HW3 - Quantum Mechanics I, Physics 3143: Assignment 3, Fall...

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Unformatted text preview: Quantum Mechanics I, Physics 3143: Assignment 3, Fall 2010 1. Griffiths problem 2.7 p. 39 ( Read examples 2.2 and 2.3 before you answer.) 2. Consider the Fourier transform of the wave function Ψ(x, t), ∞ Φ(p, t) = −∞ dx −ipx/ √ e Ψ(x, t), 2π and the inverse Fourier transform ∞ dp ipx/ √ e Φ(p, t). 2π −∞ Note that the variable p has dimensions of momentum. Ψ(x, t) = (a) Show that ∞ ∞ dp|Φ(p, t)|2 , dx|Ψ(x, t)|2 = −∞ −∞ a result known as Parseval’s theorem. Does this suggest a probability interpretation for Φ(p, t) ? Explain. (b) Show that the expectation value ∞ ∂ p(t) = dxΨ (x, t) Ψ(x, t) = i ∂x −∞ ∞ ∗ dp p |Φ(p, t)|2 . −∞ Is the probability interpretation of this result is consistent with that of part (a) ? (assume the x-integral form and show it reduces to the p-integral form ) √ 3. A particle of mass m in a box of size a is prepared in the state Ψ(x, 0) = eiQx/ / a, where the real parameter Q has dimensions of momentum. This state has maximum uncertainty in position (why ?). ( At the edge of the box, x = 0 and x = a we immagine that Ψ(x, 0) drops rapidly to zero in order to satisfy the boundary conditions. ) (a) Show that the wave function in momentum space is given by Φ(p, 0) = a −i(p−Q)a/(2 ) (p − Q)a e sinc , 2π 2 where sinc(x) ≡ sinx/x. Sketch |Φ(p, 0)|2 carefully versus p. (b) Calculate the expectation value of momentum using Φ(p, 0). (c) Calculate the uncertainty in momentum using using Φ(p, 0). Explain your answer. (d) Using part (a) show that it follows immediately that integral is over the range −∞ < ξ < ∞. 1 dξsinc2 (ξ ) = π, where the ...
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