Problem 9 - Fundamentals of Physics

Question

An electron is in a state with \(\ell=3\). (a) What multiple of \(\hbar\) gives the magnitude of \(\vec{L} ?\) (b) What multiple of \(\mu_{\mathrm{B}}\) gives the magnitude of \(\vec{\mu} ?(\mathrm{c})\) What is the largest possible value of \(m_{e}\), (d) what multiple of \(\hbar\) gives the corresponding value of \(L_{z}\), and (e) what multiple of \(\mu_{\mathrm{B}}\) gives the corresponding value of \(\mu_{\text {orb } z}\) ? (f) What is the value of the semiclassical angle \(\theta\) between the directions of \(L\), and \(\vec{L}\) ? What is the value of angle \(\theta\) for \((\mathrm{g})\) the second largest possible value of \(m_{\ell}\) and (h) the smallest (that is, most negative) possible value of \(m_{i}\) ?

Step-by-Step Solution

Answer

(a) \(2\sqrt{3}\,\hbar\), (b) \(2\sqrt{3}\,\mu_{B}\), (c) 3, (d) \(3\,\hbar\), (e) \(-3\,\mu_{B}\), (f) \(30^\circ\), (g) \(54.7^\circ\), (h) \(150^\circ\)."

Step 1: Calculate Magnitude of \( \vec{L} \)

The magnitude of the angular momentum \( \vec{L} \) is given by the formula \( L = \sqrt{\ell(\ell+1)} \cdot \hbar \). For \( \ell = 3 \), the magnitude is \( \sqrt{3(3+1)} \cdot \hbar = \sqrt{12} \cdot \hbar = 2\sqrt{3} \cdot \hbar \).

Step 2: Calculate Magnitude of \( \vec{\mu} \)

The magnetic moment \( \vec{\mu} \) is related to \( \vec{L} \) by \( \vec{\mu} = -\frac{q}{2m} \vec{L} \), where for an electron \( \vec{\mu} = - \mu_B \cdot \vec{L}/\hbar \). Thus \( \mu = 2\sqrt{3} \cdot \mu_B \).

Step 3: Determine Largest Value of \( m_{\ell} \)

The magnetic quantum number \( m_{\ell} \) ranges from \( -\ell \) to \( \ell \). Therefore, the largest possible value of \( m_{\ell} \) is 3.

Step 4: Calculate Value of \( L_z \)

The component of angular momentum \( L_z = m_{\ell} \cdot \hbar \). For the largest \( m_{\ell} = 3 \), \( L_z = 3 \cdot \hbar \).

Step 5: Calculate Value of \( \mu_{\text{orb } z} \)

The z-component of the magnetic moment is \( \mu_{\text{orb } z} = -m_{\ell} \cdot \mu_B \). For \( m_{\ell} = 3 \), \( \mu_{\text{orb } z} = -3 \cdot \mu_B \).

Step 6: Calculate Angle \( \theta \) for \( m_{\ell} = 3 \)

The semiclassical angle \( \theta \) is given by \( \cos \theta = \frac{L_z}{L} = \frac{3 \cdot \hbar}{2 \sqrt{3} \cdot \hbar} \). Simplifying, \( \cos \theta = \frac{3}{2 \sqrt{3}} = \frac{\sqrt{3}}{2} \), hence \( \theta = 30^\circ \).

Step 7: Calculate Angle \( \theta \) for Second Largest \( m_{\ell} = 2 \)

For \( m_{\ell} = 2 \), \( L_z = 2 \cdot \hbar \). The angle calculation \( \cos \theta = \frac{2}{2 \sqrt{3}} = \frac{1}{\sqrt{3}} \), gives \( \theta \approx 54.7^\circ \).

Step 8: Calculate Angle \( \theta \) for Smallest \( m_{\ell} = -3 \)

For \( m_{\ell} = -3 \), \( L_z = -3 \cdot \hbar \). Thus \( \cos \theta = \frac{-3}{2 \sqrt{3}} = -\frac{\sqrt{3}}{2} \), giving \( \theta = 150^\circ \).

Key Concepts

Angular Momentum

Angular momentum is a fundamental concept in quantum mechanics that describes the rotational motion of particles. In the quantum realm, angular momentum is quantized, meaning it can only take on discrete values. For an electron in an atom, the angular momentum is labeled by the quantum number \( \ell \), which can be any non-negative integer from 0 up to \( n-1 \), where \( n \) is the principal quantum number. This means if \( n = 4 \), \( \ell \) can be 0, 1, 2, or 3.

The magnitude of the angular momentum vector \( \vec{L} \) is calculated using the formula \( L = \sqrt{\ell(\ell+1)} \cdot \hbar \), where \( \hbar \) is the reduced Planck's constant. This formula shows how angular momentum depends on the quantum number \( \ell \), and it derives from the nature of quantum systems, where values are not continuous like in classical physics, but discrete. This quantization restricts the possible states an electron can occupy within an atom.

In practice, for \( \ell = 3 \), as in our example scenario, the angular momentum magnitude becomes \( 2\sqrt{3} \cdot \hbar \). This quantized angular momentum plays a critical role in determining the chemical properties of atoms, affecting things like the shapes of orbitals and spectroscopy results.

Magnetic Quantum Number

The magnetic quantum number, denoted as \( m_{\ell} \), is another key concept in understanding electron behavior in atoms. It relates to the orientation of the angular momentum vector \( \vec{L} \) relative to an external magnetic field, often the z-axis for simplicity. The values \( m_{\ell} \) can take range from \( -\ell \) to \( \ell \), including zero. This means each value of \( \ell \) has \( 2\ell + 1 \) possible \( m_{\ell} \) values.

For example, if \( \ell = 3 \), \( m_{\ell} \) can be -3, -2, -1, 0, 1, 2, or 3. This provides seven distinct orientations, each specifying a potential state of the electron.

• Largest \( m_{\ell} \): 3
• Smallest \( m_{\ell} \): -3

When determining the component of angular momentum along the z-axis, \( L_z \), we use \( L_z = m_{\ell} \cdot \hbar \). Thus, for \( m_{\ell} = 3 \), the angular momentum is \( 3\cdot \hbar \). The multiplicity of these \( m_{\ell} \) states explains many phenomena, like the Zeeman effect where spectral lines split under a magnetic field.

Magnetic Moment

The magnetic moment \( \vec{\mu} \) is an essential concept when discussing magnetic properties of particles with angular momentum, like electrons. It measures the tendency of a particle to align with a magnetic field. For electrons, which have charge \( -e \), the magnetic moment is opposite to the angular momentum vector due to the negative charge.

The relationship between the magnetic moment and angular momentum is described by \( \vec{\mu} = -\frac{q}{2m} \vec{L} \), which simplifies to \( \vec{\mu} = -\frac{e}{2m} \cdot \vec{L} \) for electrons. By introducing the Bohr magneton \( \mu_B = \frac{e\hbar}{2m} \), this can further simplify to \( \vec{\mu} = -\mu_B \cdot \frac{\vec{L}}{\hbar} \).

• Magnitude of Magnetic Moment: \( 2\sqrt{3} \cdot \mu_B \) when \( \ell = 3 \)
• Z-component: \( \mu_{\text{orb } z} = - m_{\ell} \cdot \mu_B \)

Understanding magnetic moments is crucial for explaining magnetic properties of materials and their interactions with magnetic fields, which is a foundation for technologies like magnetic storage and MRI.

Semiclassical Angle

The semiclassical angle \( \theta \) helps connect quantum mechanical angular momentum with classical visualizations. It represents the angle between the total angular momentum \( \vec{L} \) and its z-component \( L_z \). In quantum mechanics, the value of \( L_z \) is quantized, influencing this angle.

The angle \( \theta \) can be determined using the equation \( \cos \theta = \frac{L_z}{L} \). This highlights how the specific \( m_{\ell} \) value affects \( \theta \). For example:

• For the largest \( m_{\ell} = 3 \), \( \theta = 30^\circ \)
• For the second-largest \( m_{\ell} = 2 \), \( \theta \approx 54.7^\circ \)
• For the smallest \( m_{\ell} = -3 \), \( \theta = 150^\circ \)

This concept provides insight into how quantum numbers dictate the possible orientations of an electron's angular momentum, bridging classical and quantum mechanical interpretations.