Quantum Theory

As the wave-particle duality of light was being confirmed, in 1924 French physicist Louis de Broglie hypothesized that the same might be true of electrons. The idea that matter may also behave as a wave is called the 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. This equation can be used to calculate the wavelength of an electron if its mass and velocity are known. For example, assume an electron has $m=9.11\times10^{-31}\;\rm{kg}$ and $v=5.31\times10^6\;\rm m/\rm s.$ Austrian physicist Erwin Schrödinger built on the de Broglie hypothesis. He considered that the fact that electrons can only have specific quantized energies made them similar to a standing wave, which is a wave that exists with fixed points at either end. For example, a standing wave is created when a guitar string is plucked. The string vibrates between the nut and the bridge but remains fixed at those points. Each point on a standing wave that stays fixed and does not oscillate is a node. In 1925 Schrödinger developed an equation to describe electrons as standing waves. This equation gives information about a wave in terms of time and position. In Schrödinger's equation, $\psi$ is the 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. However, the 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 2p_x, 2p_y, and 2p_z make up the 2p subshell. Electron orbitals can be described based on quantum mechanics, the branch of science that deals with subatomic particles, their behaviors, and their interactions. A quantum number is a number that describes electrons in terms of the number of subshells, the shape of the orbital, the number of angular nodes, the energy levels, and the spin on the electrons. The first quantum number is the principal quantum number, $n$, which describes most of the energy in an electron and can be any integer but zero. The principal quantum number $n$ is proportional to the distance of the radius of the electron orbital from the nucleus. It indicates the shell where the electron is found. For example, bromine has its outermost electrons in the fourth electron shell from the nucleus, so its principal quantum number is 4. In general, higher principal quantum numbers are associated with higher energies than lower ones. The second quantum number is the orbital angular momentum number, $\ell$. This describes the subshell of the electron shell and can only be positive integer values or zero. Angular momentum limits the volume of space where an electron may be found and also relates to the shape of the subshell. Thus, $\ell$ is 0 for an 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.