The de Broglie relation λ=h/p can be rewritten in terms of the wave number k as p=kℏ. Recall that wave number is defined by k=2π/λ. Using the fact that λ=v/f, find the wave numbers k1 and k2 corresponding to frequencies f1 and f2. Express your answer as two expressions

k₁ = 2πf₁/v

k₂ = 2πf₂/v

Explanation:

Since the de Broglie relation λ=h/p (1) and momentum, p = kℏ (2)

From (1) p = h/λ = kℏ

So, h/λ = kℏ

Hence, k = h/ℏλ since ℏ = h/2π and wavelength, λ = v/f, substituting these two into k, we have

k = h/[ (h/2π) (v/f) ]

k = 2πf/v where k = wave number, f = frequency of wave and v = velocity of wave.

Now, for the first wave number k₁, k₁ = 2πf₁/v

for the second wave number k₂, k₂ = 2πf₂/v