Problem 19 - University Physics with Modern Physics

Question

A hydrogen atom in the 5\(g\) state is placed in a magnetic field of 0.600 T that is in the \(z\) -direction. (a) Into how many levels is this state split by the interaction of the atom's orbital magnetic dipole moment with the magnetic field? (b) What is the energy separation between adjacent levels? (c) What is the energy separation between the level of lowest energy and the level of highest energy?

Step-by-Step Solution

Answer

(a) 9 levels; (b) \(5.5644 \times 10^{-24} \text{ J}\); (c) \(4.45152 \times 10^{-23} \text{ J}\)."

Step 1: Determine the Quantum Numbers

The hydrogen atom in question is in the 5\(g\) state. The principal quantum number is \(n = 5\). The azimuthal quantum number \(l\) for the \(g\) state in hydrogen is \(l = 4\). Consequently, the magnetic quantum number \(m_l\) can take on values ranging from \(-l\) to \(+l\), inclusive.

Step 2: Calculate Number of Levels in the Magnetic Field

The number of levels into which the state is split by the magnetic field is determined by the possible values of \(m_l\). For \(l = 4\), \(m_l\) can be \(-4, -3, -2, -1, 0, +1, +2, +3, +4\), resulting in 9 discrete energy levels.

Step 3: Calculate Energy Separation between Adjacent Levels

The energy separation between adjacent levels in a magnetic field is given by the formula \( \Delta E = \mu_B B \), where \( \mu_B = 9.274 \times 10^{-24} \text{ J/T} \) (Bohr magneton) and \( B = 0.600 \text{ T} \). Thus, \( \Delta E = 9.274 \times 10^{-24} \times 0.600 = 5.5644 \times 10^{-24} \text{ J} \).

Step 4: Calculate Energy Separation between Extremes

The energy separation between the level of lowest energy (\(m_l = -4\)) and the level of highest energy (\(m_l = +4\)) is given by \( \Delta E_{total} = (m_l^{max} - m_l^{min}) \times \Delta E \). Here, \(m_l^{max} = +4\) and \(m_l^{min} = -4\), so \(m_l^{max} - m_l^{min} = 8\). Thus, the total separation is \(8 \times 5.5644 \times 10^{-24} = 4.45152 \times 10^{-23} \text{ J} \).

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest and most fundamental system, consisting of a single proton and a single electron. Due to its simplicity, it serves as a vital model for understanding more complex atoms and quantum mechanics. Most often, it is described by the Bohr model or through quantum mechanics, where the electron exists in specific quantized energy levels or "states." In this context, these states are determined by quantum numbers which dictate the electron's energy, position, and angular momentum. The hydrogen atom in this exercise is in the 5\(g\) state, highlighting its position in the fifth energy level and specifically within the \(g\) orbital type. Understanding the behavior of the hydrogen atom in various states, such as when interacting with magnetic fields, helps us comprehend atomic behavior and spectral lines.

Quantum Numbers

Quantum numbers are essential in the realm of quantum mechanics as they describe values of conserved quantities in the dynamics of the quantum system.

• The principal quantum number \(n\) signifies the energy level and the size of the orbital. For the 5\(g\) state, \(n = 5\).
• The azimuthal (or angular momentum) quantum number \(l\) is connected to the shape of the orbital. For \(g\) type orbitals, \(l = 4\).
• The magnetic quantum number \(m_l\) describes the orientation of the orbital in space relative to an external magnetic field. It can take values ranging from \(-l\) to \(+l\).

For a hydrogen atom in the 5\(g\) state, \(m_l\) values range from \(-4\) to \(+4\), which results in multiple energy levels when subjected to a magnetic field.

Magnetic Dipole Moment

The magnetic dipole moment is a measure of the magnetic strength and orientation of a magnetic source. For an electron in an atom, this property arises due to its angular momentum. In the presence of an external magnetic field, such as in the Zeeman effect, the magnetic dipole moment interacts with the magnetic field leading to the energy level splitting observed as spectral line splitting. The Bohr magneton \(\mu_B\), which is approximately \(9.274 \times 10^{-24} \text{ J/T}\), is the physical constant that quantifies the total magnetic moment associated with the angular momentum of the electron. This interaction causes the otherwise identical energy levels to shift depending on the orientation described by \(m_l\).

Energy Levels

Energy levels in an atom represent the different states an electron can occupy, each with a discrete energy value. In the presence of a magnetic field, these energy levels can split, a phenomenon known as the Zeeman Effect. When considering the Zeeman Effect, the energy level splitting happens due to the interaction of atomic magnetic moments with the external magnetic field. Each possible value of the magnetic quantum number \(m_l\) corresponds to a different energy level.
The energy difference between these levels can be calculated using the formula:\( \Delta E = \mu_B B \),where \(B\) is the magnetic field strength. For adjacent levels, this expression gives us the energy separation of \(5.5644 \times 10^{-24} \text{ J}\). The total shift between the extreme \(m_l\) values in a hydrogen atom in a 5\(g\) state is \(4.45152 \times 10^{-23} \text{ J}\), corresponding to a clear and measurable splitting of the spectral lines.