What is the ratio of de-Broglies wavelengths of electron and proton moving with same velocities?

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Find the ratio of the de Broglie wavelengths of an electron and a proton when both are moving with the (a) same speed, (b) the same kinetic energy, and (c) the same momentum. State which of the two will have a longer wavelength in each case.

Data: mp = 1836 me

(a) The de Broglie wavelength, λ = `"h"/"p" = "h"/"mv"`

`lambda_"e"/lambda_"p" = ("m"_"p"/"m"_"e")("v"_"p"/"v"_"e")` = 1836 as vp = ve

Thus, λe < λp.

(b) λ = `"h"/"p" = "h"/sqrt"2mK"`, where K denotes the kinetic energy `(1/2 "mv"^2)`

∴ `lambda_"e"/lambda_"p" = sqrt(("m"_"p" "K"_"p")/("m"_"e""K"_"e")) = sqrt("m"_"p"/"m"_"e") = sqrt1836 = 42.85`

as Kp = Ke

Thus, λe > λp.

(c) λ = `"h"/"p"` 

∴ `lambda_"e"/lambda_"p" = "p"_"p"/"p"_"e" = 1` as pp = pe.

Concept: De Broglie Hypothesis

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