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Quantum Chemistry/Example 26
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The square of the angular momentum of a hydrogen atom is measured to be . 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 are .