Atomic Structure : Dual nature of matter, de-Broglie’s relationship, Heisenberg uncertainty principle

Dual nature of electron

(1)     In 1924, the french physicist, Louis de Broglie suggested that if light has both particle and wave like nature, the similar duality must be true for matter. Thus an electron, behaves both as a material particle and as a wave.

(2)     This presented a new wave mechanical theory of matter. According to this theory, small particles like electrons when in motion possess wave properties.

(3)     According to de-broglie, the wavelength associated with a particle of mass m, moving with velocity v is given by the relation \[\lambda \,=\,\frac{h}{mv},  where h = Planck’s constant.

(4)     This can be derived as follows according to Planck’s equation, \[E=\,h\nu =\frac{h.c}{\lambda }\,\,\left( \therefore \,\,\,\nu =\frac{c}{\lambda } \right)   energy of  photon (on the basis of Einstein’s mass energy relationship),  E = mc equating both \[\frac{hc}{\lambda }=\,\,mc{}^{2}\,\,or\,\,\lambda =\frac{h}{mc}   which is same as de-Broglie relation. (mc = p)

(5)     This was experimentally verified by Davisson and Germer by observing diffraction effects with an electron beam. Let the electron is accelerated with a potential of V than the Kinetic energy is

\[\frac{1}{2}m{{v}^{2}}=\,\,eV\,\,;\,\,{{m}^{2}}{{v}^{2}}=\,\,\,2eVm

\[mv=\sqrt{2eVm}=\,\,P\,\,;\,\,\lambda =\frac{h}{\sqrt{2eVm}}                            

(6)     If Bohr’s theory is associated with de-Broglie’s equation then wave length of an electron can be determined in Bohr’s orbit and relate it with circumference and multiply with a whole number \[2\pi r=n\lambda \,\,or\,\,\lambda =\frac{2\pi r}{n}

From de-Broglie equation,  \[\lambda =\frac{h}{mv}.  Thus \[\frac{h}{mv}=\frac{2\pi r}{n}  or  \[mvr=\frac{nh}{2\pi }

Note :   For a proton, electron and an α-particle moving with the same velocity have de-broglie wavelength in the following order : Electron > Proton > α – particle.

(7)     The de-Broglie equation is applicable to all material objects but it has significance only in case of microscopic particles. Since, we come across macroscopic objects in our everyday life, de-broglie relationship has no significance in everyday life.

 

Heisenberg’s uncertainty principle

(1)     One of the important consequences of the dual nature of an electron is the uncertainty principle, developed by Warner Heisenberg.

(2)     According to uncertainty principle “It is impossible to specify at any given moment both the position and momentum (velocity) of an electron”.

Mathematically it is represented as,  \[\Delta x\,.\,\Delta p\ge \frac{h}{4\pi }

Where Δx = uncertainty is position of the particle, Δp = uncertainty in the momentum of the particle

Now since Δp = mΔv

So equation becomes,  \[\Delta x.\,m\Delta v\ge \frac{h}{4\pi }\,\,or\,\,\Delta x\,\times \,\Delta v\ge \frac{h}{4\pi m}

The sign ≥ means that the product of Δx and Δp (or of Δx and Δv) can be greater than, or equal to but never smaller than \[\frac{h}{4\pi }.  If Δx is made small, Δp increases and vice versa.

(3)     In terms of uncertainty in energy, ΔE and uncertainty in time Δt, this principle is written as,  \[\Delta E\,.\,\Delta t\ge \frac{h}{4\pi }

Note:      Heisenberg’s uncertainty principle cannot we apply to a stationary electron because its velocity is 0 and position can be measured accurately.