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Magazine Chapter 41: Problem 20
List the different possible combinations of \(l\) and \(j\) for a hydrogen atom in the \(n=3\) level.
Short Answer
Expert verified
\((l,j) = (0,1/2), (1,1/2), (1,3/2), (2,3/2), (2,5/2)\).
Step by step solution
01
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 \( j = l - s \).
02
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.
03
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 \( j = 3/2 \).
04
Listing Combinations
List the combinations using each value of \( l \) and the calculated \( j \):- \( (l = 0, j = 1/2) \)- \( (l = 1, j = 1/2), (l = 1, j = 3/2) \)- \( (l = 2, j = 3/2), (l = 2, j = 5/2) \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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.
