De Broglie Wavelength Formula: Equation and Derivation

De Broglie Wavelength Formula is a formula that defines the nature of a wave to that of a particle. Many experiments show that light can behave both as a wave and as a particle. The particles of light are known as photons. A French physicist named Louis de Broglie created a formula in 1924 to define the dual nature of light as a wave and as a particle. This formula is also used for electrons as well as protons. 

Key Takeaways: De Broglie wavelength, Photons, Planck’s constant, Electron, Proton, Light, Particle, Wave, Wavelength, Speeds, momentum

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De Brogile Wavelength

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Louis de Broglie stated that any particle that moves will behave like a wave. This experiment was later demonstrated by Clinton Davisson and Lester Germer in 1927. The waves that are related to matter are called matter waves. They are also known as De Broglie waves. 

Particles like electrons and protons except photons have a different de Broglie wavelength formula. At non-relativistic speeds, the momentum of a particle would be equal to its rest mass m multiplied by the velocity v.

De Broglie Wave

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De Brogile Wavelength Formula

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According to De Broglie, the wavelength which is represented by (λ) of any moving object is given by

Or

Where

λ is the de Broglie wavelength in metres

h is the Planck’s constant that is 6.63 × 10^-34 Js

p is the momentum of a particle in kg m s^-¹

m is the mass of the particle in kg

v is the velocity of a particle ms^-¹

De Broglie Wave

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De Broglie Equation Derivation

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From Einstein’s theory of relativity, we get that

E = mc² → Equation (1)

Where

E is the energy of the particle

m is the mass of the particle

c is the speed of the light

According to Planck’s theory, every quantum of a wave has a separate amount of energy associated with it, the following equation was given

E = hf → Equation (2)

Where

E is the energy of the particle

h is the Planck’s constant = 6.62607×10^-34 Js

f is the frequency

De Broglie’s Wavelength Equation suggests that particles and waves have similar behaviour. Thus, he equated the relation of energy for both the particle and the wave; by simplifying equations (1) and (2)

mc² = hf → Equation (3)

If λ is the wavelength of the wave, then the frequency will be f = v / λ

Substituting this in equation (3), we get

mv² = hv / λ

λ = h / mv

Or

λ = h / p → Equation (4)

where p is the momentum of the particle.

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De Brogile Wavelength of an Electron

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Matter waves that are related to real objects are very small and are useless to us. But for subatomic particles with negligible masses, the value of de Broglie is significant. To find the de-Broglie wavelength related to a microscopic particle.

Let us take the mass of the electron as

m = 9.1×10^-³¹kg, 

moving with the speed of light, i.e., 

c = 3 × 10⁸m/s

Then the De Broglie wavelength associated with it can be given as:

λ = h / mc

λ = 6.62607×10^-34 Js / 9.1×10^-³¹kg x 3 × 10⁸m/s

= 0.7318×10^-¹¹ m

Or

0.073 A^o

Thus, the de-Broglie wavelength associated has a significant value, and it can be found out.

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Things to Remember

• Matter can have a dual nature i.e. it can behave both as a wave and as a particle.
• De Broglie Wavelength Formula defines the nature of a wave to that of a particle.
• De Broglie Wavelength Formula: λ = h / p Or λ = h / mv
• De Broglie Wavelength is a part of quantum mechanics. 
• De Broglie Wavelength states the probable density of finding the object at a given configuration space point. 

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