Problem 47 - University Physics with Modern Physics

Question

Consider the transition from a 3\(d\) to a 2\(p\) state of hydrogen in an external magnetic field. Assume that the effects of electron spin can be ignored (which is not actually the case) so that the magnetic field interacts only with the orbital angular momentum. Identify each allowed transition by the \(m_{I}\) values of the initial and final states. For each of these allowed transitions, determine the shift of the transition energy from the zero-field value and show that there are three different transition energies.

Step-by-Step Solution

Answer

3 different transition energies occur with shifts of \( -\mu_B B\), \( 0 \), and \(+\mu_B B\).

Step 1: Understand what the quantum numbers mean

The quantum number \(m_l\) represents the magnetic quantum number. For a \(3d\) state, the principal quantum number \(n=3\) and the azimuthal quantum number \(l=2\). So the possible \(m_l\) values are \(-2, -1, 0, 1, 2\). For a \(2p\) state, \(n=2\) and \(l=1\), with possible \(m_l\) values of \(-1, 0, +1\). Transitions between these states are allowed if \(\Delta m_l = 0, \pm1\).

Step 2: Identify allowed transitions using \(m_l\) values

Considering the allowed transitions where \(\Delta m_l = 0, \pm 1\), we list all possible transitions. From \(3d\) \((-2, -1, 0, 1, 2)\) to \(2p\) \((-1, 0, +1)\), the allowed transitions are: \((-2 \rightarrow -1), (-1 \rightarrow 0), (0 \rightarrow -1), (0 \rightarrow 1), (1 \rightarrow 0), (1 \rightarrow 1), (2 \rightarrow 1)\).

Step 3: Determine energy shift for each transition due to the magnetic field

The energy shift due to a magnetic field \(B\) is given by \(\Delta E = \mu_B B \Delta m_l\), where \(\mu_B\) is the Bohr magneton. For each allowed transition, calculate \(\Delta m_l\) and thus the energy shift. \((-2 \rightarrow -1): +\mu_B B\), \((-1 \rightarrow 0): +\mu_B B\), \( (0 \rightarrow -1): -\mu_B B\), \((0 \rightarrow 1): +\mu_B B\), \((1 \rightarrow 0): -\mu_B B\), \((1 \rightarrow 1): 0\), \((2 \rightarrow 1): -\mu_B B\).

Step 4: Identify and count unique transition energies

Upon calculating the shifts, we find three unique shift values: \(-\mu_B B\), \(0\), and \(+\mu_B B\). This means that there are three distinct transition energies for these shifts, confirming the existence of three different transition energies: one shift for each value of \(\Delta E\).

Key Concepts

Quantum Numbers: Decoding the Hydrogen Atom

Quantum numbers are essential in understanding the electron's position and behavior in an atom. They define the unique state of an electron. Here's a glance at what each number represents in a hydrogen atom:

- **Principal Quantum Number \( n \)**: It determines the electron's energy level and size of the orbital. The larger the number, the higher the energy level.
- **Azimuthal (Angular Momentum) Quantum Number \( l \)**: It defines the shape of the orbital and is related to the angular momentum. For a given \( n \, l \) can range from 0 to \( n-1 \).
- **Magnetic Quantum Number \( m_l \)**: This number indicates the number of orbitals and their orientation in a magnetic field. It ranges from \( -l \) to \( +l \).

For the hydrogen atom transitioning from a \(3d \) (\(n=3, l=2\)) state to a \(2p \) (\(n=2, l=1\)) state, the magnetic quantum number (\