A hydrogen atom is in a d state. In the absence of an external magnetic field, the states with different ml 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 ml levels when the atom is put in a 0.800-T magnetic field that is in the +z@direction

Verified step by step guidance

1. Identify the quantum numbers for a hydrogen atom in a d state. The principal quantum number n can be any integer greater than or equal to 3, and the azimuthal quantum number l is 2 for a d state. The magnetic quantum number ml can take values from -l to +l, which are -2, -1, 0, 1, and 2.
2. Understand the interaction between the magnetic field and the atom's orbital magnetic dipole moment. The energy shift due to the magnetic field is given by the Zeeman effect, which can be expressed as ΔE = ml * μB * B, where μB is the Bohr magneton and B is the magnetic field strength.
3. Calculate the Bohr magneton (μB) using the formula: μB = (e * ħ) / (2 * me), where e is the elementary charge, ħ is the reduced Planck's constant, and me is the electron mass.
4. Substitute the values for the magnetic field (B = 0.800 T) and the Bohr magneton (μB) into the Zeeman effect formula for each ml value. This will give you the energy shift ΔE for each ml level.
5. Determine the energy splitting between the ml levels by calculating the difference in energy shifts for consecutive ml values. This will give you the splitting in electron volts (eV) for the hydrogen atom in the magnetic field.

Key Concepts

Zeeman Effect

The Zeeman Effect describes the splitting of a spectral line into several components in the presence of a static magnetic field. This occurs because the magnetic field interacts with the magnetic dipole moment associated with the angular momentum of electrons, causing energy level shifts. The magnitude of the splitting depends on the strength of the magnetic field and the magnetic quantum number (ml).

Magnetic Quantum Number (ml)

The magnetic quantum number, ml, is an integer that describes the orientation of an electron's orbital angular momentum in a magnetic field. It can take values from -l to +l, where l is the azimuthal quantum number. In the presence of a magnetic field, different ml values correspond to different energy levels due to the interaction with the field, leading to the splitting of degenerate states.

Orbital Magnetic Dipole Moment

The orbital magnetic dipole moment is a vector quantity associated with the motion of an electron in an atom. It arises from the electron's orbital angular momentum and is proportional to the magnetic quantum number (ml). In a magnetic field, this dipole moment interacts with the field, causing shifts in the energy levels of the atom, which is the basis for the Zeeman Effect.