Problem 50 - Physics for Scientists and Engineers with Modern Physics

Question

(II) For an electron in a \(5 g\) state, what are all the possible values of \(j, m_{j}, J,\) and \(J_{z} ?\)

Step-by-Step Solution

Answer

j = \(\frac{9}{2}\), \(\frac{7}{2}\); m_j = range of integers from -j to +j; J = j; J_z = m_j values.

Step 1: Understanding the 5g State

The exercise specifies an electron in a "\(5g\)" state. This tells us that the principal quantum number \(n = 5\) and the azimuthal quantum number \(l = 4\), because \(g\) corresponds to \(l = 4\) in spectroscopic notation.

Step 2: Finding Possible Values of j

The total angular momentum quantum number \(j\) is determined using the equation \(j = l \pm \frac{1}{2}\). For \(l = 4\), the possible \(j\) values are \(j = 4 + \frac{1}{2} = \frac{9}{2}\) and \(j = 4 - \frac{1}{2} = \frac{7}{2}\). Thus, the possible values for \(j\) are \(\frac{9}{2}\) and \(\frac{7}{2}\).

Step 3: Finding Possible Values of m_j

Each \(j\) value corresponds to a set of possible \(m_j\) values, which range from \(-j\) to \(+j\) in integer steps. For \(j = \frac{9}{2}\), \(m_j\) ranges from \(-\frac{9}{2}\) to \(+\frac{9}{2}\). For \(j = \frac{7}{2}\), \(m_j\) ranges from \(-\frac{7}{2}\) to \(+\frac{7}{2}\).

Step 4: Understanding J and J_z

For a single electron, \(J\) and \(J_z\) are often used interchangeably with \(j\) and \(m_j\). Thus, any value computed for \(j\) and \(m_j\) also holds for \(J\) and \(J_z\). So \(J\) can be \(\frac{9}{2}\) or \(\frac{7}{2}\), and \(J_z\) ranges from \(-J\) to \(+J\) as per the \(m_j\) values.

Step 5: Compiling the Answers

In the \(5g\) state, the possible values of \(j\) are \(\frac{9}{2}\) and \(\frac{7}{2}\). For \(j = \frac{9}{2}\), \(m_j\) ranges from \(-\frac{9}{2}\) to \(+\frac{9}{2}\) in integer steps. For \(j = \frac{7}{2}\), \(m_j\) ranges from \(-\frac{7}{2}\) to \(+\frac{7}{2}\). These values correspond directly to \(J\) and \(J_z\) for this single electron system.

Key Concepts

Angular Momentum

Angular momentum in quantum mechanics is a fundamental aspect related to the motion of particles. It provides insight into how electrons move around within an atom.

• Each electron has its own angular momentum due to its orbital motion and intrinsic spin.
• The total angular momentum is crucial for understanding the behavior of electrons in atoms and is given by the quantum number \( j \).

For electrons, the value of \( j \), which represents the total angular momentum quantum number, is derived from two components:

• Orbital angular momentum \( l \): Represents the electron's motion in its orbit around the nucleus.
• Spin angular momentum: An intrinsic form of angular momentum carried by electrons.

The total angular momentum is calculated as \( j = l \pm \frac{1}{2} \), indicating the combination of these two types of angular momenta. This equation shows that there are always two possible values for \( j \), reflecting the parallel or antiparallel orientation of the electron's spin relative to its orbital motion.Understanding angular momentum is essential when considering electron configurations, the spectral lines of elements, and the overall stability of atoms.

Principal Quantum Number

The principal quantum number, symbolized as \( n \), is a key concept in the quantum description of atoms. It forms the foundation for understanding the energy levels of electrons within an atom.

• The principal quantum number \( n \) determines the size and energy of the atomic orbitals, with electrons in shells with higher \( n \) values generally located farther from the nucleus.
• The value of \( n \) is always a positive integer: 1, 2, 3, etc.

In our specific case of the "5g" state, \( n = 5 \). The larger the principal quantum number, the higher the potential energy of the electron.The importance of \( n \) cannot be overstated as it influences:

• The total number of orbitals within an atom, given by the formula \( n^2 \).
• The electron's energy level. Electrons with the same \( n \) belong to the same shell but can differ in their angular momentum.

Understanding \( n \) is critical for predicting electron configurations and consequently the chemical properties and reaction tendencies of elements.

Azimuthal Quantum Number

The azimuthal quantum number, denoted as \( l \), provides details about the electron's angular momentum and the shape of the orbitals. This quantum number is crucial for classifying orbitals by their shapes and thereby influencing the atom's chemical properties.

• Each value of \( l \) corresponds to a specific orbital shape or type, with notable types being \( s, p, d, f, \) and \( g \).
• The value of \( l \) ranges from 0 to \( n-1 \). For example, if \( n = 5 \), \( l \) can be 0, 1, 2, 3, or 4, where \( l = 4 \) refers to the \( g \) orbital in this context.

In the context of our "5g" state, \( l = 4 \), indicating the electron is in a \( g \) orbital, which is one of the higher energy orbitals.Understanding the azimuthal quantum number helps in determining:

• The angular momentum of electrons, crucial for atom stability and the formation of covalent bonds.
• The subshells present in each electron shell, which further defines the permissible electron configurations.

This contributes significantly to predicting not just the electronic structure but also the resulting periodic properties and chemical behavior of elements.