Exploring the Connection: The Link between De Broglie Wavelength and Crompton Wavelength
To find out the relationship between the De.broglie Wavelength (λd) and Crompton wavelength (λc), first we need to understand their proper meaning.
When a material particle is associated with a wave is called Mater Wave and is associated with a wavelength is called De.broglie Wavelength(λd).
i.e λ = h/p = h/mv (since p=mv)
The λd can be considered in 2 cases viz. Relativistic (K.E of particle >> Rest mass Energy) and Non-relativistic (K.E of particle << Rest mass Energy). But, Relativistic case is more general in De.broglie wavelength. i.e
E = √(p²c²+m•²c⁴)
E² = p²c²+m•²c⁴
p²c²= E²- m•²c⁴
p²c²= E²- E•²
Thus,
λd = hc/pc = hc/√(E²- E•²)— — — (1)
But, according to Einstein, we have
m = γm• = m•/√(1-v²/c²) &
E = γE• = E•/√(1-v²/c²)
Now,
λd = hc/√(E²-E•²)
λd = hc/√{E•²(γ²-1)}
λd = hc/m•c {1/γ²-1}
λd = λc {1/γ²-1} — — — (2)
Where, λc = Crompton wavelength
[ we know,
γ²-1 = 1/(1-v²/c²)-1
or γ²-1 = v²/c²-v² ]
λd = λc {c²-v²/v²} — — — (3)
Equation (3) represents the relationship between Crompton wavelength and De.broglie wavelength.
