Chapter 1st:-Atomic structure and the periodic table
B.sc 1st year Book
(Page 2)

# Wave Mechanical concept of Atoms (De Broglie Hypothesis)

In 1924 a French scientist de Broglie pointed out that similar to light, the electron may also behave both as a wave as well as a particle i.e. it possesses a dual behavior. This improvement involving the wave character of electrons is introduced on the basis of a new branch of science known as quantum mechanics or wave mechanics. Hence, the fundamental idea of wave mechanics is the wave nature of matter.
Thus, according to the wave mechanical concept of matter small particles like electrons, protons or other light particles when in motion possess wave characteristics like wave frequency, wavelength, and wave amplitude besides the particle characteristics of mass and momentum.

## De-Broglie equation :

de Broglie derived and expressed tor by calculating the wavelength, ‘T'(Lamda) of the wave associated with the electron and stated that the momentum (mvr) of a particle in motion is inversely proportional to its wavelength.

Hence

$$mv\:\alpha \:\frac{1}{\lambda }\:or\:mv\:=\:\frac{h}{\lambda }$$

Where ‘h‘ is Plank constant
Proof of de Broglie’s Equation: According to de Broglie’s concept it electron behaves as a wave its energy is given as
E = hv  ……..   (Plank’s equation)
Where E is the energy, y is the frequency of the wave and ‘h’ is the Plank constant But if the electron is considered to have particle characteristics, its energy is expressed as:
E = m.c2    ……..  (Einstein’s equation)
Where ‘c’ is the velocity of light and ‘m’ is the mass of the electron
Equating the equation (i) and (ii) we have
hy = m.d2
Or
$$h\frac{c}{2}\:=\:mc^2$$
Or
$$\left(Since,\:v=\frac{c}{\pi }\right)$$
Or
$$\lambda =\frac{h}{mc}=\frac{h}{p}-….\:+\:….\left(vi\right)$$
Since photons travel in free space with a velocity of light c, its momentum (p) may be written as ;
$$\frac{hv}{mc}=\frac{mc^2}{mc}=c$$
Or
$$\frac{hv}{p}\:=\:cp\:=\:\frac{hv}{c}\:=h\frac{c}{cA^{\circ }}$$
Or
$$\left(Since,\:v=\frac{c}{\lambda }\right)\:$$
Therefore
$$\lambda =\frac{h}{p}=\frac{h}{mc}$$
If a particle of mass ‘m’ moves with velocity ‘v’ then the above equation may be written as :
$$\lambda =\frac{h}{p}=\frac{h}{mv}$$
This equation is called de Broglie’s equation and the wavelength λ is called de Broglie’s wavelength. From this equation, it is evident that the momentum (p) of the moving electron is inversely proportional to the wavelength, λ.

## de-Broglie Relationship and Bohr’s Theory :

Bohr’s theory can be modified by de-Broglie in a more natural way by considering the wave-like properties of the electron in an atom. According to de-Broglie, the electron is not a solid particle revolving around the nucleus in a circular stationary orbit, but it is a standing wave around the nucleus in a circular path. as shown in figure 1.03 .
For the wave to remain continually in a phase, the circumference of the orbit should be an integral multiple of wavelength λ i.e.
where ‘r’ is the radius of the orbit and ‘n’ is a whole number like 1,2,3……
From de-Broglie equation; $$\lambda =\frac{h}{mv}$$ we get
$$2\pi r=n\frac{h}{mv}$$
or (angular momentum) $$mvr=n\frac{h}{2\pi }$$
f(x) (Bohr’s equation).
This equation which is based on the wave nature of the electron shows that
(1) The electron can move only in those orbits for which the angular momentum must be an integral multiple of $$\frac{h}{2\pi }$$. It means the angular momentum is quantized.
Since
$$y_n=\frac{-\left(-1\right)^n\sqrt{\left(n+2\right)}}{2\left(x-1\right)^{n+3}}\:-\frac{\left(-1\right)^n\sqrt{\left(n+1\right)}}{1.\left(x-1\right)^{n+2}}-\frac{\left(-1\right)^n\sqrt{n}}{\left(x-1\right)^{n+1}}+\frac{\left(-1\right)^n\sqrt{n}}{2\left(x-2\right)^{n+1}}$$
$$=\left(-1\right)^{n+1}\sqrt{n\left[\frac{\left(n+2\right)\left(n+1\right)}{2\left(x-1\right)^{n+3}}+\frac{n+1}{\left(x-1\right)^{n+2}}\:+\frac{n}{\left(x-1\right)^{n+1}}\:-\frac{1}{\left(x-1\right)^{n+1}}\right]}$$
$$\left(iii\right)\:Let\:y\:=\frac{x^4}{\left(x-1\right)\left(x-2\right)}$$
Then y = x2 + 3x + 7 + $$\frac{15x-14}{\left(x-1\right)\left(x-2\right)}$$
$$=x^2\:+\:3x+\:7\:+\:\frac{16}{\left(x-2\right)}-\frac{1}{x-1}$$
(by P.F)

$$If\:n>2,\:D^n\left(x^2+3x+7\right)=0,\:$$

Therefore

$$y_n=\frac{16\left(-1\right)^n\sqrt{n}}{\left(x-2\right)^{n+1}}-\frac{\left(-1\right)^n\sqrt{n}}{\left(x-1\right)^{n+1}}$$

$$=\left(-1\right)^n\sqrt{n}\left(\frac{16}{\left(x-2\right)^{n+1}}-\frac{\left(1\right)}{\left(x-1\right)^{n+1}}\right)$$

(2) If the circumference is bigger or smaller than the value as given by equation (ix), the wave is not in phase as shown in the above-given figure. Thus, the de-Broglie hypothesis relation provides a theoretical basis for Bohr’s second postulate.