Explain the dual behaviour of matter. Discuss its significance to microscopic particles like electrons.

Explanation

de - Broglie's hypothesis :The light is found to exhibit wave nature as well as particle nature (dual nature ) . Based on this idea of ' duel nature of light '. De Broglie in 1924 proposed that " all micro -particles including the electron moving with high velcoity are as sociated with dual nature (i.e., ) both particle wave nature .
De- broglie an expression for the wavelength of the moving electron .
Expression for de-Broglie wavelengths :
According to 'Planck's quantum theory' , energy of a photon ,
`E=hupsilon, " But " upsilon=(c)/(lamda)`
`thereforeE=h.(c)/(lamda)......(1)`
Einstein's mass - energy equivalence equation is `E=mc^(2)" " .......(2)`
combining eqns. (1) and (2) ,
`(h.c)/(lamda)=mc^(2)" or " lamda=(h)/(mc)=(h)/(p)`
(p=mc=momentum)
This equations is applicable to photons as well as to all microparticles , moving with high speed .
`therefore` We can write in general, `lamda=(h)/(p)=(h)/(mv)`
where m = mass of the microparticle and v = its velocity
`lamda` , is called , de-Broglie wavelength or material wavelength .
Significance of de - Broglie's concept:
According to Bohr's theory , electron revolves in an orbit in which its angular momentum (mvr) is an integral multiple of `(h)/(2pi) ` . Bohr assumed electron as a particle .
Hence his equation can be taken as `"mvr"=n((h)/(2pi))` where n = a whole number .
According de Broglie , electron behaves as a standing (or stationary ) wave which extends round the nucleus in a circular orbit . If the ends of the electron wave meet to give a regular series of crests and troughs , the electron - wave is said to be 'in phase' . It means , there is constructive interfernce of electron waves and the electron motion has a character of standing wave or non-energy radiating motion . Always it is a necessary condition motion . Always it is a necessary condition to get an electron-wave in phase'such that the circumference fo the Bohr's orbit `(=2pir)` is equal to the whole number of multiple of the wavelength `(lamda)` of the electron -wave .
`nlamda=2pir`
`lamda=(2pir)/(n)`
But `lamda=(h)/("mv")` (de-Broglie)
`therefore(2pir)/(n) = (h)/("mv") " or mvr"=(nh)/(2pi)`

(a) Wave in - phase (b) Wave out -of phase
This is Bohr's equation which stipulates that "the angular momentum of an electron moving round the nuclecus is an integral multiple of `(h)/(2pi)`" . This shows that, de-Broglie's theory and Bohr's theory are in agreement with each other .