm{T}}} }}\)

where,

\(h=\) Planck constant

\(m=\) mass of a gas particle

\({{\text{k}}_{\text{B}}} = \) Boltzmann constant

De Broglie Wavelength of Day-to-Day Objects

According to the hypothesis, all particles have a wave associated with them. That is true for us humans and the objects around us. To get an idea of the de-Broglie wavelength associated with macroscopic particles:

Let us find the wavelength of a wave associated with a car of mass \(1000 \mathrm{~kg}\) moving with the velocity of \(10 \mathrm{~m} / \mathrm{s}.\)

The wavelength associated with the car will be: \(\lambda = \frac{{\rm{h}}}{{{\rm{mv}}}}\)

\(\lambda=\frac{6.62607 \times 10^{-34} \mathrm{Js}}{1000 \mathrm{~kg} \times 10 \mathrm{~m} / \mathrm{s}}=6.6 \times 10^{-30} \mathrm{~m}=6.6 \times 10^{-21} \mathrm{~nm}\)

Thus, the value of wavelength associated with this car is insignificant.

Similarly, for other macroscopic objects with large mass values, the wavelength associated with them is so small that it can not be detected.

De-Broglie Wavelength of an Electron

As we have seen above, the matter waves associated with real objects is so small that it is of no good use to us. But for sub-atomic particles with negligible masses, the value of de-Broglie wavelength is substantial. To calculate the de-Broglie wavelength associated with a microscopic particle,

Let us take an electron of mass \({\rm{m}} = 9.1 \times {10^{ – 31}}\;{\rm{kg}},\) moving with the speed of light, i.e., \(\text {c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\), then the de-Broglie wavelength associated with it can be given as:

\(\lambda = \frac{{\rm{h}}}{{{\rm{mc}}}}\)

\(\lambda=\frac{6.62607 \times 10^{-34} \mathrm{Js}}{9.1 \times 10^{-31} \mathrm{~kg} \times 3 \times 10^{8} \mathrm{~m} / \mathrm{s}}=0.7318 \times 10^{-11} \mathrm{~m}=0.073 \mathrm{~A}^{\circ}\)

This is a substantial value. Thus, the de-Broglie wavelength associated has a significant value, and it can be detected. 

Relation Between De-Broglie Wavelength and Potential (for an electron)

The expression for the de-Broglie wavelength of an electron,

\(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{mK}}} }}\)

If the electron having a charge e is moving under an external potential \(\text {V}\), then,

The kinetic energy of the electron, \({\text{K}} = {\text{eV}}\)

Substituting this expression in the above equation,

\(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{meV}}} }}\)

Put, \(h=6.62607 \times 10^{-34} \mathrm{Js}\)

\(\text {e}=1.6 \times 10^{-19} \mathrm{C}\)

\(\text {m}=9.1 \times 10^{-31} \mathrm{~kg}\)

\(\lambda = \frac{{12.27}}{{\sqrt {\rm{V}} }}{\rm{A}}^\circ \)

De Broglie Wavelength- Solved Problems

Q.1. A certain photon has a momentum of \(1.50 \times 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}\). What will be the photon’s de Broglie wavelength?

Ans:\(\text {p}=1.50 \times 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}\)
Plank’s constant, \(\text {h}=6.62607 \times 10^{-34} \mathrm{Js}\)
The de Broglie wavelength of the photon can be computed using the formula:
\(\lambda = \frac{{\rm{h}}}{{\rm{p}}}\)
\(=\frac{6.62607 \times 10^{-34} \mathrm{Js}}{1.50 \times 10^{-27} \mathrm{kgm} / \mathrm{s}}\)
\(=4.42 \times 10^{-7} \mathrm{~m}\)
\(=442 \times 10^{-9} \mathrm{~m}\)
\(=442 \mathrm{~nm}\)
The de Broglie wavelength of the photon will be \(442 \mathrm{~nm}\), and this wavelength lies in the blue-violet part of the visible light spectrum.

Q.2. What is the de Broglie wavelength of an electron which is accelerated through a potential difference of \(10\, \mathrm{kV}\)?

Ans: If the electron having a charge \(\text {e}\) is moving under an external potential \(\text {V}\), then the expression for the de-Broglie wavelength of an electron is:
\(\lambda = \frac{{12.27}}{{\sqrt {\rm{V}} }}{\rm{A}}^\circ \)
We are given, \(\text {V}=10 \mathrm{kV}=10 \times 10^{3} \mathrm{~V}=10^{4} \mathrm{~V}\)
\(\lambda = \frac{{12.27}}{{\sqrt {{{10}^4}} }}{\rm{A}}^\circ \)
\(\lambda = \frac{{12.27}}{{100}}{\rm{A}}^\circ = 0.1227{\mkern 1mu} {\rm{A}}^\circ \)

Summary

According to de-Broglie, the wavelength \(\left( \lambda \right)\) of any moving object is given by: \(\lambda = \frac{{\rm{h}}}{{\rm{p}}},\) Where \(\text {h}\) is Planck’s constant and \(\text {p}\) is the mass of the particle.

The relation between de-Broglie wavelength and the kinetic energy of an object of mass \(\text {m}\) moving with velocity \(\text {v}\) is given as: \(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{mK}}} }}\)

When a charged particle having a charge \(\text {q}\) is accelerated through an external potential difference \(\text {V}\), de-Broglie wavelength, \(\lambda = \frac{{{\rm{hv}}}}{{{\rm{qV}}}}\)

The expression for the de-Broglie wavelength of an electron, \(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{mK}}} }}\) or \(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{meV}}} }}\)

Frequently Asked Questions on De Broglie Wavelength Formula

The most commonly asked questions on De Broglie Wavelength Formula are answered here: