1. (a) Why are crystals used in the measurement of matter waves? [1] (b) Which physical phenomena do matter waves exhibit when scattered off of crystals?

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1. (a) Why are crystals used in the measurement of matter waves? [1] (b) Which physical phenomena do matter waves exhibit when scattered off of crystals? [1] 2. For de Broglie waves (of free particles), show that: (a) the angular frequency and the wave number are related by [3] ω = ~k 2 2m ; (1) (b) vph is equal to halve the speed of the particle; [3] (c) and vg is equal to the speed of the particle. [2] 3. The wavefunction of a certain quantum system is given by Ψ(x, t) = Ae− mω 2~ x 2−iEt/~ , E = 1 2 ~ω, (2) where n = 1, 2, 3, · · · , ω is a constant and −∞ < x < ∞. (a) Show that [3] A = mω π~ 1/4 (3) (b) Compute ∆x and ∆p. [Recall: ∆q = p hq 2i − hqi 2 ] [14] 1
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