Orbital Angular Momentum Quantization

In one sentence: An electron’s turning motion around the center of an atom can only take certain fixed amounts, like steps on a staircase.

$|\mathbf{L}| = \sqrt{\ell(\ell + 1)}\,\hbar \qquad L_z = m_\ell\,\hbar \qquad m_\ell = -\ell, \ldots, \ell$

Read it out loud: “The length of the vector L, written with bars around the bold letter L, equals the square root of ell times ell plus one, multiplied by h-bar. Ell is the orbital quantum number, a whole number. The z component of L, written L sub z, equals m sub ell times h-bar. Here m sub ell is the magnetic quantum number, a whole number that runs in steps from minus ell up to plus ell.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Angular momentum: Angular momentum measures how much a thing turns or orbits. A spinning top has it, and so does a planet going around the Sun. An electron moving around the nucleus, the small heavy center of an atom, also has it. For a particle moving in a circle it grows with the mass, the speed and the distance from the center. Its unit is joule seconds, J s, the same unit as Planck’s constant.
Vectors and bold letters, $\mathbf{L}$: A vector is a quantity with both a size and a direction, drawn as an arrow. A bold letter such as $\mathbf{L}$ marks a vector. The bars in $|\mathbf{L}|$ mean the length of that arrow, which is its size with no direction. A plain number, such as 5, has no direction at all.
Quantum numbers, $\ell$ and $m_\ell$: A quantum number is a whole-number label that picks out one of the allowed states of a particle. Quantum means the allowed values come in steps, not smoothly. Whole numbers are 0, 1, 2, 3 and so on, and they can be negative, like $-2$ and $-1$ . The Greek letter $\ell$ is called ell, and the subscript in $m_\ell$ is read m sub ell.
Subscripts, like $L_z$: A subscript is a small label written below a letter, which tells similar quantities apart. In $L_z$ the subscript $z$ means the component along the z axis, one chosen direction. In $m_\ell$ the subscript $\ell$ says that the number belongs to the orbital quantum number $\ell$ . The subscript is part of the name, so $L_z$ is read L sub z.
Square roots and products, like $\sqrt{\ell(\ell+1)}$: The product $\ell(\ell+1)$ multiplies a number by the next whole number. For $\ell = 2$ that is $2 \times 3 = 6$ . The square root $\sqrt{6}$ is the number that, multiplied by itself, gives 6, which is about 2.449. The formula takes the square root of the product to get the length.
Powers of ten, like $10^{-34}$: A power of ten shows how many times to divide by 10 or multiply by 10. The power $10^{-34}$ means 1 divided by $10^{34}$ , which is a decimal point followed by 33 zeros and then a 1. Multiplying powers adds exponents, and dividing subtracts them. So $2.449 \times 10^{-34}$ is a number with the same power of ten and a front number of 2.449.
The reduced Planck constant $\hbar$ and the unit J s: $\hbar$ , said h-bar, is Planck’s constant divided by $2\pi$ . Its value is $1.054571817 \times 10^{-34}$ J s, which means joule seconds. Multiplying a number in joule seconds by a plain number leaves joule seconds. The constant is tiny, which is why the steps of angular momentum are invisible in everyday objects.
Orbitals and the letters s, p, d and f: An orbital is a region where an electron in an atom is likely to be found, with a definite shape. Chemists name orbitals by the value of $\ell$ : $\ell = 0$ is an s orbital, $\ell = 1$ is p, $\ell = 2$ is d and $\ell = 3$ is f. The letters come from early descriptions of spectral lines, and they have stayed in use ever since.

What each symbol means

Symbol	Meaning
$\|\mathbf{L}\|$	The length, or magnitude, of the orbital angular momentum vector. It is measured in joule seconds, J s. Its value comes from the rule $\sqrt{\ell(\ell+1)}\,\hbar$ , which is about $2.45\,\hbar$ for $\ell = 2$ . The bars mean length, so this is a size with no direction.
$\ell$	The orbital quantum number, a whole number 0, 1, 2, 3 and so on. It has no unit, because it is only a label. It sets the size of the angular momentum, and it sets which letter, s, p, d or f, names the orbital. The d electron in the example has $\ell = 2$ .
$L_z$	The component of the angular momentum along the chosen z axis. It tells how much of the vector points along that one direction. It is measured in joule seconds, J s, like the length. It can only take the values $m_\ell \hbar$ , so it is a sharp step, not a smooth number.
$m_\ell$	The magnetic quantum number, a whole number from $-\ell$ to $+\ell$ . Negative values mean the component points opposite to the positive z direction. It has no unit, because it is a count of steps. For $\ell = 2$ its allowed values are $-2, -1, 0, 1, 2$ , which is five values.
$\hbar$	The reduced Planck constant, $1.054571817 \times 10^{-34}$ J s. It is the step size of angular momentum. Every length and component in this formula is a whole number, or a square root, times $\hbar$ .

The formula piece by piece

$\ell$: Start with the orbital quantum number, a whole number. It picks the size of the angular momentum, and it must be 0, 1, 2 and so on, never a fraction.
$\ell(\ell+1)$: Multiply $\ell$ by the next whole number. For $\ell = 2$ this gives $2 \times 3 = 6$ . This product is the standard result for the squared length, found by solving the equation for the atom, and it is not a guess.
$\sqrt{\ell(\ell+1)}$: Take the square root of the product, which turns the squared length into the length itself. For $\ell = 2$ it gives $\sqrt{6} \approx 2.449$ , which counts the length in steps of $\hbar$ .
$\sqrt{\ell(\ell+1)}\,\hbar$: Multiply by $\hbar$ to get the length in joule seconds. For $\ell = 2$ the length is about $2.58 \times 10^{-34}$ J s.
$m_\ell$: The magnetic number, which sets the component along the z axis. It is a whole number from $-\ell$ to $+\ell$ , so for $\ell = 2$ it takes five values, from $-2$ to $+2$ .
$L_z = m_\ell\,\hbar$: The z component is the magnetic number times $\hbar$ . Each value of $m_\ell$ gives one sharp z component, in steps of $\hbar$ .
$|\mathbf{L}| = \sqrt{\ell(\ell + 1)}\,\hbar \qquad L_z = m_\ell\,\hbar \qquad m_\ell = -\ell, \ldots, \ell$: The orbital angular momentum of an electron in an atom has a length set by $\ell$ , its z component is a whole-number step times $\hbar$ , and the whole-number steps run from $-\ell$ to $+\ell$ .

What the formula is doing, and why

Only certain sizes are allowed. An electron’s orbital angular momentum in an atom is quantized, which means it can take only certain fixed sizes. The size is set by the whole number $\ell$ through $|\mathbf{L}| = \sqrt{\ell(\ell+1)}\,\hbar$ . The values of $\ell$ are 0, 1, 2 and so on. Chemists label them s, p, d and f for $\ell = 0, 1, 2, 3$ .

Only one direction is sharp. Only one component, by convention the z axis, can have a sharp value at a time. Its value is $L_z = m_\ell\hbar$ . The number $m_\ell$ is a whole number from $-\ell$ to $+\ell$ , which gives $2\ell + 1$ allowed values. Why $2\ell + 1$ ? Count the whole numbers from $-\ell$ to $\ell$ : there are $\ell$ negative ones, $\ell$ positive ones and zero. For $\ell = 1$ there are three values: $-1$ , $0$ and $+1$ .

Why the length is never along z. The length is $\sqrt{\ell(\ell+1)}\,\hbar$ , and $\sqrt{\ell(\ell+1)}$ is always bigger than $\ell$ , because $\ell^2 + \ell$ is bigger than $\ell^2$ . So the vector is always longer than its largest z component. For $\ell = 1$ the length is $\sqrt{2}\,\hbar \approx 1.41\,\hbar$ , while the largest $L_z$ is $\hbar$ . The vector can never point exactly along the z axis.

Why whole-number steps. Going once around the z axis is a full circle, and the wave of the electron must come back to the same value after one full turn. Only whole-number steps of $m_\ell$ make that happen. This is one reason the values are whole numbers. The rules come from solving the Schrödinger equation for the hydrogen atom in spherical coordinates, a system that locates a point with a distance and two angles.

These rules are general. The same rules hold for any electron in a spherically symmetric potential, which means a pull or push that depends only on distance from one center. That is why the same s, p, d and f labels describe orbitals in heavier atoms too, not only in hydrogen.

An everyday picture

Think of a staircase rather than a ramp. A ramp lets you stand at any height, but a staircase lets you stand only on its steps. Orbital angular momentum stands only on steps set by $\hbar$ . The size of the step pattern comes from $\ell$ , and the z component can only sit on one of a few marked positions along the axis.

Worked example: A d electron with $\ell = 2$

A d electron has orbital quantum number $\ell = 2$ . Use $\hbar = 1.054571817 \times 10^{-34}$ J s. Find the length of its angular momentum in joule seconds, the number of allowed z values, and the largest z value in joule seconds. The letter d is the name chemists give to $\ell = 2$ .

Compute the product inside the square root. The formula needs $\ell(\ell+1)$ . With $\ell = 2$ this is $2 \times (2 + 1) = 2 \times 3 = 6$ . We multiply the number by the next whole number, because that is the rule for the squared length.
Take the square root to get the length in steps of $\hbar$ . $\sqrt{6} \approx 2.449$ , so the length is $|\mathbf{L}| = \sqrt{6}\,\hbar \approx 2.449\,\hbar$ . The length is about 2.45 times the step size. It is not a whole number of steps, which is why it is bigger than the largest z value.
Convert the length to joule seconds. Multiply $2.449 \times 1.054571817 \times 10^{-34}$ J s. The front numbers give about 2.58, and the power of ten stays $10^{-34}$ , so the length is about $2.58 \times 10^{-34}$ J s.
Count the allowed z values. The magnetic number runs from $-2$ to $+2$ : $-2, -1, 0, 1, 2$ . That is $2 \times 2 + 1 = 5$ values. Zero is included, and there is one value for each step.
Find the largest z value. The largest $m_\ell$ is $+2$ , so the largest z component is $L_z = 2\hbar = 2 \times 1.054571817 \times 10^{-34}$ J s, which is about $2.11 \times 10^{-34}$ J s.
Compare the length and the largest z value. Both numbers have the same power of ten, so we compare 2.58 with 2.11. The length is bigger than the largest z value, so the angular momentum can never line up exactly with the z axis, even at its best.

Answer: The d electron has angular momentum of length about $2.58 \times 10^{-34}$ J s. It has five allowed z values, from $-2\hbar$ to $+2\hbar$ . The largest z value, $2.11 \times 10^{-34}$ J s, is smaller than the full length, so the vector cannot point exactly along z.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Taking the length as $\ell\hbar$ . The correct value is $\sqrt{\ell(\ell+1)}\,\hbar$ , which is larger. For $\ell = 2$ , $\ell\hbar = 2\hbar$ , but the length is $2.449\hbar$ .
Counting only positive values of $m_\ell$ . There are $2\ell + 1$ values, including zero and both negative values.
Thinking the angular momentum can point exactly along the z axis. Only its z component has a sharp value, and the full vector is longer than that component.
Confusing $\ell$ with $m_\ell$ . The quantum number $\ell$ sets the size, and $m_\ell$ sets the z component.

Where you will meet it

Atomic orbitals, labeled s, p, d and f for $\ell = 0$ to $3$ .
Counting electron slots in an atom, since each subshell with quantum number $\ell$ holds $2\ell + 1$ orbitals, each with its own $m_\ell$ .
The Zeeman effect, where the energy shift in a magnetic field depends on $m_\ell$ .
Selection rules for light from atoms, which say that the usual light-emitting jumps change $\ell$ by exactly one.

Questions people ask

What is the difference between $\ell$ and $m_\ell$?

The orbital quantum number $\ell$ sets the length of the angular momentum. The magnetic quantum number $m_\ell$ sets its z component, and it can take any whole-number value from $-\ell$ to $+\ell$ .

Does the electron physically spin around the nucleus?

The picture of a little orbit is a helpful model, but the rule is about the allowed values. Orbital angular momentum has fixed sizes, and a classical orbit does not match those values exactly.

Why is the length a square root?

The squared length is $\ell(\ell+1)$ times $\hbar$ squared. Taking the square root gives the length itself, $\sqrt{\ell(\ell+1)}\,\hbar$ . The square root appears because the equations give the squared length first.

Spin One-Half: An electron has a built-in spin that, measured along any direction, gives only one of two values, called up or down.
Zeeman Effect: A magnetic field shifts an atom’s energy levels by small equal steps, so one spectral line splits into several.
Hydrogen Energy Levels: An electron in a hydrogen atom can sit only on certain energy steps, and when it jumps between steps it gives off or takes in light of one exact color.
Electron Shell Capacity: Each electron shell can hold a fixed maximum number of electrons: shell 1 holds 2, shell 2 holds 8, and shell 3 holds 18.