A hydrogen atom undergoes a transition from a 2p state to the 1s 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 wavelengths are observed for the 2p S 1s transition? What are the ml values for the initial and final states for the transition that leads to each photon wavelength?

Verified step by step guidance

1. Understand the Zeeman effect: In the presence of a magnetic field, the energy levels of atomic orbitals split due to the interaction of the magnetic field with the orbital magnetic moment. This is known as the Zeeman effect.
2. Identify the initial and final states: The initial state is 2p, which has an orbital angular momentum quantum number l = 1. The final state is 1s, which has l = 0.
3. Determine the possible magnetic quantum numbers (ml): For the 2p state, ml can take values of -1, 0, and +1. For the 1s state, ml is 0.
4. Calculate the number of transitions: Each initial ml value can transition to the final ml value of 0. Therefore, there are three possible transitions: (ml = -1 to ml = 0), (ml = 0 to ml = 0), and (ml = +1 to ml = 0).
5. Conclude the number of different photon wavelengths: Since there are three possible transitions due to the different initial ml values, three different photon wavelengths will be observed in the presence of the magnetic field.

Key Concepts

Atomic Energy Levels and Transitions

In an atom, electrons occupy specific energy levels or orbitals. When an electron transitions between these levels, it absorbs or emits a photon with energy equal to the difference between the initial and final states. For a hydrogen atom, the transition from a 2p state to a 1s state involves the emission of a photon, which can be calculated using the Rydberg formula.

Zeeman Effect

The Zeeman effect describes the splitting of atomic energy levels in the presence of a magnetic field. This occurs because the magnetic field interacts with the magnetic moment associated with the electron's orbital angular momentum. The splitting results in multiple possible transitions, each corresponding to a different photon wavelength, depending on the magnetic quantum number (ml) of the initial and final states.

Magnetic Quantum Number (ml)

The magnetic quantum number (ml) determines the orientation of an electron's orbital angular momentum in a magnetic field. It can take integer values from -l to +l, where l is the azimuthal quantum number. In the presence of a magnetic field, transitions between states with different ml values lead to the observation of multiple photon wavelengths due to the Zeeman effect.