# hw3dd - PHYS 507 Homework III Assigned Thursday Due Monday...

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PHYS 507 Homework III Assigned: October 30, 2003, Thursday. Due: November 10, 2003, Monday, at 5:00 pm. 1. We have said in class that the “equation of motion” for the average position, h x i t , of a particle with the Hamiltonian H = p 2 / 2 m + V ( x ) is m d 2 dt 2 h x i t = - h V 0 ( x ) i t , and this is the classical equation of motion if h V 0 ( x ) i t V 0 ( h x i t ). Whenever this approximation is good we are in the “classical limit” where use of quantum mechan- ics may be unnecessary. If the approximation is bad then we have to use quantum mechanics. (a) Consider a wave packet centered at h x i t = a with a width of Δ x . By carrying out a Taylor series expansion of V 0 ( x ) around a , calculate the lowest-order term in the difference between h V 0 ( x ) i t and V 0 ( h x i t ). Calculate the percentage error made by replacing quantum mechanical expression with a classical one. How large should Δ x be in order to be in the classical limit? (b) Consider specifically the potential for a one-dimensional Coulomb problem, V ( x ) = - A/ | x | . When can we say that the classical solution for h x i t is good?

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