Assignment 2
1Q
u
e
s
t
i
o
n
1
We’re given
ψ
(
x
)=
±
α
π
²
1
/
4
e
−
αx
2
/
2
.
Before proceeding let’s check that this is normalized correctly since we’ll need
that integral later on anyway. Consider some integral of the form
I
=
Z
+
∞
−∞
dxe
−
ax
2
where
a
is real and positive. The function
e
−
ax
2
is called a Gaussian and this
integral is extremely important. Integrals of this form come up all the time
in physics, especially in statistical mechanics and quantum Feld theory. The
square of
I
is
I
2
=
³ Z
+
∞
−∞
dxe
−
ax
2
´³
Z
+
∞
−∞
dye
−
ay
2
´
=
Z
dxdye
−
a
(
x
2
+
y
2
)
.
This suggests changing variables in the integrand as
x
=
r
cos
θ
,
y
=
r
sin
θ
so that
I
2
=
Z
2
π
0
dθ
Z
∞
0
re
−
ar
2
dr
which can be completed as
I
2
=2
π
"
−
e
−
ar
2
2
a
#
∞
0
=
π
a
so
I
=
r
π
a
.
or
Z
+
∞
−∞
e
−
ax
2
dx
=
r
π
a
(1)
In the case at hand the normalization integral is
Z
dxψ
(
x
)
?
ψ
(
x
)=1
Z
+
∞
−∞
r
α
π
e
−
αx
2
=1
Using (1) it’s trivial to see that
ψ
(
x
) is normalized correctly.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentWhat about the moments? It should be obvious that
Z
+
∞
−∞
xe
−
ax
2
dx
=0
(2)
because this is an odd function integrated over even limits. (Similarly for
x
3
,x
5
and so on.) What about integrals of the form
R
+
∞
−∞
x
2
n
e
−
ax
2
dx
where
n
is an integer? These can be performed with the help of a little trick: take
the derivative of both sides of (1) with respect to
a
.Ige
t
Z
+
∞
−∞
(
−
x
2
)
e
−
ax
2
dx
=
−
1
2
√
πa
−
3
/
2
so that
Z
+
∞
−∞
x
2
e
−
ax
2
dx
=
1
2
r
π
a
3
.
(3)
Youcancomputea
l
ltheevenmoments
R
+
∞
−∞
x
2
n
e
−
αx
2
dx
in this way, by suc
cessively taking derivatives with respect to
a
.
Now, on with the question at hand. We want to compute the expectation
value of the momentum
h
ˆ
p
i
=
Z
+
∞
−∞
ψ
(
x
)
?
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Vachon
 mechanics, Derivative, Trigraph, Partial differential equation, continuity equation, e−ax dx

Click to edit the document details