In quantum mechanics,
and
The angle
between
and the
axis is given by
To make
as small as possible,
must be at its maximum value,
Therefore, the minimum angle
is given by
or by
We now solve
to find
Since
will be very large, we make the approximation,
so that
Because
is large, we see that
is small, and thus we may make the approximation,
If we now substitute in
and
we obtain
This is the smallest angle that
can make with the
axis in the case of the Earth going around the sun.
In the case of a quantum particle with
, we must use the exact expression for the angle.
which gives us
This is the smallest angle that the angular momentum vector of a particle with
can make with the
axis. This angle is much larger than that for the Earth orbiting the sun, as we would expect.
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