When a body moves around a point O, the quantity that describes its rotational motion is the angular momentum. With \vec{r} the position vector from O to the body and \vec{p} the momentum of the body, the angular momentum is defined by the cross product \vec{L} = \vec{r} \times \vec{p}.
When a body of mass m goes around a circle of radius r at speed v, \vec{r} and \vec{p} are always perpendicular, so the magnitude of the angular momentum is L = mvr. Its direction is perpendicular to the plane of the circle: the direction in which a right-handed screw advances when it is turned in the direction of motion. The body in the figure moves counterclockwise as seen from above, so \vec{L} points up.
The change of angular momentum in time is set by the moment of force, \dfrac{d\vec{L}}{dt} = \vec{r} \times \vec{F}. A central force is a force that always acts along the line joining O and the body. Then \vec{r} and \vec{F} are parallel, so \vec{r} \times \vec{F} = \vec{0}, and the angular momentum does not change in time. Kepler's second law, which says that the line from the Sun to a planet sweeps out equal areas in equal times, is this conservation law expressed in terms of area.
The Coulomb force that the electron in a hydrogen atom receives from the nucleus is also a central force. So angular momentum is conserved inside the atom as well, and together with energy it is used to tell the states of the electron apart.
In Bohr's model of the atom, the angular momentum of the electron was assumed to take only integer multiples of \hbar = \dfrac{h}{2\pi}. \hbar is 1.05 \times 10^{-34} J s. A 1 kg body going around a circle of radius 1 m at 1 m/s has an angular momentum of 1 J s, about 9.5 \times 10^{33} times \hbar. In everyday rotation, the discreteness of angular momentum cannot be seen at all.
This article derives the discreteness of angular momentum from the properties of waves, and then looks in turn at what is added for motion in three dimensions.
As the simplest case, suppose an electron can move only on a ring of radius R. The position of the electron is given by the angle \varphi on the ring. Since \varphi and \varphi + 2\pi point to the same place on the ring, the wave function must take the same value there. As an equation, \psi(\varphi + 2\pi) = \psi(\varphi).
Leaving out the part that depends on time, a wave travelling along the ring can be written \psi = e^{im\varphi}. From here on, m is the number of times the wave oscillates during one turn around the ring, and the mass of the electron is written m_\mathrm{e}. For the value to come back after one turn, e^{2\pi im} = 1 is needed, and this holds only when m is an integer.
The figure draws the real part of this wave with the angle \varphi on the horizontal axis. The left end \varphi = 0 and the right end \varphi = 2\pi are the same place, so the values at the two ends must agree. They agree for m = 0, 1, 2, -2, but for m = 2.5 the value at the right end does not match the value at the left end. As time passes, a wave with positive m travels toward increasing angle, and one with negative m toward decreasing angle.
Now find the momentum of this wave. Using the length s = R\varphi measured along the ring gives \psi = e^{i(m/R)s}, so the wave number is k = \dfrac{m}{R}. From de Broglie's relation p = \dfrac{h}{\lambda} = \hbar k, the momentum is p = \dfrac{m\hbar}{R}, and the angular momentum about the center of the ring is L_z = Rp = m\hbar. Here z is the direction perpendicular to the plane of the ring.
The z component of angular momentum takes only integer multiples of \hbar. This has the same form as Bohr's condition, but here it is not an assumption: it comes from the condition that the wave function has a single value at each point of the ring.
Positive m is a state going counterclockwise as seen from above, negative m one going clockwise, and m = 0, which Bohr's model did not have, is also allowed. The wave function for m = 0 has the same value everywhere on the ring, and it describes a state that is not going around.
The energy of the electron on the ring is only kinetic energy, E = \dfrac{p^2}{2m_\mathrm{e}} = \dfrac{m^2\hbar^2}{2m_\mathrm{e}R^2}. m and -m differ only in the direction of rotation, and their energies are the same.
An electron in an atom is not limited to a ring; it can move in every direction around the nucleus. Its wave function is then written with the distance r from the nucleus and two angles that give the direction: \theta, measured from the z axis, and \varphi, the angle around the z axis.
For \varphi the situation is the same as on the ring, and the condition of returning after one turn gives L_z = m\hbar. The equation solved for \theta comes with the condition that the wave function does not diverge at \theta = 0 and \theta = \pi, that is, on the z axis. Solutions that satisfy this condition exist only for certain values of the magnitude of angular momentum. It is the same mechanism by which the energy of the harmonic oscillator became discrete when only the solutions going to 0 far away were kept.
Solving it, the square of the magnitude of angular momentum takes only the values L^2 = l(l+1)\hbar^2. Here l is an integer 0 or larger, called the azimuthal quantum number. The z component is L_z = m\hbar, where m is an integer from -l to l, giving 2l + 1 possibilities. m is called the magnetic quantum number. The reason for this name becomes clear in the last scene.
The figure draws angular momentum as an arrow of length \sqrt{l(l+1)}\,\hbar and lines up the directions in which the z component is m\hbar. For l = 2 the length is \sqrt{6}\,\hbar = 2.45\hbar, and the z component takes the five values 2\hbar, \hbar, 0, -\hbar and -2\hbar. The angle to the z axis is 35.3° for m = 2, 65.9° for m = 1 and 90° for m = 0.
The length \sqrt{l(l+1)}\,\hbar of the arrow is always larger than the largest z component l\hbar. So the arrow of angular momentum never points exactly along the z axis. The reason is explained in the next scene.
The discreteness of the direction of angular momentum is called space quantization. In 1916 Sommerfeld and Debye each arrived at it by extending Bohr's model. 2l + 1 is 1, 3, 5 and 7 for l = 0, 1, 2, 3, and this is why an atom has one s orbital, three p orbitals, five d orbitals and seven f orbitals.
In quantum mechanics, physical quantities are represented by operators that act on the wave function. The z component of angular momentum is L_z = -i\hbar\dfrac{\partial}{\partial\varphi}; acting on e^{im\varphi} it gives m\hbar e^{im\varphi}, which agrees with L_z = m\hbar found in the ring scene. The x and y components can also be written as operators that differentiate with respect to the angles.
For the operators of the three components, swapping the order in which two of them act changes the result, and L_xL_y - L_yL_x = i\hbar L_z holds. For the pairs y and z, and z and x, the same equation holds with the letters shifted in turn. For two quantities whose result depends on the order like this, the product of their spreads has a lower bound. In a state where L_z is fixed at m\hbar, \Delta L_x \Delta L_y \geq \dfrac{|m|\hbar^2}{2}, the same kind of relation as the one between position and momentum in the article on the uncertainty principle.
If m is not 0, the right side is not 0, so both L_x and L_y have a spread. Only one of the three components can have a fixed value, and it is usually chosen to be the z component. The z axis may be chosen in any direction; when a magnetic field is applied, it is chosen along the field. On the other hand, L^2 gives the same result in either order with every component, so L^2 and L_z are fixed at the same time.
This is why, in the previous scene, the arrow length \sqrt{l(l+1)}\,\hbar was larger than l\hbar. If the arrow pointed exactly along the z axis, L_z = l\hbar and L_x = L_y = 0 would be fixed at the same time. Since L^2 = L_x^2 + L_y^2 + L_z^2, even in the state m = l the average of L_x^2 + L_y^2 is l(l+1)\hbar^2 - l^2\hbar^2 = l\hbar^2, not 0.
The figure draws the state l = 2, m = 2 as an arrow on a cone around the z axis. The z component is fixed at 2\hbar, but where on the cone the arrow points is not fixed. The figure draws eight arrows on the cone and emphasizes one at a time. This is to show that the direction is not fixed; the angular momentum is not going around the cone.
In this state, the x and y components both have average 0 and spread \hbar. Their product \hbar^2 is exactly equal to the lower bound \dfrac{2\hbar^2}{2}, so the uncertainty relation holds with equality.
In the state m = 0 the z component is 0, and the spread of the x component grows to \sqrt{3}\,\hbar. Only when l = 0 is L^2 = 0, so that all three components are fixed at 0. The lowest state of the hydrogen atom is this l = 0 state, and unlike L = \hbar in Bohr's model, its angular momentum is 0.
An electron going around the nucleus is a current flowing around a circle, so it has a magnetic moment. For an electron going around a circle of radius r with period T, the size of the current is \dfrac{e}{T} and the area of the circle is \pi r^2. Using T = \dfrac{2\pi r}{v}, the size of the magnetic moment is \dfrac{e}{T}\pi r^2 = \dfrac{evr}{2} = \dfrac{e}{2m_\mathrm{e}}L. Because the charge of the electron is negative, the magnetic moment points opposite to the angular momentum.
When a magnetic field of size B is applied in the z direction, the energy of the electron changes by \dfrac{e}{2m_\mathrm{e}}BL_z. Putting in L_z = m\hbar, the change is m\mu_\mathrm{B}B. \mu_\mathrm{B} = \dfrac{e\hbar}{2m_\mathrm{e}} = 9.27 \times 10^{-24} J/T is called the Bohr magneton. A level with azimuthal quantum number l splits into 2l + 1 levels with different m, at equal spacing \mu_\mathrm{B}B. This is why m is called the magnetic quantum number.
When an atom emits light, m changes by only -1, 0 or 1, because the z component of angular momentum carried off by one photon is -\hbar, 0 or \hbar. The spacing is \mu_\mathrm{B}B for both the upper and the lower level, so the emitted light has only three energies: the value without the field, and values higher and lower than it by \mu_\mathrm{B}B. The figure shows a transition from an l = 2 level to an l = 1 level; there are nine ways to make the transition, but only three lines. The colors of the arrows and lines are set by how m changes.
In 1896 Zeeman found that the D lines of a sodium flame placed in a strong magnetic field became wider. Lorentz explained this by taking the charge oscillating in the atom to receive a force from the magnetic field, and the two received the Nobel Prize in Physics in 1902. This splitting of lines in a magnetic field is called the Zeeman effect.
The splitting is very small: even in a field of 1 T, \mu_\mathrm{B}B is 5.79 \times 10^{-5} eV. The red line of cadmium at 643.8 nm is light emitted in a transition from a level corresponding to l = 2 to a level corresponding to l = 1, and because the spins of the two electrons cancel, it splits into three lines as in the figure. At 1 T the difference in wavelength between neighboring lines is 0.019 nm, about 1/33000 of the original wavelength.
In many atoms, a line splits into more lines, and the spacing also differs from \mu_\mathrm{B}B. This is called the anomalous Zeeman effect, and the sodium D lines split in this way too. The reason is that, besides the orbital angular momentum, the electron itself has an angular momentum called spin.