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