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23 November, 11:15

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

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  1. 23 November, 11:27
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    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
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