Matter Waves

Matter Waves

According to de-Broglie a moving material particle sometimes acts as a wave and sometimes as a particle. The wave associated with moving particle is called matter wave or de-Broglie wave and it propagates in the form of wave packets with group velocity.

1) de-Broglie wavelength: According to de-Broglie theory, the wavelength of de-Broglie wave is given by, \(\lambda =\frac{h}{p}=\frac{h}{mv}=\frac{h}{\sqrt{2mE}}\)\(\Rightarrow \lambda \propto \frac{1}{p}\propto \frac{1}{v}\propto \frac{1}{\sqrt{E}}\)

Where, \(h=\) Planck’s constant, \(m=\) Mass of the particle, \(v=\) Speed of the particle, \(E=\) Energy of the particle. The smallest wavelength whose measurement is possible is that of \(\gamma \) rays. The wavelength of matter waves associated with the microscopic particles like electron, proton and neutron and \(\alpha \)– particle etc. is of the order of\({{10}^{-10}}m\).

2) De – Broglie Wavelength associated with the charged particle: The energy of a charged particle accelerated through potential difference \(V\) is\(E=\frac{1}{2}m{{v}^{2}}=qV\). Hence, de – Broglie wavelength, \(\lambda =\frac{h}{p}=\frac{h}{\sqrt{2mE}}=\frac{h}{\sqrt{2mqV}}\)

\({{\lambda }_{Electron}}=\frac{12.27}{\sqrt{V}}\overset{0}{\mathop{A}}\,\) , \({{\lambda }_{Proton}}=\frac{0.286}{\sqrt{V}}\overset{0}{\mathop{A}}\,\)  ; \({{\lambda }_{Deuteron}}=\frac{0.202}{\sqrt{V}}\overset{0}{\mathop{A}}\,\) , \({{\lambda }_{\alpha -Particle}}=\frac{0.101}{\sqrt{V}}\overset{0}{\mathop{A}}\,\)

3) De – Broglie wavelength associated with uncharged particles: For neutron de – Broglie wavelength is given as: \({{\lambda }_{Neutron}}=\frac{0.286\times {{10}^{10}}}{\sqrt{E\left( in\,eV \right)}}m=\frac{0.286}{\sqrt{E\left( in\,eV \right)}}\overset{0}{\mathop{A}}\,\)

Energy of thermal neutrons at ordinary temperature, \(E=kT\)\(\Rightarrow \lambda =\frac{h}{\sqrt{2mkT}}\)

Where, \(T=\) Absolute temperature, \(k=\) Boltzmann’s constant \(=1.38\times {{10}^{-23}}Joule/Kelvin\)

So, \({{\lambda }_{Thermal\,\,Neutron}}=\frac{6.626\times {{10}^{-34}}}{\sqrt{2\times 1.67\times {{10}^{-27}}\times 1.38\times 1{{-}^{-23}}T}}=\frac{30.83}{\sqrt{T}}\overset{0}{\mathop{A}}\,\)

4) Ratio of wavelength of photon and electron: The wavelength of a photon of energy \(E\) is given by, \({{\lambda }_{ph}}=\frac{hc}{E}\)

While the wavelength of an electron of kinetic energy \(K\) is given by, \({{\lambda }_{c}}=\frac{h}{\sqrt{2mK}}\).

Therefore, for the same energy the ratio is, \(\frac{{{\lambda }_{ph}}}{{{\lambda }_{e}}}=\frac{c}{E}\sqrt{2mK}=\sqrt{\frac{2m{{c}^{2}}K}{{{E}^{2}}}}\).