Problem 9 - University Physics with Modern Physics

Question

An electron in a hydrogen atom is in an \(s\) level, and the atom is in a magnetic field \(\vec{B}=B \hat{k} .\) Explain why the "spin up" state \(\left(m_{s}=+\frac{1}{2}\right)\) has a higher energy than the "spin down" state \(\left(m_{s}=-\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The spin up state has higher energy due to its negative alignment with the magnetic field, resulting in a lower (more negative) energy level.

Step 1: Identify the Interaction

The interaction here is between the magnetic moment of the electron and the external magnetic field. The electron's magnetic moment is influenced by its spin, and when placed in a magnetic field, different spin states will have different energy levels.

Step 2: Understand the Magnetic Moment

The magnetic moment \(\vec{\mu}\) associated with the electron's spin is given by \(-g_{s} \cdot \frac{e}{2m_{e}} \cdot \vec{S}\), where \(g_{s}\) is the spin g-factor, \(e\) is the electron charge, \(m_{e}\) is the electron mass, and \(\vec{S}\) is the spin angular momentum vector. For an electron, \(g_{s} = 2\) approximates the g-factor accurately.

Step 3: Calculate the Energy Difference

The energy difference \((\Delta E)\) due to the magnetic field is based on the equation \(\Delta E = - \vec{\mu} \cdot \vec{B}\). Since \(\vec{B} = B \hat{k}\), only the \( S_z\) component of spin needs to be considered, meaning: \(\Delta E = \frac{e \cdot B}{m_{e}} \cdot S_z\).

Step 4: Determine Spin Contributions

For a spin up state \(m_s = +\frac{1}{2}\), \(S_z = + \frac{\hbar}{2}\). For a spin down state \(m_s = -\frac{1}{2}\), \(S_z = - \frac{\hbar}{2}\). Substituting these expressions into the equation from Step 3, the energy for spin up is \(E_{\uparrow} = -\frac{e \cdot B \cdot \hbar}{2 m_{e}}\) and for spin down is \(E_{\downarrow} = \frac{e \cdot B \cdot \hbar}{2 m_{e}}\).

Step 5: Compare Energy Levels

The spin up state \(E_{\uparrow}\) has a negative energy contribution compared to the spin down state \(E_{\downarrow}\), which has a positive contribution. Since energy levels are lower when more negative, the spin up state has higher energy in this context.

Key Concepts

Electron Spin

In the world of atomic physics, one of the most wondrous discoveries is the concept of electron spin. Imagine an electron as a tiny spinning top. Spin is a fundamental property of these particles, but unlike classical spinning objects, it doesn't mean they are physically spinning. Rather, it's a quantum characteristic that helps define the electron's behavior.
The electron can exist in two spin states, often termed "spin up" and "spin down." These are represented by the spin quantum number, denoted as \( m_s \), which can take the values \(+\frac{1}{2}\) (spin up) or \(-\frac{1}{2}\) (spin down).

• "Spin up" state corresponds to \( m_s = +\frac{1}{2}\)
• "Spin down" state corresponds to \( m_s = -\frac{1}{2}\)

Spin is so crucial because it influences an electron's magnetic properties, and when placed in magnetic fields, different spin states lead to different energy levels in atoms.

Magnetic Moment

Electrons, with their spins, behave like tiny magnets. This arises due to the electron spin generating a magnetic moment, which you can think of as a measure of its magnetic strength. The magnetic moment \( \vec{\mu} \) of an electron due to its spin is given by:
\(-g_s \cdot \frac{e}{2m_e} \cdot \vec{S}, \)where:

• \( g_s \) is the spin g-factor, and for electrons, it's approximately 2.
• \( e \) is the electron's charge.
• \( m_e \) is the electron's mass.
• \( \vec{S} \) is the spin angular momentum vector.

In the presence of an external magnetic field \( \vec{B} \), the magnetic moment interacts with the field, influencing the atom's energy levels. The electron's magnetic moment attempts to align with the magnetic field, leading to variations in energy states.
This interaction is critical to understanding why "spin up" and "spin down" states have different energies, especially when the electron is subjected to an external field.

Energy Levels in Magnetic Field

The behavior of energy levels in a magnetic field is quite fascinating, particularly for the electron in a hydrogen atom. When such an atom is exposed to a magnetic field \( \vec{B} \), the interaction between the field and the electron's magnetic moment introduces splitting in energy levels, known as the Zeeman effect.
The energy difference due to this interaction is given by \( \Delta E = - \vec{\mu} \cdot \vec{B} \). For our hydrogen atom example, since the magnetic field is along the \( z \)-axis, only the \( S_z \) component matters, simplifying the equation to:
\[ \Delta E = \frac{e \cdot B}{m_e} \cdot S_z \]
For different spin states:

• For a "spin up" state \( S_z = +\frac{\hbar}{2} \), resulting in energy \( E_{\uparrow} = -\frac{e \cdot B \cdot \hbar}{2 m_e} \).
• For a "spin down" state \( S_z = -\frac{\hbar}{2} \), giving energy \( E_{\downarrow} = \frac{e \cdot B \cdot \hbar}{2 m_e} \).

In this context, the "spin up" energy, being less negative, is actually higher relative to the "spin down" energy, underscoring a fundamental principle in magnetic interactions and energy conservation.