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A free particle has the initial wave function Ψ(x, 0) = Ae−ax2 where A and a...

A free particle has the initial wave function Ψ(x, 0) = Ae−ax2 where A and a are real and positive constants. (a) Normalize it. (b) Find Ψ(x, t). (c) Find |Ψ(x, t)| 2 . Express your result in terms of the quantity w ≡ p a/ [1 + (2~at/m) 2 ]. At t = 0 plot |Ψ| 2 . Now plot |Ψ| 2 for some very large t. Qualitatively, what happens to |Ψ| 2 , as time goes on? (d) Calculate hxi,hpi,hx 2 i,hp 2 i, σx, and σp. (e) Does the uncertainty principle hold? At what time t does the system come closest to the uncertainty limit?

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