De Broglie Wavelength Formula

Light is electromagnetic radiation that acts like a wave as a particle as well. De Broglie wavelength explains this dual existence of light by explaining the nature of the wave in relation to the particle is explained by De-Broglie waves. In other words, it connects the wavelength of a wave or particle to its momentum. It states that the wavelength of a particle is inversely proportional to the mass and velocity. The wavelength of a particle is denoted by the symbol λ. Its unit of measurement meters (m) and the dimensional formula are given by [M⁰L¹T⁰]. Its formula equals the ratio of Plank’s constant to the product of mass and velocity of the particle.

De Broglie Wavelength Formula

λ = h/mv

Where,

• λ is the De Broglie wavelength,
• h is Plank’s constant with the value of 6.62 × 10⁻³⁴ Js,
• m is the mass,
• v is the velocity of the particle.

Derivation of the De Broglie Wavelength 

The De Broglie wavelength of a particle is derived by using the formulas for its energy. Consider a photon of mass m with energy as E, wavelength as λ and velocity equal to speed of light, c. The energy (E) of a photon is given as,

E = hc/λ ⇢ (1)

Also we know that,

E = mc²  ⇢ (2)

Equating (1) and (2) we get, 

hc/λ = mc²

h/λ = mc

λ = h/mc

For a particle with velocity v (less than c) the formula becomes,

λ = h/mv or λ = p

This derives the formula for De Broglie wavelength of a particle.

Sample Problems

Problem 1: Calculate the wavelength of an electron moving with a velocity of 100 m/s.

Solution:

We have,

m = 9.1 × 10^-31

v = 100

Using the formula we get,

λ = h/mv

= (6.62 × 10⁻³⁴) / (9.1 × 10^-31 × 100)

= 7281 nm

Problem 2: Calculate the wavelength of an electron moving with a velocity of 40 m/s.

Solution:

We have,

m = 9.1 × 10^-31

v = 40

Using the formula we get,

λ = h/mv

= (6.62 × 10⁻³⁴) / (9.1 × 10^-31 × 40)

= 18203.57 nm

Problem 3: Calculate the wavelength of a particle of mass 2 × 10^-29 kg moving with a velocity of 10 m/s.

Solution:

We have,

m = 2 × 10^-29

v = 10

Using the formula we get,

λ = h/mv

= (6.62 × 10⁻³⁴) / (2 × 10^-29 × 10)

= 3313.05 nm

Problem 4: Calculate the velocity of a particle of mass 2 × 10^-29 kg and wavelength of 3313 nm.

Solution:

We have,

m = 2 × 10^-29

λ = 3313 × 10⁻⁹

Using the formula we get,

λ = h/mv

=> v = h/mλ

= (6.62 × 10⁻³⁴)/(2 × 10^-29 × 3313 × 10^-9)

= 10 m/s

Problem 5: Calculate the velocity of a particle of mass 4.5 × 10^-27 kg and wavelength of 2.72 nm.

Solution:

We have,

m = 4.5 × 10^-27

λ = 2.72

Using the formula we get,

λ = h/mv

=> v = h/mλ

= (6.62 × 10⁻³⁴)/(4.5 × 10^-27 × 2.72 × 10^-9)

= 54 m/s

Problem 6: Calculate the velocity of a particle of mass 3.2 × 10^-28 kg and wavelength of 27.60 nm.

Solution:

We have,

m = 3.2 × 10^-28

λ = 27.60

Using the formula we get,

λ = h/mv

=> v = h/mλ

= (6.62 × 10⁻³⁴)/(3.2 × 10^-28 × 27.60 × 10^-9)

= 75 m/s

Problem 7: Calculate the wavelength of a particle if its momentum is 2 × 10²⁴ kg m/s.

Solution:

We have,

p = 2 × 10²⁴

Using the formula we get,

λ = h/p

= (6.62 × 10⁻³⁴)/(3.2 × 10^-28)

= 0.331305 nm