Calculate the angular momentum of an electron in a hydrogen atom.
To calculate the angular momentum of an electron in a hydrogen atom, we need to consider the quantum mechanical properties of the electron's motion within the atom. The angular momentum operator (\(L\)) is a fundamental operator in quantum mechanics, and its magnitude for an electron in a hydrogen atom can be determined using the following formula:
\[L = \sqrt{l(l+1)}\hbar\]
Where:
- \(L\) is the magnitude of the angular momentum.
- \(l\) is the orbital quantum number, which represents the shape of the electron's orbital.
- \(\hbar\) (h-bar) is the reduced Planck constant, approximately equal to \(1.0545718 \times 10^{-34}\) J·s.
In the case of a hydrogen atom, the orbital quantum number (\(l\)) can take integer values ranging from 0 to \(n-1\), where \(n\) is the principal quantum number. Each value of \(l\) corresponds to a specific orbital shape:
- \(l = 0\) corresponds to an s-orbital, which has a spherical shape.
- \(l = 1\) corresponds to a p-orbital, which has a dumbbell-like shape with three orientations along the x, y, and z axes.
- \(l = 2\) corresponds to a d-orbital, which has more complex shapes.
- \(l = 3\) corresponds to an f-orbital, which has even more complex shapes, and so on.
For a hydrogen atom with \(n = 1\), which corresponds to the ground state, the only possible value for \(l\) is 0 (s-orbital). Therefore, in this case, the angular momentum of the electron is:
\[L = \sqrt{0(0+1)}\hbar = 0\hbar = 0\]
So, for an electron in the ground state of a hydrogen atom, the magnitude of its angular momentum is zero. This implies that the electron is in an s-orbital, and its motion is spherically symmetric around the nucleus, with no preferred direction of rotation.
It's important to note that the angular momentum of an electron in a hydrogen atom is quantized, meaning it can only take on specific, discrete values determined by the quantum numbers. In the ground state, as shown above, the angular momentum is zero, but in excited states with higher values of \(n\) and \(l\), the angular momentum will be nonzero and quantized accordingly.