Problem 47 - Physics for Scientists and Engineers with Modern Physics

Question

(I) If the quantum state of an electron is specified by \(\left(n, \ell, m_{\ell}, m_{s}\right),\) estimate the energy difference between the states \(\left(1,0,0,-\frac{1}{2}\right)\) and \(\left(1,0,0,+\frac{1}{2}\right)\) of an electron in the \(1 s\) state of helium in an external magnetic field of \(2.5 \mathrm{~T}\)

Step-by-Step Solution

Answer

The energy difference is approximately \(4.637 \times 10^{-23}\) J.

Step 1: Understand the Problem Context

In this exercise, the focus is on an electron's quantum state represented by quantum numbers in an external magnetic field. We need to calculate the energy difference between two specific states in the presence of this magnetic field, using the Zeeman effect.

Step 2: Identify the Quantum Numbers

The two given states are \((1,0,0,-\frac{1}{2})\) and \((1,0,0,+\frac{1}{2})\). Both states are in the \(1s\) orbital of helium. The quantum numbers describe the principal quantum number \(n = 1\), the azimuthal quantum number \(\ell = 0\), the magnetic quantum number \(m_\ell = 0\), and the spin magnetic quantum numbers \(m_s = -\frac{1}{2}\) and \(m_s = +\frac{1}{2}\).

Step 3: Recognize the Nature of the Zeeman Effect

The normal Zeeman effect occurs in the presence of an external magnetic field and relates the shift in energy levels due to the interaction between the magnetic moment of the electron (due to its spin) and the external field. In absence of field interaction with angular momentum, we consider the spin-only Zeeman effect.

Step 4: Apply the Zeeman Effect to Determine Energy Shift

For electrons, the energy shift due to the Zeeman effect is given by \( \Delta E = g_s \mu_B B m_s \), where \( g_s \approx 2 \) is the g-factor of the electron, \( \mu_B \approx 9.274 \times 10^{-24} \) J/T is the Bohr magneton, \( B = 2.5 \) T is the external magnetic field strength, and \( m_s \) is the spin magnetic quantum number.

Step 5: Calculate Energy Difference Between States

For the states \((1,0,0,-\frac{1}{2})\) and \((1,0,0,+\frac{1}{2})\), substitute \( m_s = -\frac{1}{2} \) and \( m_s = +\frac{1}{2} \) into the energy shift formula. This results in energy shifts of \( \Delta E_- = -g_s \mu_B B \frac{1}{2} \) and \( \Delta E_+ = +g_s \mu_B B \frac{1}{2} \). The energy difference is \( \Delta E_+ - \Delta E_- = g_s \mu_B B \).

Step 6: Substitute Numerical Values and Calculate

Substituting the values, the energy difference becomes \(\Delta E = 2 \times 9.274 \times 10^{-24} \times 2.5 = 4.637 \times 10^{-23} \) J. This is the energy difference between the two electron spin states in this magnetic field.

Key Concepts

Quantum Numbers

Every electron in an atom is described by a unique set of quantum numbers, which essentially dictate the "address" of the electron within the atom. These numbers outline various properties of the electron, helping us make sense of its behavior and interactions.

• **Principal Quantum Number (n):** Describes the main energy level or shell of the electron. In our problem, both electrons are in the n=1 state, indicating they occupy the innermost shell, closest to the nucleus.
• **Azimuthal Quantum Number (ℓ):** Associated with the subshell or type of orbital. A value of ℓ=0 specifies an s-orbital, like in the two given states \( (1,0,0,-\frac{1}{2}) \) and \( (1,0,0,+\frac{1}{2}) \).
• **Magnetic Quantum Number (m_ℓ):** Defines the orientation of the orbital in space within a magnetic field. This value is m_ℓ=0 since s-orbitals are spherical and have no directional preference.
• **Spin Magnetic Quantum Number (m_s):** Depicts the intrinsic spin of the electron, which can either be +1/2 or -1/2, often referred to as spin 'up' or 'down'. The Zeeman effect specifically considers variations in this number when an external magnetic field is applied.

Understanding these numbers is essential, as they lay the foundation for analyzing how an electron's energy changes when it is exposed to certain conditions, like a magnetic field.

Energy Shift

The Zeeman effect discusses how an electron's energy levels can shift when exposed to an external magnetic field. This is because the magnetic field interacts with the magnetic moments of electrons.
The formula for the energy shift due to the Zeeman effect is:\[\Delta E = g_s \mu_B B m_s\]where:

• \( g_s \approx 2 \) is the electron's g-factor.
• \( \mu_B \approx 9.274 \times 10^{-24} \) J/T is the Bohr magneton, a physical constant that plays a central role in this effect.
• \( B \) is the strength of the external magnetic field.
• \( m_s \) is the electron spin number.

For our calculation:

• The electron in state \( (1,0,0,-\frac{1}{2}) \) experiences an energy shift \( \Delta E_- = -g_s \mu_B B \frac{1}{2} \).
• The electron in state \( (1,0,0,+\frac{1}{2}) \) undergoes an energy shift \( \Delta E_+ = +g_s \mu_B B \frac{1}{2} \).

Subtracting these shifts provides the energy difference between the states, labeled as \( \Delta E_+ - \Delta E_- = g_s \mu_B B \). It is pivotal to comprehend that this energy shift directly correlates to the electron's ability to transition between spin states.

Magnetic Field

The magnetic field is the external force field that initiates the Zeeman effect. It influences electron energy levels by interacting with their magnetic moments. The strength of this field, measured in Tesla (T), is pivotal in determining the magnitude of the energy shift.
In this problem, the external magnetic field has a specified strength of 2.5 T. Its presence is crucial because it creates a separation between energies of electron spin states, which are otherwise identical in the absence of such a field.
Various factors are associated with the magnetic field:

• **Orientation:** The orientation of the magnetic field with respect to the electron's spin can affect the degree of interaction.
• **Magnitude:** As seen in our exercise, the 2.5 T magnetic field causes a notable energy difference between the states. Larger fields would result in even more pronounced shifts.

Such fields find extensive use in spectroscopy and other quantum physics phenomena. Understanding the effects of a magnetic field is fundamental for analyzing variations in atomic energy levels, as demonstrated in our problem.