hw5 - PHYS 507 Homework V(Fall'05 Assigned November 7 2005...

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PHYS 507 Homework V (Fall ’05) Assigned: November 7, 2005, Monday. Due: November 16, 2005, Wednesday, at 5:00 pm. Note: First Midterm exam is on November 19, Saturday at 14:00 somewhere in the Physics Dept. 1. The Virial Theorem: Consider the Hamiltonian for a particle in 1D, H = p 2 2 m + V ( x ) = T + V , where T is the kinetic energy operator. We assume that the potential V ( x ) is of such a nature that H has bounded eigenstates | ϕ n i with discrete energies E n . In that case | ϕ n i can be normalized to unity, h ϕ n | ϕ n i = 1. Denote the expectation values in state | ϕ n i by h i n . (a) Show that h [ H, A ] i n =0 for any operator A . (b) Consider the operator A = xp + px . Compute [ T, A ] and [ V, A ]. Using these expressions and part (a), find a relationship between h T i n and h xV 0 ( x ) i n (this is called the virial theorem). (c) Apply the theorem to potentials of the form V ( x ) = c | x | α , where c and α are constants, and express h T i n and h V i n in terms of the energy eigenvalue E n .
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