Louis de-Broglie thought that similar to the dual nature of light, material particles must also possess the dual character of particle and wave. This means that material particles sometimes behave as particle nature and sometimes behave like a wave nature.
According to de-Broglie –
A moving particle is always associated with a wave, called as de-Broglie matter-wave, whose wavelengths depend upon the mass of the particle and its velocity.
According to Planck’s theory of radiation–
$E=h\nu \qquad(1) $
Where
h – Planck’s constant
$\nu $ - frequency
According to Einstein’s mass-energy relation –
$E=mc^ {2} \qquad (2)$
According to de Broglie's hypothesis equation $ (1)$ and equation $(2)$ can be written as –
$mc^ {2} = h \nu$
$mc^ {2} = \frac{hc}{\lambda }$
$\lambda =\frac{h}{mc}\qquad(3) $
$\lambda =\frac{h}{P}$
Where $P$ –Momentum of Photon
Similarly from equation $(3)$ the expression for matter waves can be written as
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$\lambda=\frac{h}{mv}=\frac{h}{P}\qquad(4)$
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Here $P$ is the momentum of the moving particle.
1.) de-Broglie Wavelength in terms of Kinetic Energy
$K=\frac{1}{2} mv ^{2}$
$K=\frac{m^{2}v^{2}}{2m}$
$K=\frac{P^{2}}{2m}$
$P=\sqrt{2mK}$
Now substitute the value of $P$ in equation $ (4)$ so
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$\lambda =\frac{h}{\sqrt{2mK}} \qquad (5)$
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2.) de-Broglie Wavelength for a Charged particle
The kinetic energy of a charged particle is $K = qv$
Now substitute the value of $K$ in equation$(5)$ so
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$\lambda =\frac{h}{\sqrt{2mqv}}$
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3.) de-Broglie Wavelength for an Electron
The kinetic energy of an electron
$K=ev$
If the relativistic variation of mass with a velocity of the electron is ignored then $m=m_{0}$ wavelength
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$\lambda =\frac{h}{\sqrt{2m_{0}ev}}$
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So wavelength of de-Broglie wave associated with the electron in non-relativistic cases
4.) de-Broglie wavelength for a particle in Thermal Equilibrium
For a particle of mass $m$ in thermal equilibrium at temperature $T@
$K=\frac{3}{2}kT$
Where $K$ – Boltzmann Constant
$\lambda =\frac{h}{\sqrt{2m.\frac{3}{2}kt}}$
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$\lambda =\frac{h}{\sqrt{3mKT}}$
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Properties of matter wave →
- Matter waves are generated only if the material's particles are in motion.
- Matter-wave is produced whether the particles are charged or uncharged.
- The velocity of the matter wave is constant; it depends on the velocity of material particles.
- For the velocity of a given particle, the wavelength of matter waves will be shorter for a particle of large mass and vice-versa.
- The matter waves are not electromagnetic waves.
- The speed of matter waves is greater than the speed of light.
According to Einstein’s mass-energy relation
$E=mc^{2}$
$h\nu = mc^{2}$
$\nu =\frac{mc^{2}}{h}$
Where $\nu$ is the frequency of matter-wave.
We know that the velocity of matter-wave
$ u =\nu \lambda $
Substitute the value of $\nu$ in the above equation
$u =\frac{mc^{2}}{h}. \lambda $
$u =\frac{mc^{2}}{h} . \frac{h}{mv}$
Where $v$ → particle velocity which is less than the velocity of light.
- The wave and particle nature of moving bodies can never be observed simultaneously.
