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Unformatted text preview: University of Colorado, Department of Physics PHYS3220, Fall 09, HW#7 due Wed, Oct 7, 2PM at start of class 1. Momentum operator (Total: 10 pts) Since the momentum p is an observable, its expectation value < p > should be a real value. However, the complex factor (- i ~ ) of the momentum operator ˆ p =- i ~ ∂ ∂x raises the question whether the expectation value of p in some quantum states could be complex. Show that, in fact, the imaginary part of the expectation value of p is always zero. (Hint: How do you write the imaginary part of a number z with the help of z and z * ?) 2. Probability current density (Total: 10 pts) Show the following relations for the probability current density J ( x,t ) = i ~ 2 m Ψ( x,t ) ∂ Ψ * ( x,t ) ∂x- Ψ * ( x,t ) ∂ Ψ( x,t ) ∂x ¶ a) J ( x,t ) = ~ m Im (Ψ * ( x,t ) ∂ Ψ( x,t ) ∂x ) b) J ( x,t ) = 1 m Re (Ψ * ( x,t )ˆ p x Ψ( x,t )) 3. Griffiths, problem 1.16 (Total: 10 pts) Consider any two normalizable solutions of the time-dependent Schr¨...
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- Fall '08
- Momentum, expectation value, time-dependent schr¨dinger equation, probability current density