Dual Nature Of Electron

**Category : **JEE Main & Advanced

(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, \[E=\,h\nu =\frac{h.c}{\lambda }\] \[\left( \because \ \ \nu =\frac{c}{\lambda } \right)\]

energy of photon (on the basis of Einstein’s mass energy relationship), \[E=\,mc{}^{2}\]

Equating both \[\frac{hc}{\lambda }=\,\,mc{}^{2}\,\,or\,\,\lambda =\frac{h}{mc}\] which is same as de-Broglie relation. \[\left( \because \ \ mc=p \right)\]

(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

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

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

(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 \[mvr=\frac{nh}{2\pi }\]

(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.

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