A hydrogen atom is in a d state. In the absence of an...

A hydrogen atom is in a $d$ state. In the absence of an external magnetic field, the states with different $m_{l}$ 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 $0.400 \mathrm{~T}$ 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.

A hydrogen atom is in a $d$ state. In the absence of an external magnetic field, the states with different $m_{l}$ 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 $0.400 \mathrm{~T}$ 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.

Key Concepts

Quantum Numbers

Quantum numbers such as the orbital quantum number (l) and the magnetic quantum number (m?) characterize the discrete energy states and orientations of electrons within an atom. These numbers are essential in determining the structure of atomic energy levels and predicting how these levels split under external influences like magnetic fields.

Orbital Angular Momentum

This quantum property of electrons characterizes the electron's motion around the nucleus and is quantified by the orbital quantum number (l). It determines the shape and symmetry of the electron's probability distribution and plays a crucial role in how the electron interacts with external magnetic fields through its contribution to the magnetic dipole moment.

Magnetic Dipole Moment

This is the property of a system that causes it to experience a torque in the presence of a magnetic field. For electrons in atoms, the orbital motion generates a magnetic dipole moment proportional to the angular momentum. This moment is what interacts with an external magnetic field to cause energy level shifts, as seen in phenomena like the Zeeman effect.

Bohr Magneton

The Bohr magneton is a fundamental physical constant that provides the natural unit for the magnetic dipole moment of an electron due to its orbital or spin motion. It is used to quantify the contribution of the electron’s motion to its magnetic dipole moment and appears in formulas calculating energy shifts in a magnetic field.

Zeeman Effect

This effect refers to the splitting of atomic energy levels when an external magnetic field is applied. The energy splitting arises from the interaction between the magnetic dipole moment of the electron (which depends on the orbital angular momentum and characterized by quantum numbers) and the external magnetic field, resulting in different energy shifts for different magnetic quantum numbers.

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