# Orbital Angular Momentum Quantization

> An electron's turning motion around the center of an atom can only take certain fixed amounts, like steps on a staircase.

Page: https://formulabro.yunusemrevurgun.com/quantum-mechanics/angular-momentum-quantization/  
Category: Quantum Mechanics  
Published: 2026-10-22  
Updated: 2026-10-22

## Formula

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

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

- **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.

More notation help: https://formulabro.yunusemrevurgun.com/math-basics/

## What each symbol means

- $|\mathbf{L}|$: 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 $\ell = 2$.

1. **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.
2. **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.
3. **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.
4. **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.
5. **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.
6. **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.

## 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.

## Related formulas

- [Spin One-Half](https://formulabro.yunusemrevurgun.com/quantum-mechanics/spin-one-half/)
- [Zeeman Effect](https://formulabro.yunusemrevurgun.com/quantum-mechanics/zeeman-effect/)
- [Hydrogen Energy Levels](https://formulabro.yunusemrevurgun.com/quantum-mechanics/hydrogen-energy-levels/)
- [Electron Shell Capacity](https://formulabro.yunusemrevurgun.com/quantum-mechanics/electron-shell-capacity/)

## Sources

- [Angular momentum operator, Wikipedia](https://en.wikipedia.org/wiki/Angular_momentum_operator)
- [Azimuthal quantum number, Wikipedia](https://en.wikipedia.org/wiki/Azimuthal_quantum_number)

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