Problem 48 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Silver atoms \(\left(\operatorname{spin}=\frac{1}{2}\right)\) are placed in a \(1.0-\) T magnetic field which splits the ground state into two close levels. (a) What is the difference in energy between these two levels, and \((b)\) what wavelength photon could cause a transition from the lower level to the upper one? (c) How would your answer differ if the atoms were hydrogen?

Step-by-Step Solution

Answer

(a) 1.855 x 10^-23 J; (b) ~1.073 cm; (c) Energy difference same for hydrogen, but transition wavelength may vary.

Step 1: Understand the Zeeman Effect

When an atom with spin is placed in a magnetic field, its energy levels split due to the Zeeman effect. The energy difference between these levels depends on the magnetic field strength (B) and the magnetic moment of the particle (μ).The formula for the energy difference is given by:\( \Delta E = g \cdot \mu_B \cdot B \cdot m_s \)where \(\mu_B\) is the Bohr magneton, \(g\) is the g-factor (2 for electrons), and \(m_s\) is the magnetic spin quantum number difference. For silver atoms, \(\Delta m_s = 1\).

Step 2: Calculate Energy Difference for Silver

Given that silver atoms have a spin of \(\frac{1}{2}\), the Bohr magneton \(\mu_B = 9.274 \times 10^{-24} \) J/T, and \(g = 2\), we use the following values:\[ \Delta E = 2 \times 9.274 \times 10^{-24} \times 1.0 \times 1 = 1.855 \times 10^{-23}\, \text{J} \]

Step 3: Calculate Wavelength for Silver

The energy of a photon is related to its wavelength (\(\lambda\)) by the equation:\[ E = \frac{hc}{\lambda} \]where \(h = 6.626 \times 10^{-34} \) J·s is Planck's constant and \(c = 3 \times 10^8 \) m/s is the speed of light. Rearranging for \(\lambda\), we find:\[ \lambda = \frac{hc}{\Delta E} = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{1.855 \times 10^{-23}} \approx 1.073 \times 10^{-2} \text{m} \]

Step 4: Consider Hydrogen Atom

For hydrogen atoms, the process is similar, but instead of using the g-factor for electrons, we consider the fact that they behave similarly under a magnetic field due to also having a half-integer spin value. The energy difference would remain the same, but since hydrogen atoms have a different fundamental frequency spectrum, the exact wavelengths involved in transitions may differ.

Key Concepts

Zeeman Effect

The Zeeman effect occurs when an atom is placed in a magnetic field and its energy levels split due to interactions between the magnetic field and the magnetic moments of electrons. This effect is essential in understanding how energy states in atoms change. In the case of silver atoms with spin of \( \frac{1}{2} \), the presence of a magnetic field causes the splitting of the ground state into closely spaced energy levels.
Understanding this phenomenon helps in analyzing various physical effects in quantum mechanics as well as in practical applications such as magnetic resonance imaging (MRI).
The formula to calculate the energy difference due to the Zeeman effect is:
\[ \Delta E = g \cdot \mu_B \cdot B \cdot m_s \]

• \( \Delta E \) is the energy difference between the split levels.
• \( g \) is the g-factor, which is typically 2 for electrons.
• \( \mu_B \) is the Bohr magneton, a physical constant related to the electron's magnetic moment.
• \( B \) is the strength of the magnetic field.
• \( m_s \) is the difference in magnetic quantum number. For silver, this is 1.

Energy Levels

Energy levels in atoms are like steps on a ladder, where electrons reside at different "steps" or energies. When a magnetic field is applied, these energy levels may split. The Zeeman effect is a prominent example that illustrates this phenomenon. This splitting is crucial for understanding the behavior of atoms in magnetic fields.
For atoms like silver, the energy difference between the split levels helps us understand the potential transitions electrons can make. Calculating this difference involves understanding the factors, such as the g-factor and other constants, that affect electron movement.Let's look at how we calculate the energy difference for silver atoms in a magnetic field of 1.0 T using the equation:
\[ \Delta E = 2 \times 9.274 \times 10^{-24} \times 1.0 \times 1 = 1.855 \times 10^{-23} \text{ J} \]
This calculation informs us of how strong the magnetic manipulation of the atom's energy levels is.

Photon Wavelength

Photon wavelength is directly related to the energy difference between two energy levels in atom transitions. When an electron jumps from one energy level to another, it emits or absorbs a photon. The wavelength of this photon provides us with clues about the energy change.
How do we connect energy to wavelength? Using the relation:
\[ E = \frac{hc}{\lambda} \]

• \( E \) is the energy difference, specifically the one calculated using the results of the Zeeman effect.
• \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J·s} \).
• \( c \) is the speed of light \( 3 \times 10^8 \text{ m/s} \).
• \( \lambda \) represents the wavelength.

By rearranging and calculating for \( \lambda \), we find:
\[ \lambda = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{1.855 \times 10^{-23}} \approx 1.073 \times 10^{-2} \text{ m} \]
This photon wavelength results from the atoms' transition between split energy levels due to the magnetic field.

Magnetic Field

A magnetic field applies a force that can alter the trajectories and energy levels of charged particles such as electrons. This manipulation forms the basis of the Zeeman effect, where energy levels split based on the interaction with the magnetic field.
How the magnetic field affects atoms depends on its strength. The formula for energy difference \( \Delta E \) reveals how the magnetic field \( B \) influences this splitting. In practical scenarios, including silver atoms in a magnetic field of 1.0 T, we see firsthand how energy levels change.
The manipulation of atom energy states by magnetic fields is key in multiple scientific fields, enhancing our understanding of atom behavior in various conditions. This knowledge facilitates advancements in technologies relying on magnetic field interactions, such asdata storage and MRI. By studying these interactions, we gain insight into the detailed workings of atomic and subatomic particles.