Problem 14 - University Physics with Modern Physics

Question

A hydrogen atom is in a \(d\) state. In the absence of an external magnetic field the states with different \(m_{I}\) values have (approximately) the same energy. Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. (a) Calculate the splitting (in electron volts) of the \(m_{l}\) levels when the atom is put in a 0.400 - T magnetic field that is in the \(+\) z-direction. (b) Which \(m_{l}\) level will have the lowest energy? (c) Draw an energy-level diagram that shows the \(d\) levels with and without the external magnetic field.

Step-by-Step Solution

Answer

(a) \(0.400\) T splits levels by up to \(4.63 \times 10^{-5}\) eV. (b) \(m_l = -2\) has the lowest energy. (c) Diagram shows splitting by \(m_l\) value.

Step 1: Understanding the Zeeman Effect

The interaction of an external magnetic field with the magnetic dipole moment of an atom leads to the Zeeman effect, which causes the energy levels to split. The change in energy is given by the formula: \[\Delta E = m_l \mu_B B\]where \(m_l\) is the magnetic quantum number, \(\mu_B\) is the Bohr magneton (approximately \(9.274 \times 10^{-24}\) J/T), and \(B\) is the magnetic field strength. For a \(d\) state, \(l = 2\), so \(m_l\) can be \(-2, -1, 0, 1, 2\).

Step 2: Calculate the energy splitting for each \(m_l\) level

Substitute the given magnetic field \(B = 0.400\) T into the formula for each \(m_l\) value:For \(m_l = 2\):\[\Delta E = 2 \times 9.274 \times 10^{-24} \times 0.400 = 7.4192 \times 10^{-24} \text{ J}\]For \(m_l = 1\):\[\Delta E = 1 \times 9.274 \times 10^{-24} \times 0.400 = 3.7096 \times 10^{-24} \text{ J}\]For \(m_l = 0\):\[\Delta E = 0 \text{ J}\]For \(m_l = -1\):\[\Delta E = -1 \times 9.274 \times 10^{-24} \times 0.400 = -3.7096 \times 10^{-24} \text{ J}\]For \(m_l = -2\):\[\Delta E = -2 \times 9.274 \times 10^{-24} \times 0.400 = -7.4192 \times 10^{-24} \text{ J}\]

Step 3: Convert energy splitting to electron volts

To convert energy from joules to electron volts, use the conversion factor:\[1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \]For \(m_l = \pm 2\):\[\Delta E_{eV} = \frac{7.4192 \times 10^{-24}}{1.602 \times 10^{-19}} \approx 4.63 \times 10^{-5} \text{ eV}\]For \(m_l = \pm 1\):\[\Delta E_{eV} = \frac{3.7096 \times 10^{-24}}{1.602 \times 10^{-19}} \approx 2.31 \times 10^{-5} \text{ eV}\]

Step 4: Determine the lowest energy level

Since the energy shift \(\Delta E = m_l \mu_B B\), the lowest energy corresponds to the most negative \(\Delta E\). For \(m_l = -2\), the energy shift is the most negative at \(-7.4192 \times 10^{-24}\) J (or \(-4.63 \times 10^{-5}\) eV). Thus, the \(m_l = -2\) level will have the lowest energy.

Step 5: Sketch the energy-level diagram

Draw the energy levels with \(m_l = -2, -1, 0, 1, 2\) both without and with the magnetic field. Without the field, these levels will be degenerate (same energy). With the field, the levels will split according to their \(m_l\) values, with spacing consistent with calculated energy shifts:- \(m_l = 2\) at the highest energy- \(m_l = 1\) slightly lower- \(m_l = 0\) at the reference line- \(m_l = -1\) slightly lower than 0- \(m_l = -2\) at the lowest energy

Key Concepts

Magnetic Quantum Number

The magnetic quantum number, denoted as \(m_l\), plays a crucial role in atomic physics. It is part of the quantum numbers used to define the unique quantum state of an electron in an atom. The value of \(m_l\) determines the orientation of the orbital's angular momentum in a magnetic field.
\[m_l\] can range from \(-l\) to \(+l\), where \(l\) is the azimuthal or orbital quantum number. For a \(d\) state, \(l = 2\), which results in five possible values for \(m_l\): \(-2, -1, 0, 1, 2\).
This diversity of \(m_l\) values allows the electron’s angular momentum to align differently in an external magnetic field, contributing to variations in energy levels.

• The presence of a magnetic field results in energy differences due to Zeeman effect.
• These differences are determined by the formula \(\Delta E = m_l \mu_B B\), where \(\mu_B\) is the Bohr magneton and \(B\) is the magnetic field strength.

Bohr Magneton

The Bohr magneton, symbolized as \(\mu_B\), is a fundamental physical constant in quantum mechanics. It measures the magnetic moment of an electron caused by its angular momentum.
It quantifies the magnetic dipole moment due to an electron's orbital motion around the nucleus. For most calculations related to electron magnetic properties, the Bohr magneton serves as the standard unit. Its approximate value is \(9.274 \times 10^{-24}\) Joules per Tesla (J/T).
Understanding \(\mu_B\) is important because it links the magnetic quantum number and magnetic field strength to energy changes in a system. This linkage is critical in understanding phenomena like the Zeeman effect.

Hydrogen Atom Energy Levels

Energy levels of the hydrogen atom are quantized, meaning only certain discrete energy values are possible. These levels depend on quantum numbers, particularly the principal quantum number \(n\), which can take positive integer values \(n = 1, 2, 3, \ldots\).
Since hydrogen has only one electron, the energy levels are well-defined and provide clear insight into atomic structure. In the context of the Zeeman effect:

• Energy levels which are normally degenerate (same energy) in the absence of a magnetic field are split into distinct levels.
• The splitting is determined by the orientation of \(m_l\) in the external magnetic field.

This splitting must be considered especially when analyzing atomic spectra or electron interactions in more complex environments.

Electron Volt Conversion

One common unit of energy in quantum mechanics is the electron volt (eV). It is preferred in these discussions due to its convenience in quantifying small energy changes and particle interactions.
1 electron volt is equal to \(1.602 \times 10^{-19}\) Joules. This conversion factor is crucial when translating between the standard energy unit used in physics (Joules) and the more manageable unit applied in atomic studies (electron volts).
Converting from Joules to eV involves dividing the energy in Joules by this conversion factor, allowing clearer representation of energy changes such as those in the Zeeman effect. This scaling provides significant insights for physicists into small-scale energy phenomena found within atomic structures.