Estimate the ratio of the term in (39.9) that is linear in \(\mathrm{H}\) to the kinetic-energy term, for a hydrogen atom in a magnetic field of \(10^{5}\) gauss.

The Zeeman effect is an important phenomenon in atomic physics that occurs when an atom is exposed to a magnetic field. This magnetic field causes the energy levels of the atom's electrons to split into multiple closely spaced levels. These new energy levels depend on the magnetic field's strength and the specific characteristics of the electrons, such as their magnetic moment. This splitting is known as the Zeeman effect.
In the case of the hydrogen atom, the term linear in the magnetic field (\text{H}) represents the interaction between the magnetic field and the electron's magnetic moment. The mathematical representation of this interaction can be given by: \(\frac{e \text{H}}{2m_e c} L_z\). Here,

• \text{e} is the electron charge.
• \text{H} is the magnetic field strength.
• \text{m}_\text{e} is the electron mass.
• \text{c} is the speed of light.
• \text{L}_\text{z} represents the z-component of the electron's angular momentum.

Understanding the Zeeman effect helps in exploring how magnetic fields influence atomic energy levels and can be observed in various astronomical and laboratory contexts.

Kinetic energy is the energy possessed by an object due to its motion. In the context of an electron in a hydrogen atom, kinetic energy represents the energy due to the electron's movement around the nucleus. The expression for the kinetic energy of an electron in a hydrogen atom is given by: \(\text{T} = \frac{p^2}{2m_e}\). Here,

• \text{T} is the kinetic energy.
• \text{p} is the momentum of the electron.
• \text{m}_\text{e} stands for the mass of the electron.

Understanding kinetic energy is essential for grasping how electrons behave in atoms. The kinetic energy contributes to the total energy of the electron and is central to calculations involving electron dynamics and interactions within an atom.

A magnetic field (\text{H}) is a field produced by moving electric charges or magnetic dipoles and affects the behavior of charged particles within its vicinity. In the exercise, the magnetic field strength is given as 10⁵ gauss, which is equivalent to 0.1 tesla. This magnetic field impacts the hydrogen atom, causing the split in the energy levels of the electrons, which is observed as the Zeeman effect.
The strength and orientation of the magnetic field determine the degree of splitting and the energy shifts within the atom. The expression linking the magnetic field to the Zeeman effect term is: \(\frac{e \text{H}}{2m_e c} L_z\) . Here,

• \text{H} is the magnetic field strength.
• \text{e} is the electron charge.
• \text{m}_\text{e} is the electron mass.
• \text{c} is the speed of light.
• \text{L}_\text{z} is the z-component of the angular momentum.

Understanding the magnetic field's influence on atomic systems is crucial in many areas of physics, including spectroscopy and quantum mechanics.

Momentum (\text{p}) is a fundamental physical quantity representing the product of an object's mass and velocity. For an electron in a hydrogen atom, the momentum plays a key role in describing its motion around the nucleus. The relationship between momentum and kinetic energy is given by the formula: \(\text{T} = \frac{p^2}{2m_e}\) . Here,

• \text{T} is the kinetic energy.
• \text{p} represents the momentum.
• \text{m}_\text{e} is the mass of the electron.

The estimation of momentum is crucial for calculating electron dynamics within atoms. Momentum is influenced by various quantum factors, such as the electron's principal quantum number. Accurate calculations of momentum provide deeper insights into electron behavior and interactions within a hydrogen atom.

Angular momentum (\text{L}) is a measure of the amount of rotation an object has, taking into account its mass distribution and rotational velocity. For electrons in atoms, angular momentum is quantized, meaning it can only take on discrete values. The z-component of the electron's angular momentum (\text{L}_\text{z}) plays a vital role in the Zeeman effect.
In the context of the given problem, the z-component of angular momentum, \text{L}_\text{z}, is given by the relation: \(\text{L}_z = \frac{h}{2\text{π}} m\) . Here,

• \text{h} is Planck's constant.
• \text{π} is pi, a mathematical constant.
• \text{m} represents the magnetic quantum number.

Understanding angular momentum and its components is crucial for exploring electron dynamics and interactions, particularly in the presence of external magnetic fields.