de-Broglie Hypothesis and Wavelength of Matter Wave:
A natural question arises : As radiation has dual (wave particle) nature, the particles of nature (the electrons, protons, etc), also exhibit wave-like character? In 1924, the French Physicist de-Broglie (pronounced as de brog) (1892-1987) put forward the bold hypothesis that moving particles of matter should display wave like properties under suitable conditions. He reasoned that nature was symmetrical and that the two basic physical entities-matter and energy, must have symmetrical character. If radiation shows dual aspects, so should matter. de-Broglie proposed that the wavelength λ. associated with a particle of p momentum is given by the relation
λ = \(\frac{h}{p}=\frac{h}{m v}\) …………….. (1)
Where m is the mass of the particle and v its speed. Equation (1) is known as the de-Broglie relation and the wavelength A of the matter wave is called de-Broglie wavelength. The dual aspect of matter is evident in the de-Broglie relation (1), A is the attribute of a wave while on the right hand side the momentum p is a typical attribute of a particle. Planck’s constant h relates the two attributes. Equation (1) for a material particle is basically a hypothesis whose validity can be tested only by experiment. However, it is interesting to see that it is satisfied also by a photon for a photon, as we have seen
Clearly from equation (1), λ is smaller for a heavier particle (large m) or more energetic particle (large v). For example, the de-Broglie wavelength of a ball of mass 10 gm moving with a speed of 2 ms-1 is easily calculated as
This wavelength is so small that it is beyond any measurement. This is the reason why macroscopic objects in our daily life do not show wave-like properties, on the other hand, in the sub-atomic domain, the wave character of particles is significant and measurable.
Consider an electron [mass (m) charge, (e)] accelerated from rest through a potential V. The kinetic energy K of the electron equal to the work done (eV) on it.
By the electric field
The variation of potential with wave length
Wavelength of Matter Waves Associated with Different Types of Particles
Now we will find out the wavelength of matter waves related to different particles.
(a) If m is the mass and q is charge on any particle and it is accelerated by potential V, then K = qV
(b) For uncharged particles like neutron and gas atoms
Mean kinetic energy of electron
K = \(\frac{3}{2}\)kT
where k = 1.38 × 10-23 J/K is Boltzmann constant
