Louis de Broglie postulated that electrons can behave as both particles and waves. He derived the de Broglie relation that shows the wavelength of a particle is inversely proportional to its momentum. This relationship uses Planck's constant and can be applied to electrons or any moving matter. The de Broglie wavelength allows relating the circumference of an electron's orbit to an integer multiple of its wavelength, providing insight into quantized electron orbits around an atom's nucleus.

1. de Broglie Wavelength
Dr. Mithil Fal Desai
Shree Mallikarjun and Shri
Chetan Manju Desai College
Canacona Goa

2. Dual Nature of electron
Louis de Broglie (1924)
postulated that some
time electron act as
particle and some
time as wave
e-

3. Calculating de Broglie wavelenth
Planck’s quantum theory,
E = ℎ𝑣
Einstein
E = mC2
Thus,
𝐸 = ℎ𝑣 = 𝑚𝐶2
𝜆 =
ℎ
𝑚𝐶
Here, h is plank constant, 𝑣 is frequency, m is mass and C is speed of light

4. Calculating de Broglie wavelength
Mass and speed of light can be replaced by a mass of matter and velocity, we have or momentum
(p)
𝜆 =
ℎ
𝑚𝑣
=
ℎ
𝑝
… de Broglie relation
A circular wave is completely in a phase, the circumference of the orbit essentially be equivalent
to the integral multiple of wavelength (λ)
Therefore Circumference,
2πr = nλ
here n is integer and r is radius
λ =
2πr
𝑛
=
ℎ
𝑚𝑣
λ =
2πr
𝑛
=
ℎ
𝑚𝑣
Rearranging
mvr =
𝑛ℎ
2π
2πr = nλ
2πr ≠nλ

5. Summary
𝒎𝒗𝒓 =
𝒏𝒉
𝟐𝝅
Bohr suggests that n is the principal quantum
number, the number of waves an electron creates in a
particular orbit is equal to the principal quantum
number of the orbit.