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Unformatted text preview: Physics 131 Problem Set 4— Due February 8, 2008 Problem 1 (10 points) Consider a harmonic oscillator with a Hamiltonian of 2 2 2
p mw a:
H = — 1
2m 2 ( )
using a trial wavefunction
N
we) — $2 + b2 (2) a.) Using dimensional analysis, determine how N and b depend on the dimensionful param—
eters of the Hamiltonian (and h). b.) Determine N by demanding that 1/2(a:) is normalized.
c.) Determine b for the ground state. (1.) How does the ground state energy compare with the exact answer. Problem 2 (10 points) Following Section 7.2 in Grifﬁths, ShOW that the ansatz used for He leads to the fact that
H “ is not stable and would decay to H + 6‘. Problem 3 (10 points)
a.) Following the discussion in class, show that the (90?) WKB approximation satisﬁes 1 1
060’; + 5012 + 50,1, 3 0 (3) b.) Using that F = pp’/m (where F = —%¥), show that 1mF 1 m2F2
=——— — d 4
02 4p3+8/ p61” () Notice that the condition that we found in class for the consistency of the WKB approxi—
mation hmF/p3 << 1. c.) Show that this leads to a modiﬁed WKB wave function of the form N 2' hmF 2' #77121172 pdcc 2'
= — 1 — — — — — — d %
g
t
n
E
ti
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by H = 1’1 + ve) mm) = {2” :3 (6) a.) Solve the Schroedinger equation exactly using Airy functions. There is no need to
normalize the wavefunctions. b.) For a neutron, what are the ﬁrst four energy eigenvalues? (This experiment has been
done at a reactor in Grenoble). c.) Using the WKB approximation and implenting the connection formula discussed in class,
solve for the energy eigenvalues for the ﬁrst four levels. d.) Compare the errors between the WKB results with the exact answer to hmF/p3. Problem 5 (10 points) Classically, show that the probability to ﬁnd a particle in a harmonic oscillator potential is
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