Question

An electron in an atom has an angular momentum quantum number of \(2 .\) (a) What is the magnitude of the angular momentum of this electron in terms of \(\hbar ?\) (b) What are the possible values for the \(z\) -components of this electron's angular momentum? (c) Draw a diagram showing possible orientations of the angular momentum vector \(\overrightarrow{\mathbf{L}}\) relative to the z-axis. Indicate the angles with respect to the z-axis.

Short answer

Answer: The magnitude of the angular momentum of the electron is \(\sqrt{6}\hbar\), and the possible values of z-components of its angular momentum are \(-2\hbar\), \(-\hbar\), \(0\), \(\hbar\), and \(2\hbar\).

Step-by-step solution

Calculate the magnitude of angular momentum

To calculate the magnitude of the angular momentum, use the formula for the magnitude of the angular momentum vector: \(L = \sqrt{l(l + 1)}\hbar\) Since l = 2, plug in this value: \(L = \sqrt{2(2 + 1)}\hbar\)

Simplify the magnitude of angular momentum

Simplify the expression to get the magnitude of angular momentum: \(L = \sqrt{2(3)}\hbar = \sqrt{6}\hbar\) So, the magnitude of the angular momentum of this electron is \(\sqrt{6}\hbar\).

Determine possible values of z-components

To determine the possible values of the z-components of the electron's angular momentum, use the formula: \(m_l\hbar\) where \(m_l\) ranges from \(-l\) to \(l\) in integer steps. In our case, \(l=2\), so the possible values for \(m_l\) are -2, -1, 0, 1, and 2. Thus, the possible values for the z-components of angular momentum are: \(-2\hbar, -\hbar, 0, \hbar,\) and \(2\hbar\)

Draw the diagram

To draw the diagram, first sketch the z-axis, then place the possible orientations of the angular momentum vector relative to the z-axis. There are five possible orientations corresponding to the z-components of angular momentum (\(-2\hbar, -\hbar, 0, \hbar,\) and \(2\hbar\)). The angles will depend on the orientation of the vectors relative to the z-axis, but generally, the angles range between \(0^\circ\) and \(180^\circ\). In this diagram, it is essential to show the different orientations of the angular momentum vector \(\overrightarrow{\mathbf{L}}\) corresponding to the different z-components of angular momentum. Indicate the vectors and the angles between the angular momentum vectors and the z-axis to represent the various orientations of the electron's angular momentum.