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FreeParticle - FREE PARTICLES I One-dimensional wave...

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FREE PARTICLES 1 I: One-dimensional wave functions A. Free particle with momentum p : 1. Write down the de Broglie relations relating momentum ( ~ p ) to wavelength ( λ ) and energy to frequency. Momentum is a vector but wavelength is not. How do you reconcile this discrepancy? 2. Physicists tend to introduce new terms in order to simplify working with equations. To- day, we will use the following terms: ~ k = (2 π/λ r : wave vector where ˆ r is the direction of propagation of the wave ω = 2 πf : angular frequency ( f is the frequency of a wave) ~ = h/ (2 pi ): A convenient term (pronounced ‘h bar’) which helps keep track of many of the factors of 2 π which appear in quantum mechanics Write the deBroglie relations using the wave vector, the angular frequency, and ~ . 3. What is the general form in one dimension of the wave function for a free particle of mass m and momentum p ? Recall that the wave function has to satisfy the one-dimensional Schr¨ odinger equation: i ~ Ψ( x, t ) ∂t = - ~ 2 2 m 2 Ψ( x, t ) ∂x 2
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