CP A hydrogen atom undergoes a transition from a 2p state to...

CP A hydrogen atom undergoes a transition from a 2$p$ state to the 1 s ground state. In the absence of a magnetic field, the energy of the photon emitted is 122 nm. The atom is then placed in a strong magnetic field in the $z$ -direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. (a) How many different photon wave- lengths are observed for the 2$p \rightarrow 1 s$ transition? What are the $m_{l}$ values for the initial and final states for the transition that leads to each photon wavelength? (b) One observed wavelength is exactly the same with the magnetic field as without. What are the initial and final $m_{l}$ values for the transition that produces a photon of this wavelength? (c) One observed wavelength with the field is longer than the wavelength without the field. What are the initial and final $m_{l}$ values for the transition that produces a photon of this wave-length? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field.

CP A hydrogen atom undergoes a transition from a 2$p$ state to the 1 s ground state. In the absence of a magnetic field, the energy of the photon emitted is 122 nm. The atom is then placed in
a strong magnetic field in the $z$ -direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. (a) How many different photon wave-
lengths are observed for the 2$p \rightarrow 1 s$ transition? What are the $m_{l}$ values for the initial and final states for the transition that leads to each photon wavelength? (b) One observed wavelength is exactly the same with the magnetic field as without. What are the initial and final $m_{l}$ values for the transition that produces a photon of this wavelength? (c) One observed wavelength with the field is longer than the wavelength without the field. What are the initial and final $m_{l}$ values for the transition that produces a photon of this wave-length? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field.

Key Concepts

Hydrogen Atom Energy Levels

Hydrogen atom energy levels are determined by quantum numbers, including the principal quantum number n, which dictates the energy level, and the orbital quantum number l, which indicates the shape of the electron's orbital. Transitions between these quantized energy levels involve the emission or absorption of photons whose energies correspond to the difference in the electron’s energy. This forms the basis for understanding spectral lines in atomic physics.

Orbital Angular Momentum and Magnetic Quantum Number

The orbital angular momentum of an electron is quantized and described by the quantum number l, while its orientation in space is given by the magnetic quantum number m_l. The magnetic quantum number takes integer values from -l to +l, and it plays a crucial role in determining how the atom interacts with an external magnetic field, especially regarding the splitting of degenerate energy levels in phenomena such as the Zeeman effect.

Zeeman Effect

The Zeeman effect refers to the splitting of atomic energy levels when an external magnetic field is applied. In the presence of a magnetic field, the degeneracy of the energy levels is lifted due to the interaction of the magnetic field with the atom's orbital magnetic moment. This results in multiple closely spaced spectral lines rather than a single line, with each line corresponding to transitions between levels split by the magnetic field.

Interaction of the Magnetic Field with the Orbital Magnetic Moment

When an atom is subjected to a magnetic field, its orbital magnetic moment interacts with the field, causing shifts in the energy levels. These shifts depend on the magnetic quantum number m_l and the strength of the magnetic field. The differing energy shifts for each m_l state mean that transitions between energy levels can occur at several distinct energies, leading to the observation of multiple photon wavelengths for a given electronic transition.

Selection Rules for Electric Dipole Transitions

Electric dipole transitions in atoms are governed by specific selection rules, which include the requirement that the change in the magnetic quantum number ?m_l is 0 or ±1. These rules determine which transitions between split energy levels are allowed. In the context of the Zeeman effect, the allowed transitions are dictated by these rules, thereby directly influencing which photon wavelengths are observable when the atom is in an external magnetic field.

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