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5.2: The Quantum Mechanical Model

Difficulty Level: At Grade Created by: CK-12
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Lesson Objectives

  • Understand the de Broglie wave equation and how it illustrates the wave nature of the electron.
  • Explain the difference between quantum mechanics and classical mechanics.
  • Understand how the Heisenberg uncertainty principle and Schrödinger’s wave equation led to the idea of atomic orbitals.
  • Know the four quantum numbers and how they are related to the arrangement of electrons in an atom.
  • Describe the interrelationships between principal energy level, sublevel, orbital and electron spin and how they relate to the number of electrons of an atom.

Lesson Vocabulary

  • angular momentum quantum number
  • Heisenberg uncertainty principle
  • magnetic quantum number
  • orbital
  • principal quantum number
  • spin quantum number
  • quantum mechanical model
  • quantum mechanics
  • quantum numbers

Wave Nature of the Electron

Bohr’s model of the atom was valuable in demonstrating how electrons are capable of absorbing and releasing energy and how atomic emission spectra are created. However, the model did not really explain why electrons should exist only in fixed circular orbits or why there would not be a limitless number of orbits with a continuum of possible energies. In order to explain why atomic energy states are quantized, scientists needed to rethink the way in which they viewed the nature of the electron and its movement.

de Broglie Wave Equation

Planck’s investigations into the emission spectra of hot objects and subsequent studies on the photoelectric effect had proven that light was capable of behaving as a particle, even though it is also known to behave as a wave. It seemed reasonable to wonder if matter that primarily acts as a particle, such as electrons, could also sometimes exhibit behavior that is typical of waves. In 1924, French scientist Louis de Broglie (1892-1987) derived an equation that described the wave nature of any particle. He determined that the wavelength (\begin{align*}\lambda\end{align*}) of any moving object is given by:

\begin{align*}\lambda = \dfrac{h}{mv}\end{align*}

In this equation, \begin{align*}h\end{align*} is Planck’s constant, \begin{align*}m\end{align*} is the mass of the particle in kg, and \begin{align*}v\end{align*} is the velocity of the particle in m/s. The problem below shows how to calculate the wavelength of an electron.

Sample Problem 5.4: de Broglie Equation

An electron with a mass of 9.11 × 10-31 kg is moving at nearly the speed of light. Using a velocity of 3.00 × 108 m/s, calculate the wavelength of this electron.

Step 1: List the known quantities and plan the problem.


  • mass (\begin{align*}m\end{align*}) = 9.11 × 10-31 kg
  • Planck's constant (\begin{align*}h\end{align*}) = 6.626 × 10-34 J•s
  • velocity (\begin{align*}v\end{align*}) = 3.00 × 108 m/s


  • wavelength (\begin{align*}\lambda\end{align*})

Apply the de Broglie wave equation \begin{align*}\lambda = \dfrac{h}{mv}\end{align*} to solve for the wavelength of the moving electron.

Step 2: Calculate

\begin{align*}\lambda = \dfrac{h}{mv} = \dfrac{6.626 \times 10^{-34} \ \text{J} \cdot \ \text{s}}{(9.11 \times 10^{-31} \ \text{kg}) \times (3.00 \times 10^8 \ \text{m/s})} = 2.42 \times 10^{-12} \ \text{m}\end{align*}

Step 3: Think about your result.

This very small wavelength is about 1/20th of the diameter of a hydrogen atom. Looking at the equation, we can see that the wavelengths of everyday objects will be even smaller, because their masses will be much larger.

Practice Problem
  1. Calculate the wavelength of a 0.145 kg baseball thrown at a speed of 40. m/s.

The above practice problem results in an extremely short wavelength on the order of 10−34 m. This wavelength is impossible to detect even with advanced scientific equipment. Indeed, while all objects move with wavelike motion, we never notice it because the wavelengths are far too short. On the other hand, particles that are extremely small, such as the electron, can have measurable wavelengths. The wave nature of the electron proved to be a key insight that led to a new way of understanding how the electron functions. An electron that is confined to a particular space around the nucleus of an atom can only move around that atom in such a way that its electron wave “fits” the size of the atom correctly (Figure below). This means that the frequencies of electron waves are quantized. Based on the \begin{align*}E=h\nu\end{align*} equation, the fact that only certain quantized frequencies are allowed for a given electron means that electrons can only exist in an atom at specific energies, as Bohr had previously theorized.

The circumference of the orbit in (A) allows the electron wave to fit perfectly into the orbit. This is an allowed orbit. In (B), the electron wave does not fit properly into the orbit, so this orbit is not allowed.

The study of motion in terms of large objects like baseballs and cars is called mechanics, or more specifically, classical mechanics. Because of the quantum nature of the electron and other tiny particles moving at high speeds, classical mechanics is inadequate to accurately describe their motion. Quantum mechanics is the study of the motion of objects that are atomic or subatomic in size and thus demonstrate wave-particle duality. In classical mechanics, the size and mass of the objects involved effectively obscures any quantum effects, so such objects appear to be capable of gaining or losing energy in any amount. Particles whose motion is better described by quantum mechanics can only gain or lose energy in discrete units called quanta.

Heisenberg Uncertainty Principle

Another feature that is unique to quantum mechanics is the uncertainty principle. The Heisenberg Uncertainty Principle states that it is impossible to determine simultaneously both the position and the velocity of a particle. The detection of an electron, for example, would be made by way of its interaction with photons of light. Since photons and electrons have nearly the same energy, any attempt to locate an electron with a photon will knock the electron off course, resulting in uncertainty about where the electron is headed (Figure below). We do not have to worry about the uncertainty principle with large everyday objects because of their mass. If you are looking for something with a flashlight, the photons coming from the flashlight are not going to cause the thing you are looking for to move. This is not the case with atomic-sized particles, so scientists needed a new way to think about how to envision the location of electrons within atoms.

Heisenberg Uncertainty Principle: The observation of an electron with a microscope requires reflection of a photon off of the electron. This reflected photon causes a change in the path of the electron.

You can see a funny, animated explanation of Heisenberg's Uncertainty Principle at http://video.pbs.org/video/18121247.

Quantum Mechanical Model

In 1926, Austrian physicist Erwin Schrödinger (1887-1961) used the wave-particle duality of the electron to develop and solve a complex mathematical equation that accurately described the behavior of the electron in a hydrogen atom. The quantum mechanical model of the atom comes from the solution to Schrödinger’s equation. Quantization of electron energies is a requirement in order to solve the equation. This is unlike the Bohr model, in which quantization was simply assumed with no mathematical basis.

Recall that in the Bohr model, the exact path of the electron was restricted to very well defined circular orbits around the nucleus. The quantum mechanical model is a radical departure from that. Solutions to the Schrödinger wave equation, called wave functions, give only the probability of finding an electron at a given point around the nucleus. Electrons do not travel around the nucleus in simple circular orbits.

The location of the electrons in the quantum mechanical model of the atom is often referred to as an electron cloud. The electron cloud can be thought of in the following way. Imagine placing a square piece of paper on the floor with a dot in the circle representing the nucleus. Now take a marker and drop it onto the paper repeatedly, making small marks at each point the marker hits. If you drop the marker many, many times, the overall pattern of dots will be roughly circular. If you aim toward the center reasonably well, there will be more dots near the nucleus and progressively fewer dots as you move away from it. Each dot represents a location where the electron could be at any given moment. Because of the uncertainty principle, there is no way to know exactly where the electron is. An electron cloud has variable densities: a high density where the electron is most likely to be and a low density where the electron is least likely to be (Figure below).

An electron cloud: the darker region near the nucleus indicates a high probability of finding the electron, while the lighter region further from the nucleus indicates a lower probability of finding the electron.

In order to specifically define the shape of the cloud, it is customary to refer to the region of space within which there is a 90% probability of finding the electron. This is called an orbital, which can be defined as a three-dimensional region of space in which there is a high probability of finding an electron.

Atomic Orbitals and Quantum Numbers

Solutions to the Schrödinger wave equation place limits on the energies that an electron is allowed to have. The mathematical representation of those energies results in regions of space called orbitals, which, as we will soon see, can have different sizes and shapes. In order to distinguish between various orbitals and the electrons that occupy them, scientists use quantum numbers. Quantum numbers specify the properties of the atomic orbitals and the electrons in those orbitals. Understanding quantum numbers is helped by an analogy. Let’s say you are attending a basketball game. Your ticket may specify a gate number, a section number, a row, and a seat number. No other ticket can have the same four parts to it. It may have the same gate, section, and seat number, but if so, it would have to be in a different row. There are also four quantum numbers which describe the range of possible locations for every electron in every atom. No two electrons in a given atom can have the same four quantum numbers. We will describe each of these quantum numbers separately.

Principal Quantum Number

The principal quantum number is symbolized by the letter n and designates the principal or main energy level occupied by the electron. The possible values of n begin with n = 1, which is the lowest energy level and is located closest to the nucleus. As the value of n increases to n = 2, 3, and so on, the distance from the nucleus increases. The principal quantum number is essentially the same as the energy levels in the Bohr model of the atom that were used to explain atomic emission spectra. More than one electron may occupy a given principal energy level, but the specific number of electrons that can be held by each energy level varies depending on the value of n.

Angular Momentum Quantum Number

The angular momentum quantum number is symbolized by the letter l and indicates the shape of the orbital. Orbitals of different shapes exist for most principal energy levels, and for atoms that contain more than one electron, the different shapes result in slightly different energies for the electrons in each orbital. The various orbital shapes split each principal energy level into one or more energy sublevels. The number of possible sublevels in a given principal energy level is equal to the value of n. For example, when n = 1, there is only one sublevel, but when n = 2, there are two sublevels. The quantum number l is an integer that varies from 0 up to a value equal to n - 1. If n = 1, the only possible value of l is 0, whereas if n = 4, l can have a value of 0, 1, 2, or 3. The orbitals of each sublevel are also designated by a particular letter, such as s, p, d, or f. If l = 0, the orbital is called an s orbital, and if l = 1, the orbital is a p orbital. See Table below for a summary.

Principal Energy Levels and Sublevels
Principal Energy Level Number of Possible Sublevels Possible Angular Momentum Quantum Numbers Orbital Designation by Principal Energy Level and Sublevel
n = 1 1 l = 0 1s
n = 2 2

l = 0

l = 1



n = 3 3

l = 0

l = 1

l = 2




n = 4 4

l = 0

l = 1

l = 2

l = 3





From the table, you can see that in the 1st principal energy level (n = 1) there is only one sublevel possible – an s sublevel. In the 2nd principal energy level (n = 2), there are two sublevels possible – the s and p sublevels. This continues through the 3rd and 4th principal energy levels, in which we add the d and the f sublevels. In general, for the nth principal energy level there are n sublevels available. The order of the sublevels is always the same.

Magnetic Quantum Number

As mentioned above, each of the different orbital types has a different shape. The s orbitals are spherical in shape (Figure below), the p orbitals are dumbbell shaped (Figure below), and the d and f orbitals are more complex shapes that include multiple lobes (Figure below).

The s orbitals are spherical in shape and centered on the nucleus. The larger the value of the principal quantum number n, the larger the corresponding s orbital will be.

The p orbitals are dumbbell-shaped. (a) shows the electron density distribution for one of the orbitals, while (b) shows how the p orbitals are typically drawn. The three different p orbitals are identical in shape, but they can be oriented along the x, y, or z axis.

View the p orbitals at http://www.dlt.ncssm.edu/core/Chapter8-Atomic_Str_Part2/chapter8-Animations/P-orbitalDiagram.html.

The five d orbitals have more complex shapes. Four of them have a 4-lobed appearance with different orientations, while the dz2 orbital has a more complex shape.

The magnetic quantum number is symbolized by the letter ml and indicates the orientation of the orbital around the nucleus. Because an s orbital is spherical in shape and is centered on the nucleus, it only has one possible orientation. If l = 0, as would be the case for an s orbital, there is only one possible value for the magnetic quantum number, ml = 0. In the s sublevel of each principal energy level (n), there is just one s orbital. The larger the value of n, the larger the corresponding s orbital will be. As seen in Figure above, the dumbbell-shaped p orbitals have three possible orientations. In one orientation, called px, the lobes of the orbital lie along the axis defined as x. In the py orbital, they lie along the y-axis, which is at a 90° angle to the px orbital. Finally, the pz orbital lies along the z-axis, which makes a 90° angle relative to the other two orbitals. In any p sublevel, there are always three possible orbitals, so there are also three possible values of the magnetic quantum number: ml = −1, ml = 0, and ml = +1. There is no particular relationship between the coordinates (x, y, and z) and the ml values.

There are five different d orbitals within each d sublevel. The corresponding magnetic quantum numbers are ml = −2, ml = −1, ml = 0, ml = +1, and ml = +2. Finally, the pattern continues with the f sublevel, which contains seven possible f orbitals and ml values ranging from −3 to +3. If we only consider atoms that are in their ground state, all of the electrons in the atoms of every known element reside in orbitals that belong to one of these four sublevels.

Spin Quantum Number

Experiments show that electrons spin on their own internal axis, much as Earth does. The spinning of a charged particle creates a magnetic field. The orientation of that magnetic field depends upon the direction that the electron is spinning, either clockwise or counterclockwise. The spin quantum number is symbolized by the letter ms and indicates the direction of electron spin. The only possible values are \begin{align*}m_s = +\frac{1}{2}\end{align*} and \begin{align*}m_s = - \frac{1}{2}\end{align*}. Each orbital, regardless of its shape and orientation, can hold a maximum of two electrons, and any two electrons that occupy the same orbital must have opposite spins. Table below shows a summary of the first few energy levels and sublevels along with the number of electrons that can potentially be contained by each state.

Electron Arrangement Within Energy Levels
Principal Quantum Number (n) Allowable Sublevels Number of Orbitals per Sublevel Number of Orbitals per Principal Energy Level Number of Electrons per Sublevel Number of Electrons per Principal Energy Level
1 s 1 1 2 2


































Notice that the total number of allowable orbitals in each principal energy level (n) is equal to n2. That is, when n = 1, there is 12 = 1 orbital possible. When n = 2, there are 22 = 4 orbitals possible, and so on. Since each orbital holds two electrons, the number of electrons that can exist in a given principal energy level is equal to 2n2.

Lesson Summary

  • de Broglie showed that electrons exhibit characteristics of both waves and particles.
  • The behavior of atomic and subatomic sized particles is explained by quantum mechanics, where energy is gained and lost in small discrete amounts.
  • The Heisenberg uncertainty principle states that is not possible to simultaneously know the location of an electron and its velocity at any precise moment.
  • The Schrödinger wave equation proved mathematically that the energy of an electron must be quantized.
  • The quantum mechanical model of the atom describes the probability that an atom’s electrons will be located within certain regions called orbitals.
  • The arrangement of electrons in an atom is governed by four quantum numbers, which designate a principal energy level, an energy sublevel, an orbital orientation, and a spin for each electron in the atom.

Lesson Review Questions

Reviewing Concepts

  1. How is the wavelength of a moving object related to its mass?
  2. Why is the de Broglie wave equation meaningful only for submicroscopic particles such as atoms and electrons but not for larger everyday objects?
  3. How does the Heisenberg uncertainty principle affect the way in which electron locations are viewed in the quantum mechanical model as compared to the Bohr model?
  4. What is an atomic orbital?
  5. How many quantum numbers are used to describe each electron in an atom?
  6. Name two ways in which an electron that occupies the n = 2 principal energy level is different from an electron that occupies the n = 1 principal energy level.
  7. Identify which quantum number describes each of the following.
    1. the orientation of an orbital in space
    2. the direction of electron spin
    3. the main energy of an electron
    4. the shape of an orbital


  1. What is the wavelength (in nm) of an electron moving at 250. m/s? If this wavelength belonged to a photon, in what region of the electromagnetic spectrum would it belong?
  2. What are the possible values of l that an electron in the n = 3 principal energy level can have? Which sublevel does each of those l values represent?
  3. What are the possible values of ml that an electron in the p sublevel can have?
  4. Which of the following combinations of principal energy level and sublevel cannot exist?
    1. 4d
    2. 3f
    3. 1p
    4. 2s
    5. 3p
    6. 2d
  5. How is a pz orbital different from a px orbital? How are they the same?
  6. How many orbitals are found in each of the following?
    1. any s sublevel
    2. any f sublevel
    3. the n = 2 principal energy level
    4. the 4p sublevel
    5. the n = 3 principal energy level
  7. What is the maximum number of electrons that can occupy each of the following?
    1. any single orbital
    2. the n = 2 principal energy level
    3. the 3d sublevel
    4. the n = 4 principal energy level
    5. the 2p sublevel

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