ECE 306
Homework Set #1
Spring 2005
(Due 12 noon, Wednesday, February 2, 2005.)
1. Consider the following hypothetic wave function for a particle confined in the region
6
4
≤
≤
−
x
:
a.
Sketch the wave function
b.
Normalize this wave function over the range the particle is confined in (i.e.
find A)
c.
Determine the expectation values < x >, < x
2
> and
σ
2
= < (x  < x >)
2
> using
the normalized wave function.
d.
Again, using the normalized wave function, calculate the expectation value of
the kinetic energy of the particle.
2. Consider the wave function
t
i
x
e
Ae
t
x
ω
λ
ψ
−
−
=
)
,
(
where A,
λ
and
ω
are positive real constants.
a.
Normalize
Ψ
b.
Determine the expectation values of x and x
2
c.
Find the standard deviation of x. Sketch the graph of 
Ψ

2
, as a function of, and
mark the points (< x > +
σ
) and (< x > 
σ
) to illustrate the sense in which
σ
represents the “spread” in x. What is the probability that the particle would be
found outside this range?
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