Erwin Schrödinger (1887 – 1961) was an Austrian physicist who achieved fame for his contributions to quantum mechanics, especially the Schrödinger equation, for which he received the Nobel Prize in 1933. His development of what is known as Schrödinger's wave equation was made during the first half of 1926. It came as a result of his dissatisfaction with the quantum condition in Bohr's orbit theory and his belief that atomic spectra should really be determined by some kind of discrete energy value.

### The Quantum Mechanical Model of the Atom

Most definitions of **quantum theory** and quantum mechanics offer the same description for both. These definitions essentially describe quantum theory as a theory in which both energy and matter have characteristics of waves under some conditions and characteristics of particles under other conditions.

Quantum theory suggests that energy comes in discrete packages called **quanta** (or, in the case of electromagnetic radiation, **photons**). Quantum theory has some mathematical development, often referred to as quantum mechanics, that offers explanations for the behavior of electrons inside the electron clouds of atoms.

The wave-particle duality of electrons within the electron cloud limits our ability to measure both the energy and the position of an electron simultaneously. The more accurately we measure either the energy or the position of an electron, the less we know about the other. Our fact that we cannot accurately know both the position and the momentum of an electron at the same time causes an inability to predict a trajectory for an electron. Consequently, electron behavior is described differently than the behavior of normal sized particles. The trajectory that we normally associate with macroscopic objects is replaced for electrons in electron clouds, with statistical descriptions that show, not the electron path, but the region where it is most likely to be found. Since it is the electron in the electron cloud of an atom that determines its chemical behavior, the quantum mechanics description of electron configuration is necessary to understanding chemistry.

The most common way to describe electrons in atoms according to quantum mechanics is to solve the **Schrodinger equation** for the energy states of the electrons within the electron cloud. When the electron is in these states, its energy is well-defined but its position is not. The position is described by a **probability distribution** map called an **orbital**.

Schrodinger’s equation is shown below.

\begin{align*}i \ \hbar \frac{\partial}{\partial t}\psi (r,t)=-\frac{\hbar}{2m}\nabla ^2\psi (r,t)+V(r,t) \psi (r,t)\end{align*}

where \begin{align*}i\end{align*}

\begin{align*}\hbar\end{align*} is Planck’s constant divided by \begin{align*}2 \pi\end{align*}

\begin{align*}\psi \ (r, t)\end{align*} is the wave function

\begin{align*}m\end{align*} is the mass of the particle

\begin{align*}\nabla ^2\end{align*} is the Laplacian operator, \begin{align*} \frac{\partial ^2}{ \partial x^2}+ \frac{\partial ^2}{ \partial y^2}+ \frac{\partial ^2}{ \partial z^2}\end{align*} (these refer to partial second derivatives)

\begin{align*}V(r, t)\end{align*} is the potential energy influencing the particle

It should be quite clear that without years of high level mathematics, just seeing the equation is no value at all. Without understanding the math, the equation makes no sense.

The stable energy levels for an electron in an electron cloud are those that have integer values in three positions in the equation. Schrodinger found that having integer values in these three places in the equation produced a wave function that described a standing wave. These three integers are called quantum numbers and are represented by the letters \begin{align*}n\end{align*}, \begin{align*}l\end{align*}, and \begin{align*}m\end{align*}.

Solutions to Schrödinger’s equation involve four special numbers called **quantum numbers**. (Three of the numbers, \begin{align*}n\end{align*}, \begin{align*}l\end{align*}, and \begin{align*}m\end{align*}, come from Schrödinger’s equation, and the fourth one comes from an extension of the theory). These four numbers completely describe the energy of an electron. Each electron has exactly four quantum numbers, and no two electrons have the same four numbers. The statement that no two electrons can have the same four quantum numbers is known as the **Pauli exclusion principle**.

The principal quantum number, \begin{align*}n\end{align*}, is a positive integer \begin{align*}(1, 2, 3, \ldots n)\end{align*} that indicates the main energy level of an electron within an atom. According to quantum mechanics, every principal energy level has one or more sub-levels within it. The number of sub-levels in a given energy level is equal to the number assigned to that energy level. That is, principal energy level 1 will have 1 sub-level, principal energy level 2 will have two sub-levels, principal energy level 3 will have three sub-levels, and so on. In any energy level, the maximum number of electrons possible is \begin{align*}2n^2\end{align*}. Therefore, the maximum number of electrons that can occupy the first energy level is \begin{align*}2 (2 \times 1^2)\end{align*}. For energy level 2, the maximum number of electrons is \begin{align*}8 (2 \times 2^2)\end{align*}, and for the 3rd energy level, the maximum number of electrons is \begin{align*}18 (2 \times 3^2)\end{align*}. The **Table** below lists the number of sub-levels and electrons for the first four principal quantum numbers.

Principal Quantum Number |
Number of Sub-Levels |
Total Number of Electrons |

1 | 1 | 2 |

2 | 2 | 8 |

3 | 3 | 18 |

4 | 4 | 32 |

The largest known atom contains slightly more than 100 electrons. Quantum mechanics sets no limit as to how many energy levels exist, but no more than 7 principal energy levels are needed to describe all the electrons of all the known atoms. Each energy level can have as many sub-levels as the principal quantum number, as discussed above, and each sub-level is identified by a letter. Beginning with the lowest energy sub-level, the sub-levels are identified by the letters \begin{align*}s\end{align*}, \begin{align*}p\end{align*}, \begin{align*}d\end{align*}, \begin{align*}f\end{align*}, \begin{align*}g\end{align*}, \begin{align*}h\end{align*}, \begin{align*}i\end{align*}, and so on. Every energy level will have an \begin{align*}s\end{align*} sub-level, but only energy levels 2 and above will have \begin{align*}p\end{align*} sub-levels. Similarly, \begin{align*}d\end{align*} sub-levels occur in energy level 3 and above, and \begin{align*}f\end{align*} sub-levels occur in energy level 4 and above. Energy level 5 could have a fifth sub-energy level named \begin{align*}g\end{align*}, but all the known atoms can have their electrons described without ever using the \begin{align*}g\end{align*} sub-level. Therefore, we often say there are only four sub-energy levels, although theoretically there can be more than four sub-levels.

The principal energy levels and sub-levels are shown in the following diagram. The principal energy levels and sub-levels that we use to describe electrons are in red. The energy levels and sub-levels in black are theoretically present but are never used for known atoms.

The sub-energy levels are identified by the azimuthal quantum number, \begin{align*}l\end{align*}.

When the azimuthal quantum number \begin{align*}l=0\end{align*}, the sub-level = \begin{align*}s\end{align*}.

When the azimuthal quantum number \begin{align*}l=1\end{align*}, the sub-level = \begin{align*}p\end{align*}.

When the azimuthal quantum number \begin{align*}l=2\end{align*}, the sub-level = \begin{align*}d\end{align*}.

When the azimuthal quantum number \begin{align*}l=3\end{align*}, the sub-level = \begin{align*}f\end{align*}.

Quantum mechanics also tells us how many orbitals are in each sub-level. In Bohr’s model, an orbit was a circular path that the electron followed around the nucleus. In quantum mechanics, an orbital is defined as an area in the electron cloud where the probability of finding the electron is high. The number of orbitals in an energy level is equal to the square of the principal quantum number. Hence, energy level 1 will have 1 orbital (1^{2}), energy level 2 will have 4 orbitals (2^{2}), energy level 3 will have 9 orbitals (3^{2}), and energy level 4 will have 16 orbitals (4^{2}).

The \begin{align*}s\end{align*} sub-level has only one orbital. Each of the \begin{align*}p\end{align*} sub-levels has three orbitals. The \begin{align*}d\end{align*} sub-levels have five orbitals, and the \begin{align*}f\end{align*} sub-levels have seven orbitals. If we wished to assign the number of orbitals to the unused sub-levels, \begin{align*}g\end{align*} would have nine orbitals and \begin{align*}h\end{align*} would have eleven. You might note that the number of orbitals in the sub-levels increases by odd numbers \begin{align*}(1, 3, 5, 7, 9, 11, \ldots)\end{align*}. As a result, the single orbital in energy level 1 is the \begin{align*}s\end{align*} orbital. The four orbitals in energy level 2 are a single \begin{align*}2s\end{align*} orbital and three \begin{align*}2p\end{align*} orbitals. The nine orbitals in energy level 3 are a single \begin{align*}3s\end{align*} orbital, three \begin{align*}3p\end{align*} orbitals, and five \begin{align*}3d\end{align*} orbitals. The sixteen orbitals in energy level 4 are the single \begin{align*}4s\end{align*} orbital, three \begin{align*}4p\end{align*} orbitals, five \begin{align*}4d\end{align*} orbitals, and seven \begin{align*}4f\end{align*} orbitals.

Principal Energy Level (n) |
\begin{align*}s \quad \ p \quad \ d \quad \ f\end{align*} |
Total Number of Orbitals (n^{2}) |
Maximum Number of Electrons (2n^{2}) |

1 | \begin{align*}1 \quad - \quad - \quad -\end{align*} | 1 | 2 |

2 | \begin{align*}1 \quad \ 3 \quad - \quad -\end{align*} | 4 | 8 |

3 | \begin{align*}1 \quad \ 3 \quad \ 5 \quad \ -\end{align*} | 9 | 18 |

4 | \begin{align*}1 \quad \ 3 \quad \ 5 \quad \ 7\end{align*} | 16 | 32 |

The **Table** above shows the relationship between \begin{align*}n\end{align*} (the principal quantum number), the number of orbitals, and the maximum number of electrons in a principal energy level. Theoretically, the number of orbitals and number of electrons continue to increase for higher values of \begin{align*}n\end{align*}. However, no atom actually has more than 32 electrons in any of its principal levels.

Each orbital will also have a probability pattern that is determined by interpreting Schrödinger’s equation. The 3-dimensional probability pattern for the single orbital in the \begin{align*}s\end{align*} sub-level is a sphere. The probability patterns for the three orbitals in the \begin{align*}p\end{align*} sub-levels are shown below. The three images on the left show the probability pattern for the three \begin{align*}p\end{align*} orbitals in each of the three dimensions. On the far right is an image of all three \begin{align*}p\end{align*} orbitals together. These \begin{align*}p\end{align*} orbitals are said to be shaped like dumbbells (named after the objects weight lifters use), water wings (named after the floating balloons young children use in the swimming pool), and various other objects.

The probability patterns for the five \begin{align*}d\end{align*} orbitals are more complicated and are shown below.

The seven \begin{align*}f\end{align*} orbitals shown below are even more complicated.

You should keep in mind that no matter how complicated the probability pattern is, each shape represents a single orbital, and the entire probability pattern is the result of the various positions that either one or two electrons can take.

### Summary

- Energy comes in discrete packages called quanta.
- Electrons in the electron cloud of atoms have a limited number of energy levels due to the quantization of energy.
- The most common way to describe electrons in atoms according to quantum mechanics is to solve the Schrodinger equation for the energy states of the electrons within the electron cloud.
- When the electron is in these states, its energy is well-defined but the electron position is not.
- The position is described by a probability distribution map called an orbital.
- Solutions to Schrödinger’s equation involve four special numbers called quantum numbers. (Three of the numbers, \begin{align*}n\end{align*}, \begin{align*}l\end{align*}, and \begin{align*}m\end{align*}, come from Schrödinger’s equation, and the fourth one comes from an extension of the theory.)
- These four numbers completely describe the energy of an electron.
- Each electron has exactly four quantum numbers, and no two electrons have the same four numbers.
- The principal quantum number, \begin{align*}n\end{align*}, is a positive integer \begin{align*}(1, 2, 3, \ldots n)\end{align*} that indicates the main energy level of an electron within an atom.

### Review

- If we were dealing with an atom that had a total of 20 electrons in its electron cloud, how many of those electrons would have quantum number \begin{align*}n=1\end{align*}.
- What is the total number of electrons in the \begin{align*}p\end{align*}-orbitals of the second energy level?
- How many energy levels are necessary to contain the first 10 electrons in an electron cloud?

### Explore More

Use this resource to answer the questions that follow.

- Who first suggested that matter also might exhibit the properties of both particles and waves?
- When the concept of describing the trajectory of an electron was given up, what description of the electron replaced it?
- If the shape of the electron orbital is spherical, how many values are possible for the quantum number \begin{align*}l\end{align*}?