The square of the angular momentum of a hydrogen atom is measured to be ![{\displaystyle L^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912)
. What are the possible values of the z-component of the orbital angular momentum,
, that could be measured for this atom?
Solution:
The eigenvalues of the square of the angular momentum operator (
2) for a quantum mechanical system, such as an electron in a hydrogen atom, are given by:
where
are the spherical harmonics, which are eigenfunctions of
2,
is the orbital quantum number, and
is the magnetic quantum number. The given value for
= 20 ħ2 , so we set up the equation:
Dividing by
and simplifying, we get:
In this quadratic equation,
can be factored to get:
Since
must be a non-negative integer. The magnetic quantum number
can take on any integer value from
to
, thus for
,
can be:
The z-component of the angular momentum,
, is quantized in units of
and given by:
.
Therefore, the possible values of the z-component of the orbital angular momentum,
, that could be measured for the atom with a given ![{\displaystyle L^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912)
are
.