qmhw2 - PHY4221: Homework 2 Due Sep. 18, 2008 1. A particle...

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Unformatted text preview: PHY4221: Homework 2 Due Sep. 18, 2008 1. A particle of mass m is confined inside the box −a ≤ x ≤ a by infinite potential walls at the boundaries. The state of the particle is in the ground state. At time t = 0 the box suddenly expands (symmetrically) to twice its size. (a) Calculate the wave function of the particle as a function of time. Your answer may be expressed in terms of a summation of energy eigenfunctions of the new box. (b) What is the (time-dependent) probability of finding the particle in the ground state of the old box? (c) Describe how fast the walls should move in order to achieve the sudden expansion. 2. Show that a bound state exists for an attractive 1D finite square well problem, i.e., V (x) = −|V0 |θ(a − |x|). Then show that a bound state also exists in any (arbitrary) 1D attractive potential with the properties: V (x) = −|V (x)| and V (x) → 0 as x → ±∞. 3. For the double delta-potential problem with V (x) = gδ (x − a) + gδ (x + a). (a) Find the bound state if g < 0, if exists, (b) Find some (not all) free eigenstates of the system. 4. A 1D quantum system can be considered as a limiting case of certain 3D problems. Consider a particle confined in an infinitely long hollow tube with a square cross section of length L inside. If the walls of the tube are treated as infinite potentials, calculate the energy eigenvalues and the corresponding eigenfunctions of the 3D system. Then try to set up a 1D time-dependent Schr¨dinger equation if the energy particle is sufficiently low. o Indicate how low the energy is needed. 1 ...
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