Problem 18 - College Physics

Question

(a) How many possible sets of quantum numbers are there for the \(n=1\) and \(n=2\) shells? (b) Write the explicit values of all the quantum numbers \(\left(n, \ell, m_{\ell}, m_{\mathrm{s}}\right)\) for these levels.

Step-by-Step Solution

Answer

For \(n=1\), there are 2 sets: \((1,0,0,+\frac{1}{2})\) and \((1,0,0,-\frac{1}{2})\). For \(n=2\), there are 8 sets: \((2,0,0,\pm\frac{1}{2})\) and \((2,1,m_\ell,\pm\frac{1}{2})\) for \(m_\ell=-1,0,1\).

Step 1: Understanding Quantum Numbers

Quantum numbers describe the properties of electrons in an atom. For a shell with principal quantum number \(n\), the possible values for the angular momentum quantum number \(\ell\) range from \(0\) to \(n-1\). Each \(\ell\) has magnetic quantum numbers \(m_\ell\) from \(-\ell\) to \(\ell\). The spin quantum number \(m_s\) can be \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Step 2: Identify Quantum Numbers for \(n=1\)

For \(n=1\), the only possible value of \(\ell\) is \(0\). For \(\ell=0\), \(m_{\ell}=0\). The spin quantum number \(m_s\) can be \(+\frac{1}{2}\) or \(-\frac{1}{2}\), giving two sets: \((1,0,0,+\frac{1}{2})\) and \((1,0,0,-\frac{1}{2})\).

Step 3: Calculate Possible Sets for \(n=1\)

There are 2 possible sets of quantum numbers for \(n=1\) because there are only two combinations of \(m_s\).

Step 4: Identify Quantum Numbers for \(n=2\)

For \(n=2\), the possible values of \(\ell\) are \(0\) and \(1\). For \(\ell=0\), \(m_{\ell}=0\) with spins \(+\frac{1}{2}\) and \(-\frac{1}{2}\), similar to the \(n=1\) case. Thus, these sets are \((2,0,0,+\frac{1}{2})\) and \((2,0,0,-\frac{1}{2})\).

Step 5: Quantum Numbers for \(\ell=1\) in \(n=2\)

For \(\ell=1\), \(m_\ell\) can be \(-1, 0, \) or \(1\). Each \(m_\ell\) has two possible spin states \(m_s=+\frac{1}{2}\) and \(-\frac{1}{2}\). This results in six sets: \((2,1,-1,+\frac{1}{2})\), \((2,1,-1,-\frac{1}{2})\), \((2,1,0,+\frac{1}{2})\), \((2,1,0,-\frac{1}{2})\), \((2,1,1,+\frac{1}{2})\), and \((2,1,1,-\frac{1}{2})\).

Step 6: Calculate Possible Sets for \(n=2\)

Adding the sets from \(\ell=0\), and \(\ell=1\) gives a total of 8 sets for \(n=2\) (2 from \(\ell=0\) and 6 from \(\ell=1\)).

Step 7: Compile Total Quantum Sets

Overall, there are 2 possible sets for \(n=1\) and 8 sets for \(n=2\). We have listed these explicitly in the previous steps.

Key Concepts

Principal Quantum Number

The principal quantum number, symbolized as \(n\), denotes the primary energy level of an electron within an atom. It is the most crucial quantum number as it dictates the size and energy of the electron's orbit. Think of \(n\) as the shell in which an electron can be found. The value of \(n\) is always a positive integer:

• Starts at \(n=1\).
• For each increase in \(n\), the orbital becomes larger and holds more energy.

Understanding \(n\) is essential because each different \(n\) value means a different electron shell, leading to unique electron configurations. For example, the \(n=1\) shell can only hold a limited set of electron quantum numbers, unlike \(n=2\), which offers more variety and options.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted as \(\ell\), defines the shape of an electron's orbital within a given shell with principal quantum number \(n\). This number can take integer values from 0 to \(n-1\), meaning the number of possible \(\ell\) values increases with each larger shell:

• \(\ell = 0\): s-orbital, spherical in shape.
• \(\ell = 1\): p-orbital, dumbbell-shaped.
• Higher \(\ell\) values (2, 3,...) represent d and f orbitals.

Each different value of \(\ell\) brings a distinct type of orbital and contributes to the atom's electronic structure. For \(n = 2\), \(\ell\) can be both 0 and 1, allowing for s and p orbitals within this shell.

Magnetic Quantum Number

The magnetic quantum number \(m_{\ell}\) gives us information about the orientation of the orbital in space. For each value of \(\ell\), \(m_{\ell}\) takes integer values from \(-\ell \) to \( +\ell \), including zero. This means each type of orbital shape (\(\ell\)) has a different number of orientations:

• For \(\ell = 0\), \(m_{\ell} = 0\) as there is only one orientation for s-orbitals.
• For \(\ell = 1\), \(m_{\ell}\) can be \(-1, 0,\) or \( +1 \), representing the three different orientations of p-orbitals.

The various orientations specified by \(m_{\ell}\) allow orbitals to hold more electrons while maintaining stability, by accommodating different spatial alignments.

Spin Quantum Number

The spin quantum number, denoted by \(m_s\), flags the inherent spin of an electron, a fundamental quantum property analogous to angular momentum. However, instead of orbiting around the nucleus, this spin refers to the idea of the electron behaving like a tiny magnet:

• \(m_s\) can have only two possible values: \(+\frac{1}{2}\) and \(-\frac{1}{2}\).
• These values represent the two possible spins: "up" and "down".

Spin quantum numbers account for the electron's magnetic behavior and, interestingly, each orbital can hold a maximum of 2 electrons, each with opposite spins. This characteristic aids in minimizing repulsion between electrons sharing the same spatial region. Understanding \(m_s\) is crucial for grasping how electrons fill orbitals in an energy-efficient and stable manner.