de Broglie’s hypothesis

de Broglie proposed that the dual nature i.e., both particle nature and wave nature is associated with all the particles in motion. de Broglie suggested that just as light has both particle and wave nature, matter or electrons shall also have dual nature (particle nature and wave nature).

Correlating Einstein’s equation with Planck’s quantum theory, de Broglie concluded his theory. Using de Broglie’s equation he calculated the wavelength of an electron.

λ = \frac{h}{mv}

Where λ = wavelength

m = mass of an electron

v = velocity of electron

h = Planck’s constant

According to de Broglie electrons behave as matter waves of a wavelength λ = \frac{h}{mv} and atomic orbitals they exist as standing waves.

Derivation of de Broglie’s equation

De Broglie’s equation can easily be derived from Planck’s quantum theory and Einstein’s equation of mass-energy equivalence.

According to Planck’s quantum theory, the energy of radiation with frequency “\nu” is given by E = h\nu

Where ‘h’ is Planck’s constant

Einstein’s equation for mass energy equivalence is E = mc^{2}

Where m = mass of light photon

C =  velocity of light

Combining equations (1) and (2)

H\nu = mc^{2}  of \frac{h}{mc} = \frac{c}{v}

But the velocity of light c = \nu \lambda (or) v = \frac{c}{\lambda}

Hence \frac{h}{mc} = \frac{c}{c/ \lambda} or \lambda = \frac{h}{mc}

Similarly for other particles having a velocity ‘v’

\lambda = \frac{h}{mv} = \frac{h}{p}

where p = momentum = mv

Bohr’s theory and de Broglie concept

Bohr assumed electron as a particle and postulated that an electron can revolve only in an orbit in which its angular momentum (mvr) is equal to \frac{nh}{2 \pi}.

According to de Broglie an electron moving around the nucleus in a circular orbit, behaves as a standing or stationary wave. To behave in such away the circumference of the Bohr’s orbit (= 2\pi r) should be equal to the whole number multiple of the wavelength \left ( \lambda \right ) of the electron wave.

2 \pi r = n \lambda

\therefore \lambda = \frac{2 \pi r}{n}

n = whole number

According to de Broglie

\lambda = \frac{h}{mv}

\therefore \frac{2 \pi r}{n}  = \frac{h}{mv} or

mvr = \frac{nh}{2 \pi}

Hence de Broglie’s theory and Bohr’s theory are in agreement with each other.