Physics 471 Problem Set 6 Fall 2004
22. A particle of mass mmoves in the harmonic oscillator potential V(x) = 1
2mω2x2.
(a) Show that the motion classical motion is restricted to the region −q2E/mω2≤x≤
q2E/mω2.
(b) Using the harmonic oscillator wave functions given in Eq. (2.85), find an expression
for the probability that a particle in the state nwill be found outside the classically
allowed region. Express your result as a dimensionless integral over the variable
ξ=qmω/¯h x.
(c) Use Mathematica to evaluate this probability for n= 0,1,2. Mathematica denotes
the Hermite polynomials Hn(x) by HermiteH[n,x].
23. Using the harmonic oscillator wave functions ψn(x), evaluate
xmn =Z∞
−∞
dxψ∗
m(x)x ψn(x),
with the help of recurrence relation
Hn+1(ξ) = 2ξHn(ξ)−2nHn−1(ξ),
and the orthogonality properties of the ψn(x). Answer:
xmn =s¯h
2mω h√n+ 1 δm n+1 +√n δm n−1i.
24. A free particle has an initial wave function given by
Ψ(x, 0) = Ae−|x|
a,
where aand Aare positive constants.
(a) Determine Ain terms of aby normalizing Ψ(x, 0)
(b) Determine C(k) from
C(k) = 1
√2πZ∞
−∞
dx e −ikxΨ(x, 0) .
(c) Compute
Z∞
−∞
dk |C(k)|2.
25. Compute the average value of the energy of the particle described by the wave function in
the previous problem using
hEi=Z∞
−∞
dk ¯h2k2
2m|C(k)|2.
26. Griffiths Problem 2.23.