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Wavelength of an Electron

According to de Broglie the wavelength of matter waves is given by

lambda = h/(mnu) = h/p`

Here h is Planck's constant , m is the mass , `nu` is the velocity of the particle , p is the momentum of the particle.

Lets derive the de Broglie wavelength of an electron using the above formula.


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de Broglie wavelength of an electron - derivation


Whenan electron of mass m and charge e is accelerated through a potential difference V, then the energy eV is equal to kinetic energy if the electron.

1/2 mnu^2 = eV `

or    `rArr nu = sqrt( (2eV)/m)`--------------------------> ( 1)

The de Broglie wavelength is `lambda = h/(m nu)`

Substituting the value of `nu`

lambda = h/(m sqrt( (2eV)/m)) 

rArr lambda = h/(sqrt(2meV))`              ------------------------> (2)

Substituting the known values in equation (2) , we get

lambda = 12.27/sqrt(V)` A0

If V = 100 voltrs, then `lambda ` = 1.227 Ao , the wavelength associated with an electron accelerated by 100 volts is 1.227 Ao .

Since E =eV is kinetic energy associated with the electron, the equation (2) becomes

rArr lambda = h/sqrt(2mE)`


de Broglie wavelength of an electron - problems


1) What is tjhe de Broglie wavelength of an electron , if the speed is 105 ms-1. (Given m = 9.1 x 10-31 kg ; v = 105 ms-1 ; h = 6.626 x 10-34 Js)

Solution :

Given:Mass of the electron is m = 9.1 x 10-31 kg ;

Velocity of the electron is `nu` = 105 ms-1 ;

Planck's constant h = 6.626 x 10-34 Js

Formula `lambda = h/(m nu)`

Plugging in the values we get `lambda = (6.626 xx 10 ^(-34))/(9.1 xx 10^(-31) xx 10^5)`

rArr lambda = 72.81 A^0`


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Pro : 2 What is the de Broglie wave length of an electron of kinetic energy 120 eV?

Solution : Given Kinetic e energy = 120 eV = 120 x 1.6 x 10-19 J.

Also for an electron we know that

m = 9.1 x 10-31 kg ;

h = 6.626 x 10-34 Js

Formula `lambda = h/sqrt(2mE)`

Plugging in the known values we get,

rArr lambda = (6.626 xx 10^(-34) )/( sqrt ( 2 xx 9.1 xx 10^(-31) xx 120 xx 1.6 xx 10^(-19)))`

rArr lambda="1.121" xx 10^(-10) m`