Problem 51 - Physics for Scientists and Engineers with Modern Physics

Question

(II) What are the possible values of \(j\) for an electron in (a) the \(4 p,\) (b) the \(4 f,\) and \((c)\) the \(3 d\) state of hydrogen? (d) What is \(J\) in each case?

Step-by-Step Solution

Answer

(a) \( j = \frac{3}{2}, \frac{1}{2} \); (b) \( j = \frac{7}{2}, \frac{5}{2} \); (c) \( j = \frac{5}{2}, \frac{3}{2} \); \( J = j \).

Step 1: Understanding Quantum Numbers

Electrons in atoms are characterized by quantum numbers: the principal quantum number \( n \), the azimuthal quantum number (angular momentum) \( l \), the magnetic quantum number \( m_l \), and the spin quantum number \( m_s \). The total angular momentum quantum number, \( j \), is given by \( j = l + m_s \) or \( j = l - m_s \). For electrons, \( m_s = \pm \frac{1}{2} \).

Step 2: Evaluating 4p State

For the 4p state, the principal quantum number \( n = 4 \) and the azimuthal quantum number \( l = 1 \). Using \( j = l + m_s \) and \( j = l - m_s \), we have possible values of \( j \) as \( j = 1 + \frac{1}{2} = \frac{3}{2} \) and \( j = 1 - \frac{1}{2} = \frac{1}{2} \).

Step 3: Evaluating 4f State

For the 4f state, \( n = 4 \) and \( l = 3 \). Possible values of \( j \) are \( j = 3 + \frac{1}{2} = \frac{7}{2} \) and \( j = 3 - \frac{1}{2} = \frac{5}{2} \).

Step 4: Evaluating 3d State

For the 3d state, \( n = 3 \) and \( l = 2 \). The possible values of \( j \) are \( j = 2 + \frac{1}{2} = \frac{5}{2} \) and \( j = 2 - \frac{1}{2} = \frac{3}{2} \).

Step 5: Determine Total Angular Momentum \( J \)

In this context, \( J \) is typically the same as \( j \), representing the quantum states of the electron. Thus, \( J \) takes values of \( j \) for each state: (a) \( \frac{3}{2} \) and \( \frac{1}{2} \) for 4p, (b) \( \frac{7}{2} \) and \( \frac{5}{2} \) for 4f, and (c) \( \frac{5}{2} \) and \( \frac{3}{2} \) for 3d.

Key Concepts

Angular Momentum in Quantum Mechanics

Angular momentum is a fundamental property in quantum mechanics, just like in classical mechanics. However, when dealing with subatomic particles like electrons, angular momentum has some unique characteristics. In quantum mechanics:

• The angular momentum is quantized, meaning it can only take on certain discrete values.
• It is defined by the azimuthal quantum number, symbolized by \( l \).
• For a given principal quantum number \( n \), \( l \) can take on integer values from 0 to \( n-1 \).

Each value of \( l \) corresponds to a different kind of subshell or orbital type in an atom:

• \( l = 0 \) corresponds to an s subshell.
• \( l = 1 \) corresponds to a p subshell.
• \( l = 2 \) corresponds to a d subshell.
• \( l = 3 \) corresponds to an f subshell.

These subshells help determine an electron's magnetic and chemical properties. In physics, quantum angular momentum is crucial to understanding electron configuration and behavior in atoms.

Total Angular Momentum: \( j \)

Total angular momentum, denoted by \( j \), is a combination of an electron's orbital angular momentum \( l \) and its spin angular momentum \( s \). An electron's spin quantum number, \( m_s \), can be either \( +\frac{1}{2} \) or \( -\frac{1}{2} \).The possible values of total angular momentum \( j \) are determined using the equation:

• \( j = l + m_s \)
• \( j = l - m_s \)

For each \( l \) value, there are typically two corresponding \( j \) values. For example, in the 4f state where \( l = 3 \):

• If \( m_s = +\frac{1}{2} \), then \( j = 3 + \frac{1}{2} = \frac{7}{2} \)
• If \( m_s = -\frac{1}{2} \), then \( j = 3 - \frac{1}{2} = \frac{5}{2} \)

The total angular momentum is an important quantum property because it describes the quantum states accessible to an electron, impacting its energy, magnetic behavior, and interactions within the atom.

Quantum States and Their Importance

Quantum states describe the unique configurations of electrons within an atom. These are defined by a set of quantum numbers: principal (\( n \)), angular momentum (\( l \)), magnetic (\( m_l \)), and spin (\( m_s \)). The total angular momentum \( j \) plays a vital role in defining these states.Each quantum state corresponds to a specific energy level and sublevel, sequestering electrons into distinct orbitals. This determines:

• How electrons fill the available orbitals in an atom (electron configuration).
• The atom's chemical properties and how it bonds with other atoms.
• The atom's response to external electromagnetic fields (spectroscopic behavior).

Using quantum mechanics to analyze quantum states helps predict chemical reactions and physical properties of materials. It also provides deeper insights into the electronic structure of matter, crucial for fields like quantum chemistry and condensed matter physics.