18.4 Wave Nature of Particles
It has been observed that light displays a dual nature, it acts as a wave and it acts as a particle. Assuming symmetry in nature, the French physicist, Louis de Broglie proposed in 1924 that particles should also possess wave-like properties. As momentum p of photon is:
p=λh (Eq. 18.3)
de Broglie suggested that momentum of a material particle of mass m moving with velocity v should be given by the same expression. Thus,
p=λh=mv
or
λ=ph=mvh…(18.10)
where λ is the wavelength associated with particle waves. Hence, an electron can be considered to be a particle and a wave. Equation 18.9 is called de Broglie relation.
An object of large mass and ordinary speed has such a small wavelength that its wave effects such as interference and diffraction are negligible. For example, a rifle bullet of mass 20g flying with speed 330ms−1 will have a wavelength λ given by
λ=mvh=2×10−2kg×330ms−16.63×10−34Js=1×10−34m
This wavelength is so small that it is not measurable or detectable by any of its effects.
On the other hands, for an electron moving with a speed of 1×106ms−1:
λ=9.1×10−31kg×1×106ms−16.63×10−34Js=7×10−10m
This wavelength is in the X-rays range. Thus, diffraction effects for electrons are measurable whereas interference effects for bullets are not.
Example 18.3
Find the wavelength of two photons produced when a positron-electron pair annihilates. The rest mass energy of each photon is 0.51MeV.
Solution
E=0.51MeV
E=0.51×106ev×1.6×10−19J/eV=8.16×10−14J
Wavelength λ=?
As E=hf=λhc or λ=Ehc
Putting the values
λ=8.16×10−14J6.63×10−34Js×3×108ms−1
λ=2.43×10−12m=2.43pm
