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Unformatted text preview: University of Colorado, Department of Physics PHYS3220, Fall 09, HW#6 due Wed, Sep 30, 2PM at start of class 1. Momentum space (Total: 15 pts) In class we have defined the momentum space wave function Φ( p,t ) as Φ( p,t ) = 1 √ 2 π ∞ Z-∞ Ψ( x,t )exp(- ipx/ ~ ) dx where Ψ( x,t ) = 1 ~ √ 2 π ∞ Z-∞ Φ( p,t )exp( ipx/ ~ ) dp is a solution of the time-dependent Schr¨ odinger equation in configuration (position) space, that is i ~ ∂ ∂t Ψ( x,t ) = •- ~ 2 2 m ∂ 2 ∂x 2 + V ( x,t ) ‚ Ψ( x,t ) Show that the expectation values of x and p can be written in terms of Φ( p,t ) as: < x > = 1 ~ ∞ Z-∞ Φ * ( p,t ) i ~ ∂ ∂p ¶ Φ( p,t ) dp (1) < p > = 1 ~ ∞ Z-∞ Φ * ( p,t ) p Φ( p,t ) dp (2) 2. Uncertainty Principle Δ x Δ p ≥ ~ / 2 (Total: 10 pts) We never notice the Uncertainty Principle for macroscopic objects. Let’s see how big an effect the Uncertainty Principle produces for an object that is very small but still large as compared to atoms. Consider a 1compared to atoms....
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- Fall '08
- Momentum, Uncertainty Principle, Fundamental physics concepts, wave function, time-dependent schr¨dinger equation