There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers ${\displaystyle n}$ , ${\displaystyle \ell }$ , ${\displaystyle m_{l}}$ , and ${\displaystyle m_{s}}$  specify the complete quantum state of a single electron in an atom called its wavefunction or orbital. The Schrödinger equation for the wavefunction of an atom with one electron is a separable partial differential equation. (This is not the case for the neutral helium atom or other atoms with mutually interacting electrons, which require more sophisticated methods for solution^[2]) This means that the wavefunction as expressed in spherical coordinates can be broken down into the product of three functions of the radius, colatitude (or polar) angle, and azimuth:^[3]

${\displaystyle \psi (r,\theta ,\phi )=R(r)P(\theta )F(\phi )}$ 

The differential equation for ${\displaystyle F}$  can be solved in the form ${\displaystyle F(\phi )=Ae^{\lambda \phi }}$ . Because values of the azimuth angle ${\displaystyle \phi }$  differing by 2${\displaystyle \pi }$  radians (360 degrees) represent the same position in space, and the overall magnitude of ${\displaystyle F}$  does not grow with arbitrarily large ${\displaystyle \phi }$  as it would for a real exponent, the coefficient ${\displaystyle \lambda }$  must be quantized to integer multiples of ${\displaystyle i}$ , producing an imaginary exponent: ${\displaystyle \lambda =im_{l}}$ .^[4] These integers are the magnetic quantum numbers. The same constant appears in the colatitude equation, where larger values of ${\displaystyle {m_{l}}^{2}}$  tend to decrease the magnitude of ${\displaystyle P(\theta ),}$  and values of ${\displaystyle m_{l}}$  greater than the azimuthal quantum number ${\displaystyle \ell }$  do not permit any solution for ${\displaystyle P(\theta ).}$ 

The axis used for the polar coordinates in this analysis is chosen arbitrarily. The quantum number ${\displaystyle m_{l}}$  refers to the projection of the angular momentum in this arbitrarily-chosen direction, conventionally called the ${\displaystyle z}$ -direction or quantization axis. ${\displaystyle L_{z}}$ , the magnitude of the angular momentum in the ${\displaystyle z}$ -direction, is given by the formula:^[5]

${\displaystyle L_{z}=m_{l}\hbar }$ .

This is a component of the atomic electron's total orbital angular momentum ${\displaystyle \mathbf {L} }$ , whose magnitude is related to the azimuthal quantum number of its subshell ${\displaystyle \ell }$  by the equation:

${\displaystyle L=\hbar {\sqrt {\ell (\ell +1)}}}$ ,

where ${\displaystyle \hbar }$  is the reduced Planck constant. Note that this ${\displaystyle L=0}$  for ${\displaystyle \ell =0}$  and approximates ${\displaystyle L=\left(\ell +{\tfrac {1}{2}}\right)\hbar }$  for high ${\displaystyle \ell }$ . It is not possible to measure the angular momentum of the electron along all three axes simultaneously. These properties were first demonstrated in the Stern–Gerlach experiment, by Otto Stern and Walther Gerlach.^[6]

The energy of any wave is its frequency multiplied by Planck's constant. The wave displays particle-like packets of energy called quanta. The formula for the quantum number of each quantum state uses Planck's reduced constant, which only allows particular or discrete or quantized energy levels.^[5]

The quantum number ${\displaystyle m_{l}}$  refers, loosely, to the direction of the angular momentum vector. The magnetic quantum number ${\displaystyle m_{l}}$  only affects the electron's energy if it is in a magnetic field because in the absence of one, all spherical harmonics corresponding to the different arbitrary values of ${\displaystyle m_{l}}$  are equivalent. The magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field (the Zeeman effect) — hence the name magnetic quantum number. However, the actual magnetic dipole moment of an electron in an atomic orbital arises not only from the electron angular momentum but also from the electron spin, expressed in the spin quantum number.

Since each electron has a magnetic moment in a magnetic field, it will be subject to a torque which tends to make the vector ${\displaystyle \mathbf {L} }$  parallel to the field, a phenomenon known as Larmor precession.

• Quantum number
  • Azimuthal quantum number
  • Principal quantum number
  • Spin quantum number
  • Total angular momentum quantum number
• Electron shell
• Basic quantum mechanics
• Bohr atom
• Schrödinger equation