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Magnetic quantum number

physical science Maturity 11-13 Next article →

In the study of atomic physics, scientists use specific numbers to describe the state of an electron. These are called quantum numbers. One very important type is the magnetic quantum number. This number helps distinguish the quantum states of an electron or other particles. It specifically identifies the angular momentum along a chosen axis in space. This chosen direction is usually called the z-axis.

Vector model of orbital angular momentum
Vector model of orbital angular momentum

By using this number, we can understand how a particle is oriented in space.

There are two main types of magnetic quantum numbers. The first is the orbital magnetic quantum number, often written as $m_l$. This number tells us which orbitals are available within a specific subshell of an atom. It describes the component of the orbital angular momentum that lies along the z-axis. The second type is the spin magnetic quantum number, written as $m_s$. This describes the z-axis component of the spin angular momentum for a particle. For an electron, the spin quantum number is $1/2$. The spin magnetic quantum number can be either +1/2 or -1/2. Scientists often call these "spin-up" and "spin-down" states.

Atomic orbitals spdf m-eigenstates
Atomic orbitals spdf m-eigenstates

To understand the math, we look at the wavefunction of an electron. The wavefunction describes the complete quantum state of a single electron in an atom. This is often solved using the Schrödinger equation. For an atom with only one electron, this equation is a separable partial differential equation. This means the wavefunction can be broken into three separate functions. These functions represent the radius, the colatitude angle, and the azimuth angle. The magnetic quantum number arises from the math of the azimuth angle. Because the position repeats every 360 degrees, the coefficient must be an integer multiple of $i$. These integers are the magnetic quantum numbers.

These numbers define the structure of the periodic table. Electrons live in subshells like s, p, d, or f. These subshells are defined by the azimuthal quantum number, $l$. The orbital magnetic quantum number $m_l$ takes integer values from $-l$ to $+l$. For example, an s subshell has an $l$ value of 0. This means it has only 1 orbital. A p subshell has $l=1$, which allows for 3 orbitals. A d subshell has $l=2$, which allows for 5 orbitals. An f subshell has $l=3$, which allows for 7 orbitals. Each of these orbitals can hold up to two electrons. These electrons must have opposite spins to fit in the same orbital.

Atomic orbitals spdf m-eigenstates
Atomic orbitals spdf m-eigenstates

We can see the relationship between these numbers in a clear pattern. The number of orbitals in a subshell increases as the azimuthal number grows. An s subshell holds 2 electrons. A p subshell holds 6 electrons. A d subshell holds 10 electrons. An f subshell holds 14 electrons. If we look even further, a g subshell has $l=4$. This would result in 9 orbitals and 18 electrons. This organized structure is how we understand the arrangement of all elements.

Atomic orbitals spdf m-eigenstates
Atomic orbitals spdf m-eigenstates

The name "magnetic" comes from how these particles react to magnetic fields. Every type of angular momentum has an associated magnetic dipole moment. When an atom is placed in an external magnetic field, its energy levels shift. This specific phenomenon is known as the Zeeman effect. The magnetic field also exerts a force called torque on the electron. This torque causes a movement called Larmor precession. This happens because the electron's magnetic moment tries to align with the field.

Vector model of orbital angular momentum
Vector model of orbital angular momentum

History shows us how we first proved these tiny behaviors. Two scientists, Otto Stern and Walther Gerlach, performed a famous experiment. It is known as the Stern-Gerlach experiment. Their work demonstrated these properties of particles in space. This helped confirm how angular momentum and spin actually work. Today, we use these concepts to study everything from basic atoms to complex nuclear spin. We even use capitalized versions of these numbers to describe the total angular momentum of an entire system of particles.

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File:Atomic orbitals spdf m-eigenstates.png
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File:Vector model of orbital angular momentum.svg
Vector model of orbital angular momentum.svg
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