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.
Indicate how low the energy is needed. 1 ...
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