Problem 20 - University Physics with Modern Physics

Question

List the different possible combinations of \(l\) and \(j\) for a hydrogen atom in the \(n=3\) level.

Step-by-Step Solution

Answer

\((l,j) = (0,1/2), (1,1/2), (1,3/2), (2,3/2), (2,5/2)\).

Step 1: Understanding Quantum Numbers

In a hydrogen atom, the quantum numbers are used to describe the allowed states of electrons. The principal quantum number, denoted as \( n \), indicates the energy level. The azimuthal quantum number, denoted as \( l \), relates to the subshells within an energy level and ranges from 0 to \( n-1 \). The magnetic quantum number \( m_l \), which is not directly needed here, would range from \(-l\) to \( l \). The spin quantum number \( s \) is either +1/2 or -1/2 but generally does not affect combinations of \( l \) and \( j \). Finally, \( j \) is the total angular momentum quantum number and is described by \( j = l + s \) or \( j = l - s \).

Step 2: Identifying Values for \(l\)

For the \( n=3 \) level, the values that \( l \) can take are \( l = 0, 1, 2 \). These correspond to different subshells known as 3s, 3p, and 3d.

Step 3: Calculating \(j\) for each \(l\)

For each value of \( l \), calculate the possible values of \( j \), which are determined by \( j = l + s \) and \( j = l - s \), where \( s = 1/2 \). - For \( l = 0 \), \( j = 1/2 \).- For \( l = 1 \), \( j = 3/2 \) and \( j = 1/2 \).- For \( l = 2 \), \( j = 5/2 \) and \( j = 3/2 \).

Step 4: Listing Combinations

List the combinations using each value of \( l \) and the calculated \( j \):- \( (l = 0, j = 1/2) \)- \( (l = 1, j = 1/2), (l = 1, j = 3/2) \)- \( (l = 2, j = 3/2), (l = 2, j = 5/2) \)

Key Concepts

Principal Quantum Number

The principal quantum number is a fundamental concept in quantum mechanics, especially when studying the behavior of electrons in a hydrogen atom. Designated by the symbol \( n \), the principal quantum number indicates the main energy level that an electron occupies.

• It takes on positive integer values: \( n = 1, 2, 3, \ldots \).
• Higher values of \( n \) mean the electron is further from the nucleus and has a higher energy.
• For each value of \( n \), there are specific subshells where the electron can reside.

Understanding the role of the principal quantum number is crucial because it determines the size and energy of the electron's orbitals. For example, in the exercise, we are dealing with electrons in the \( n=3 \) energy level.

Azimuthal Quantum Number

The azimuthal quantum number, often denoted as \( l \), gives information about the shape of an electron's orbital within an energy level. This quantum number defines the subshells in which electrons can exist.

• The possible values of \( l \) are integers ranging from 0 to \( n-1 \), where \( n \) is the principal quantum number.
• For example, if \( n = 3 \), then \( l \) could be 0, 1, or 2.
• Each value of \( l \) corresponds to a particular type of subshell represented by letters: 0 is \( s \), 1 is \( p \), 2 is \( d \), and so on.

Therefore, for \( n=3 \), electrons can be in the 3s, 3p, or 3d subshells, reflecting different shapes and angular momentum characteristics.

Total Angular Momentum Quantum Number

The total angular momentum quantum number, symbolized by \( j \), integrates both the orbital angular momentum and the intrinsic spin of the electron.

• For each azimuthal quantum number \( l \), the possible values of total angular momentum \( j \) are calculated using \( j = l + s \) and \( j = l - s \), where \( s = 1/2 \) for electrons.
• This means for \( l = 0, 1, 2 \), you have possible \( j \) values that describe how the electron's orbit aligns with its spin components.
• This results in values such as \( j = 1/2 \), \( j = 3/2 \), and \( j = 5/2 \) for the \( n=3 \) level when evaluated with different \( l \) values.

The existence of different \( j \) values for a given \( l \) manifests in fine structure splitting of spectral lines, a refined aspect of the atomic spectral study.

Energy Level Subshells

Energy level subshells are divisions of electron shells based on quantum numbers, especially the azimuthal quantum number \( l \). Each principal energy level can contain multiple subshells, and these subshells have varying energy and shape characteristics.

• For each energy level \( n \), there are \( n \) possible subshells because \( l \) ranges from 0 to \( n-1 \).
• In a hydrogen atom's \( n=3 \) energy level, there are the 3s, 3p, and 3d subshells.
• These subshells each have a distinct number of orbitals, and therefore, can hold different numbers of electrons: \( s \) has 1 orbital, \( p \) has 3 orbitals, and \( d \) has 5 orbitals.

The concept of energy level subshells helps explain the electron distribution around an atom and is pivotal in understanding atomic structure and chemical bonding.