Question

A hydrogen atom in a 3\(p\) state is placed in a uniform external magnetic field \(\vec B\). Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. (a) What field magnitude \(B\) is required to split the 3\(p\) state into multiple levels with an energy difference of 2.71 \(\times\) 10\(^{-5}\) eV between adjacent levels? (b) How many levels will there be?

Short answer

(a) 0.47 T is needed. (b) There will be 3 levels.

Step-by-step solution

Understanding the Zeeman effect

In the presence of an external magnetic field, the orbital energy levels of an atom split due to the Zeeman effect. This effect specifically operates on the magnetic component associated with the orbital motion of electrons.

Identifying the relevant formula

The energy difference between levels in the Zeeman effect can be calculated by \( \Delta E = \mu_B B m_l \), where \( \mu_B \) is the Bohr magneton and \( m_l \) is the magnetic quantum number.

Details of the 3p state

For a 3\(p\) state, the orbital angular momentum quantum number \( l = 1 \), so \( m_l \) can be \(-1, 0, +1\). This gives three possible orientations relative to the magnetic field.

Calculate the required magnetic field

Given that the energy difference \( \Delta E = 2.71 \times 10^{-5} \) eV between adjacent levels, we use the formula \( \Delta E = \mu_B B \), solving for \( B \). The Bohr magneton \( \mu_B \) is approximately \( 5.79 \times 10^{-5} \) eV/T. Therefore, \( B = \frac{\Delta E}{\mu_B} = \frac{2.71 \times 10^{-5} \text{ eV}}{5.79 \times 10^{-5} \text{ eV/T}} \approx 0.47 \text{ T} \).

Determine the number of levels

Since \( m_l \) can take values of \(-1, 0, +1\) for \( l = 1 \), the 3\(p\) state splits into three levels in the magnetic field.

Key concepts

Magnetic Quantum Number

The magnetic quantum number, denoted as \( m_l \), is a key player in understanding atomic structure, especially when atoms are exposed to external magnetic fields. This number arises from the solutions to the quantum mechanical model of an atom. It signifies the orientation of the orbital angular momentum of an electron. In simpler terms, \( m_l \) dictates how an electron's orbital is oriented in space.

For any given electron in a hydrogen atom with an orbital angular momentum quantum number \( l \), \( m_l \) can take on integer values ranging from \(-l\) to \(+l\). This means that if \( l = 1 \), such as in a \(p\) orbital, \( m_l \) can be \(-1, 0,\) or \(+1\).

• \( m_l = -1 \) represents one orientation.
• \( m_l = 0 \) is another possible orientation.
• \( m_l = +1 \) is the third orientation.

These different orientations correspond to how the orbital aligns relative to an external magnetic field. The different values lead to different energy levels due to the Zeeman effect.

Orbital Angular Momentum

Orbital angular momentum, in the context of quantum mechanics, describes the angular momentum of electrons orbiting an atomic nucleus. It is an important property that influences the structure and energy levels of atoms. Represented by the quantum number \( l \), it determines the shape of an electron's orbital.

For instance, a \(3p\) state in a hydrogen atom has an orbital angular momentum quantum number \( l = 1 \). This means that the electron can be found in a "\(p\)" shaped orbital, which has a specific type of symmetry and complexity compared to other types like "\(s\)" or "\(d\)."

• "\(s\)" orbitals (\(l = 0\)) are spherical.
• "\(p\)" orbitals (\(l = 1\)) have a dumbbell shape.
• "\(d\)" orbitals (\(l = 2\)) are more complex with various orientations.

The value of \( l \) also limits the possible values of \( m_l \), the magnetic quantum number. This relationship is crucial when considering how energy levels split in a magnetic field through the Zeeman effect.

Bohr Magneton

The Bohr magneton \( \mu_B \) is a fundamental physical constant that measures the magnetic moment of an electron due to its orbital or spin motion. It's named after the physicist Niels Bohr and is defined as \( \mu_B = \frac{e \hbar}{2m_e} \), where \( e \) is the electron charge, \( \hbar \) is the reduced Planck's constant, and \( m_e \) is the electron mass.

In practical terms, the Bohr magneton is used to express the magnetic moment of electrons in atoms. In the study of magnetic effects like the Zeeman effect, \( \mu_B \) is crucial for determining how much the energy levels of an atom change when subjected to an external magnetic field.

For example, when a hydrogen atom in the \(3p\) state is placed in a magnetic field, the shifts in energy levels can be calculated using the formula \( \Delta E = \mu_B B m_l \). The magnitude of change depends on both the strength of the magnetic field \( B \) and the value of \( m_l \). With \( \mu_B \) approximately equal to \( 5.79 \times 10^{-5} \) eV/T, it becomes clear how these calculations are systematically connected to physics constants.