Quantum Mechanics Solution Manual |
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© Leon van Dommelen |
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2.3.6 Solution dot-f
Question:
Verify that the most general multiple of
that is normalized on the interval 0
is ![$e^{{\rm i}\alpha}\sin(x)$](img54.gif)
where
is any arbitrary real number. So, using the Euler formula, the following multiples of
are all normalized: ![$\sin(x)$](img44.gif)
, (for
0), ![$-\sin(x)$](img56.gif)
, (for
), and ![${\rm i}\sin(x)$](img58.gif)
, (for
![$\pi$](img57.gif)
2).
Answer:
A multiple of
means
, where
is some complex constant, so the magnitude is
You can always write
as
where
is some real angle, and then you get for the norm:
So for the multiple to be normalized, the magnitude of
must be
1/
, but the angle
can be arbitrary.