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Unformatted text preview: PHYS4221 Quantum Mechanics I Fall term of 2010 Problem Set No.3 (Due on October 15, 2010) 1. If we define the probability particle density ρ and probability current density J as follows: ρ ( x,t ) = | ψ ( x,t ) | 2 J ( x,t ) = ¯ h 2 mi ( ψ * ( x,t ) ∂ ∂x ψ ( x,t )- ψ ( x,t ) ∂ ∂x ψ * ( x,t ) ) , then show that they satisfy the continuity equation: ∂ρ ∂t + ∂J ∂x = 0 . [Note that the above definitions and continuity equation can be generalized to the 3-D case in a straightforward manner.] 2. A free particle has the initial wave function Ψ( x, 0) = A exp n- ax 2 o , (a) Normalize Ψ( x, 0). (b) Find Ψ( x,t ). (c) Sketch | Ψ( x,t ) | 2 (as a function of x ) at t = 0, and again for very large t . Qualitatively, what happens to | Ψ( x,t ) | 2 as time goes on? (d) Find h x i , h p i , h x 2 i , h p 2 i , σ x and σ p . (e) Does the uncertainty principle hold? At what time t does the system come closest to the uncertainty limit?...
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