**de Broglie hypothesis**. He also developed an equation for the wavelength ($\lambda$) of a particle having mass

*m*and traveling at velocity

*v*using Planck's constant,

*h*, equal to $6.62607004\times10^{-34}\,\rm{kg}\cdot{\rm{m}}^2\rm{/s}$, which is called the

**de Broglie wavelength**.

**node**.

#### Nodes in a Standing Wave

**wave function**, which is a mathematical expression that gives information about measurable properties of a system, such as energy, momentum, and position. $\widehat H$ is the Hamiltonian operator that determines the change of quantum states, and

*E*is the energy of the system.

**Heisenberg uncertainty principle**states that it is impossible to simultaneously measure the position and the momentum of a particle. It is only possible to know a range of probabilities where an electron might appear, not where it actually is. The range of probabilities where an electron might appear is an

**electron shell**, which is one or more electron subshells that have the same quantum number $n$

*.*Electron shells are determined by an electron's distance from the nucleus. However, an electron's angular position relative to the nucleus also affects where it might be found. An

**electron orbital**is the area of an atom in which an electron has the greatest probability of being located. Each orbital can contain at most two electrons. An

**electron subshell**is a group of electron energy levels with the same size and shape that have the same quantum numbers $n$ and $\ell$

*.*For example, the orbitals 2

*p*, 2

_{x}*p*, and 2

_{y}*p*make up the 2

_{z}*p*subshell.

#### Shapes of Electron Orbitals

#### Energy Levels of Principal Quantum Number

*s*subshell, 1 for a

*p*subshell, 2 for a

*d*subshell, 3 for an

*f*subshell, and so on. This number is also related to $n$. The angular momentum number $\ell$ consists of integers that range from 0 to $n-1$. For example, bromine has $n=4$, so its $\ell$ values are 0, 1, 2, and 3.

The third quantum number is the magnetic quantum number, $m$, which indicates the orientation in space of a particular orbital. The value of $m$ can be any integer from $-\ell$ to $+\ell$, including 0. So, for $\ell=2$ (*d* subshell), the range for $m$ includes –2, –1, 0, 1, and 2. These define the $d_{{x^2}-{y^2}},d_{z^2},d_{xy},d_{xz}$*,* and $d_{yz}$ orbitals.

The fourth quantum number is the spin quantum number, $s$. Its value is either $+1\rm{/}2$ or $-1\rm{/}2$ and relates to the electron's spin as either up or down. Paired electrons may never have the same spin value, which means they cannot have the same four quantum numbers, according to the to **Pauli exclusion principle**. An electron with $s=+1/2$ is called an alpha electron, while an electron with $s=-\rm{1/2}$ is called a beta electron. Electrons with opposite spins are depicted as one arrow pointing upward and the other arrow pointing downward.