Lesson 2: The Quantum Mechanical Model
Part a: Schrodinger's Wave Mechanical Model
Part 2a: Schrodinger's Wave Mechanical Model
Part 2b: Orbitals
Part 2c: Energy Levels
Part 2d: Quantum Numbers
The Schrodinger Wave Equation
Louis de Broglie’s treatment of the electron as a wave was the precursor to the modern model of the atom. Referred to as the Wave Mechanical Model, or more commonly the Quantum Mechanical Model, the modern model is a highly mathematical model that describes electrons by a wave function, Ψ. 
Proposed by Austrian physicist Erwin Schrodinger in 1926, the wave mechanical model is a complex mathematical description of the atom and its electrons. The x-dimension version of the Schrodinger wave equation is shown at the right. There is also a y-dimension and z-dimension equation. Given the complexity of the mathematics, we will avoid the equations and instead discuss the conclusions that were derived from the use of the equations to describe the electrons.
Schrodinger’s approach to modeling the atom was to specify the location of each electron by a probability function. Instead of locating the electron at a point, Schrodinger described the electron as being located within orbitals. An orbital is not an orbit. An orbit describes the electron moving along a well-defined path a precise distance from the nucleus. An orbital describes a region of space in which there is a 90% probability that the electron will be located. There is no effort to describe where the electron is within the space, nor how it arrived at its current location, nor which direction it is heading and where it will be next. The Wave Mechanical Model describes electrons as having a high likelihood of being in an orbital that has a (relatively) precisely known energy. These orbitals are often plotted in three dimensions as an electron cloud.
Quantum Numbers
The solutions to the Schrodinger wave equation result in four quantum numbers that describe the orbitals and the electrons that are inside of them. Each electron in the atom has its own unique set of four quantum numbers. The quantum numbers and their significance are described below:
| First Quantum Number |
n |
Principal Energy Level |
| The energy level of an electron depends mostly on the first quantum number. Values for this number are whole numbers beginning with 1. The energy level is greater for greater values of n. The energy of an electron is affected to a lesser degree by the second quantum number. The value of n also affects the size of the orbitals. Orbitals are larger for larger values of n. |
| Second Quantum Number |
l |
Orbital Type/Shape |
Every principal energy level consists of one or more types of orbitals. The different types of orbitals have unique shape. When l has a value of 0, the orbital type is spherical (s orbital). When l has a value of 1, the orbital type can be described as a dumbbell with two lobes on opposite sides of the nucleus (p orbital).
Image Credit: https://commons.wikimedia.org/wiki/File:Single_electron_orbitals.jpg
There are also d orbital types and f orbital types. Their shapes are more complex. Orbital types and shapes will be discussed in greater detail on the next page of this lesson.
Different orbital types in the same principal energy level will have slightly different energies. Thus the orbital types are often referred to as energy sublevels. While the n quantum number is the main factor affecting the energy level, the l quantum number has a small influence, affects the energy sublevel. The s-type orbitals are lower in energy than the p-type orbitals. The p-type orbitals are lower in energy than the d-type orbitals which are lower in energy than the f-type orbitals. |
| Third Quantum Number |
ml |
Orbital Orientation |
The s-orbitals are symmetrical about the origin in all directions. There is only one way by which it can be oriented. There are three different p-orbital types. They each consist of three lobes. The lobes are aligned along one of the three axes. The third quantum number indicated the axis along which the lobes are aligned. The d orbitals and f orbitals also have specific orientations relative to the axes. The third quantum number describes their orientation of the orbitals.
|
| Fourth Quantum Number |
ml |
Electron Spin |
| The first three quantum numbers describe the relative size, type, and orientation of the orbitals. These orbitals are the region of space in which the electrons are located. There can be at most two electrons in any one of the orbitals. The fourth quantum number describes the electrons. Two electrons in the same orbital are distinguished from one another by their spin direction – clockwise or counter-clockwise. |
Here is a summary of the four quantum numbers:
As An Example
You may be curious and wondering: what would be an example of the set of quantum numbers for an electron in an atom? If you are curious, here’s a few examples:
For the one electron in the hydrogen atom, the set of four quantum numbers might be:
[n, l, ml, ms] = [1, 0, 0, +½]
For two electrons in the helium atom, the set of four quantum numbers might be:
[n, l, ml, ms] = [1, 0, 0, +½] and
[n, l, ml, ms] = [1, 0, 0, -½]
For three electrons in the lithium atom, the set of four quantum numbers might be:
[n, l, ml, ms] = [1, 0, 0, +½] and
[n, l, ml, ms] = [1, 0, 0, -½] and
[n, l, ml, ms] = [2, 0, 0, +½]
For five electrons in the lithium atom, the set of four quantum numbers might be:
[n, l, ml, ms] = [1, 0, 0, +½] and
[n, l, ml, ms] = [1, 0, 0, -½] and
[n, l, ml, ms] = [2, 0, 0, +½] and
[n, l, ml, ms] = [2, 0, 0, -½] and
[n, l, ml, ms] = [2, 1, 0, +½]
Some people claim "Curiosity killed the cat." We say "Keep being curious! It’s never killed any cats." After all, nobody knows whether the cat was dead or alive before curiosity opened the box.
For those who remain curious, we will revisit quantum numbers in much greater detail on
the last page of Lesson 2.
Before You Leave
- Need more details on quantum numbers? Visit our Quantum Numbers page.
- Practice. Try our Quantum Mechanics Concept Builder. The first of the three activities would make great practice!
- The Check Your Understanding section below include questions with answers and explanations. It provides a great chance to self-assess your understanding.
Check Your Understanding
Use the following questions to assess your understanding. Tap the Check Answer buttons when ready.
1. Identify the following statements as being TRUE or FALSE in terms of their description of Schrodinger’s Wave Mechanical Model of the atom.
- The model is derived from experimental findings regarding electrons and their energy state.
- The model describes electrons as orbiting the nucleus with quantized energy levels.
- The model provides distinct coordinates of the position of every electron.
- The model is derived from mathematical equations.
- The model is a quantum model; electrons can only have certain, discrete energy levels.
- The model does not describe the electron’s location with much precision.
2. Given the following statements about two electrons, identify which quantum number - first, second, third, or fourth - is guaranteed to be different for the two electrons.
- Electron A and electron B are located inside the same orbital - that is the same size, type, and orientation.
- Electron A and electron B have significantly different energy levels.
- Electron A and electron B are located in orbitals that have a different shape.
- Electron A and electron B have slightly different energy levels.
- Electron A and electron B are in the same type of orbital in the same principal energy level but the orientation of the orbitals about the axes are different.
- Electron A is in an s orbital and electron B is in a p orbital.
- Electron A and electron B are located in two different p orbitals, each having the same energy.
- Electron A and electron B are both located in two different s orbitals that have different sizes.