Question

Assuming that the spin-orbit interaction is not overwhelmed by an extemal magnetic field, what is the minimum angle the toral angular momentum vector J may make with the \(z\) -axis in a \(3 d\) state of hydrogen?

Short answer

The minimum angle that the total angular momentum vector, J, may make with the z-axis in a 3d state of hydrogen is 0 degrees.

Step-by-step solution

Identify the relevant quantum numbers

The quantum number corresponding to the orbital shape for a 3d state of hydrogen is the azimuthal quantum number, \(l\), which is equal to 2. The d letter in 3d state is the conventional designation for \(l = 2\). The total angular momentum is given by \(J = |l - s| \) to \(l + s.\) For an electron, the spin quantum number \(s\) is equal to 1/2.

Apply the projection of J over z-axis

The maximum value for the projection of the total angular momentum along the z-axis is given by \(m_j\) = J. The value of \(m_j\) can range from -J to +J. In the minimum angle case, \(m_j\) will be equal to J, so the total angular momentum J will be projected in the same direction of the z-axis. For now, and since spin orbit interaction is not overwhelmed, let's assume that this value of J, let's call it \(J_{max}\), is the actual maximum possible for J, so it corresponds to \(J_{max} = l + s.\)

Calculate \(J_{max}\)

The calculation is done by adding \(l\) and \(s\), which gives \(J_{max} = l + s = 2 + 1/2 = 5/2\).

Calculate minimum angle theta

The minimum angle theta is given by the relation: \(cos(\theta) = m_j / J_{max}\). Note that all \(J_{max}\) are projected along the z-axis. plugging in our calculated values, we have \(cos(\theta) = 5/2 / 5/2 = 1\), so \(\theta = arccos(1) = 0\) degrees. This means that at the minimum angle, the total angular momentum vector J is aligned along the z-axis.

Key concepts

Spin-Orbit Interaction

The spin-orbit interaction in quantum mechanics is a complex phenomenon where the intrinsic spin of a particle, such as an electron, interacts with its orbital motion around the nucleus. This interaction is a pivotal aspect of atomic physics, affecting energy levels and the magnetic properties of atoms.

Imagine an electron orbiting a nucleus, much like a planet orbits the sun. Now, besides this orbital movement, the electron also has a spin, a form of angular momentum where it's 'twisting' on its own axis. The electron’s spin interacts with the magnetic field generated by its orbital motion, leading to slight shifts in the energy levels known as fine structure.

This interaction is stronger in heavier atoms, where the nuclear charge is greater, leading to more significant shifts. It plays a crucial role in determining the atomic spectra and the magnetic moments of particles. While it might seem abstract, understanding the spin-orbit interaction is essential for explaining the structure of complex atoms and the behavior of electrons in various materials, including semiconductors and superconductors.

Quantum Numbers

Quantum numbers are the digital fingerprints of an electron, giving us a detailed ID of its energy, position, and spin within an atom. These numbers are the bread and butter for anyone studying quantum mechanics, as they describe the state of an electron in a way that's entirely unique.

There are four key quantum numbers:

• The principal quantum number ( \(n\)), which indicates an electron's energy level and distance from the nucleus.
• The azimuthal quantum number ( \(l\)), denoting the shape of the electron's orbital. For instance, \(l = 2\) is associated with a 'd' orbital, where we find lobed patterns of electron probability.
• The magnetic quantum number ( \(m_l\)), which tells us the orientation of the orbital in space.
• The spin quantum number ( \(s\)), which represents the direction of the electron's intrinsic spin, either up ( \(\frac{1}{2}\)) or down ( \(-\frac{1}{2}\)).

It's these numbers that define the permissible states electrons can have in an atom and are fundamental in solving quantum mechanical problems. Each electron in an atom has its own set of quantum numbers — no two can be identical, thanks to the Pauli Exclusion Principle.

Angular Momentum Projection

Angular momentum projection is about understanding how much of a particle's angular momentum is pointing in a particular direction — most often, the z-axis, which is chosen by convention. This aspect of quantum mechanics is often visualized by picturing a top spinning while tilted; the spin represents the total angular momentum, and its shadow on the ground represents the angular momentum's projection on the z-axis.

In quantum mechanics, the projection of an electron's total angular momentum ( \(J\)) is represented by a quantum number ( \(m_j\)). The value of \(m_j\) ranges between the negative and positive values of \(J\), and the magnitude of this projection is what determines the angular relationship to the chosen axis.

Understanding this concept is critical when we tackle problems involving quantum states and the orientation of particles within a magnetic field. For instance, the exercise above asks for the minimum angle between the total angular momentum vector \(J\) and the z-axis in a \(3d\) state of hydrogen, reflecting the direct link between angular momentum projection and the physical orientation of quantum objects in space.