To normalize the wavefunction
�
−
�
�
e
−ax
in the range
�
≤
�
≤
∞
θ≤x≤∞, where
�
>
0
a>0, we need to find the normalization constant. The normalization condition for a wavefunction
�
(
�
)
ψ(x) is given by:
∫
−
∞
∞
∣
�
(
�
)
∣
2
�
�
=
1
∫
−∞
∞
∣ψ(x)∣
2
dx=1
For
�
−
�
�
e
−ax
, we have:
∫
�
∞
∣
�
−
�
�
∣
2
�
�
=
1
∫
θ
∞
∣e
−ax
∣
2
dx=1
∫
�
∞
�
−
2
�
�
�
�
=
1
∫
θ
∞
e
−2ax
dx=1
To evaluate this integral, we can use the integral of an exponential function:
∫
�
�
�
�
�
=
1
�
�
�
�
+
�
∫e
cx
dx=
c
1
e
cx
+C
Applying this to our integral:
∫
�
∞
�
−
2
�
�
�
�
=
1
−
2
�
�
−
2
�
�
∣
�
∞
∫
θ
∞
e
−2ax
dx=
−2a
1
e
−2ax
∣
∣
θ
∞
Since
�
>
0
a>0, as
�
x approaches
∞
∞,
�
−
2
�
�
e
−2ax
approaches 0. Therefore, the upper limit of the integral is 0:
1
−
2
�
�
−
2
�
�
∣
�
∞
=
1
−
2
�
�
−
2
�
�
−2a
1
e
−2ax
∣
∣
θ
∞
=
−2a
1
e
−2aθ
For the normalization condition to hold, this expression should equal 1:
1
−
2
�
�
−
2
�
�
=
1
−2a
1
e
−2aθ
=1
Rearranging the equation:
−
2
�
=
�
2
�
�
−2a=e
2aθ
Taking the natural logarithm of both sides:
2
�
�
=
ln
(
−
2
�
)
2aθ=ln(−2a)
�
�
=
1
2
ln
(
−
2
�
)
aθ=
2
1
ln(−2a)
Since
�
>
0
a>0, it is not possible to obtain a real value for
�
�
aθ using this wavefunction. Therefore, the wavefunction
�
−
�
�
e
−ax
cannot be normalized in the given range
�
≤
�
≤
∞
θ≤x≤∞.
Now let's examine the two functions provided:
(i) sin(ax)
(ii) cos(ax)e^(-x^2)
For a function to be a valid wavefunction, it must satisfy certain criteria, such as being continuous, single-valued, and square-integrable. To determine if these functions can be normalized, we need to check if their square integrals converge.
(i) sin(ax):
∫
−
∞
∞
∣
sin
(
�
�
)
∣
2
�
�
∫
−∞
∞
∣sin(ax)∣
2
dx
This integral is finite and can be normalized, as the square of the sine function integrates to a constant value.
(ii) cos(ax)e^(-x^2):
∫
−
∞
∞
∣
cos
(
�
�
)
�
−
�
2
∣
2
�
�
∫
−∞
∞
∣cos(ax)e
−x
2
∣
2
dx
This integral is also finite and can be normalized, as the exponential decay of
�
−
�
2
e
−x
2
compensates for the oscillations of the cosine function.
In conclusion, both functions sin(ax) and cos(ax)e^(-x^2) can be normalized and are acceptable as wavefunctions.