The quantum mechanical model of the atom

Introduction to the quantum mechanical model of the atom: Thinking about electrons as probabilistic matter waves using the de Broglie wavelength, the Schrödinger equation, and the Heisenberg uncertainty principle. Electron spin and the Stern-Gerlach experiment.

Key points

• Louis de Broglie proposed that all particles could be treated as matter waves with a wavelength lambda, given by the following equation:
lambda, equals, start fraction, h, divided by, m, v, end fraction
• Erwin Schrödinger proposed the quantum mechanical model of the atom, which treats electrons as matter waves.
• Schrödinger's equation, $\hat{H}\psi=E\psi$, can be solved to yield a series of wave function $\psi$, each of which is associated with an electron binding energy, E.
• The square of the wave function, $\psi^2$, represents the probability of finding an electron in a given region within the atom.
• An atomic orbital is defined as the region within an atom that encloses where the electron is likely to be 90% of the time.
• The Heisenberg uncertainty principle states that we can't know both the energy and position of an electron. Therefore, as we learn more about the electron's position, we know less about its energy, and vice versa.
• Electrons have an intrinsic property called spin, and an electron can have one of two possible spin values: spin-up or spin-down.
• Any two electrons occupying the same orbital must have opposite spins.

Introduction to the quantum mechanical model

"We must be clear that when it comes to atoms, language can only be used as in poetry." —Niels Bohr
Matter begins to behave very strangely at the subatomic level. Some of this behavior is so counterintuitive that we can only talk about it with symbols and metaphors—like in poetry. For example, what does it mean to say an electron behaves like a particle and a wave? Or that an electron does not exist in any one particular location, but that it is spread out throughout the entire atom?
If these questions strike you as odd, they should! As it turns out, we are in good company. The physicist Niels Bohr also said, "Anyone who is not shocked by quantum theory has not understood it." So if you feel confused when learning about quantum mechanics, know that the scientists who originally developed it were just as befuddled.
We will start by briefly reviewing Bohr's model of hydrogen, the first non-classical model of the atom.

Review of Bohr's model of hydrogen

As we have seen in a previous article on the Bohr model, the emission spectra of different elements contain discrete lines. The following image shows the visible region of the emission spectra for hydrogen.
Emission spectrum for hydrogen showing purplish blue lines at 410 and 434 nm, a light blue line at 486 nm, and a red line at 656 nm. All lines are against a black background.
Hydrogen emits four wavelengths of light in the visible region. Image credit: Emission spectrum from Wikimedia Commons, CC0 1.0
The quantized emission spectra indicated to Bohr that perhaps electrons could only exist within the atom at certain atomic radii and energies. Recall that quantized refers to the fact that energy can only be absorbed and emitted in a range of allowable values rather than with any possible value. The following diagram of the Bohr model shows the electron existing in a finite number of allowed orbits or shells around the nucleus.
Diagram showing the the first three levels—n=1, 2, and 3—for Bohr's model of hydrogen. An electron is relaxing from n-3 to n=2, as indicated by an arrow starting at n=3 and going to n=2. The loss of a photon is shown for the electronic transition with an energy of hf.
A diagram of the Bohr model of the hydrogen atom. Electrons move in circular orbits that are at fixed distances from the nucleus. Light is emitted when excited electrons, n, is greater than, 1, relax back to a lower energy level. Image credit: from Wikimedia Commons, CC BY-SA 3.0
From this model, Bohr derived an equation that correctly predicted the various energy levels in the hydrogen atom, which corresponded directly to the emission lines in the hydrogen spectrum. Bohr's model was also successful at predicting the energy levels in other one-electron systems, such as H, e, start superscript, plus, end superscript. However, it failed to explain the electronic structure in atoms that contained more than one electron.
While some physicists initially tried to adapt Bohr's model to make it useful for more complicated systems, they eventually concluded that a completely different model was needed.

Wave-particle duality and the de Broglie wavelength

Another major development in quantum mechanics was pioneered by French physicist Louis de Broglie. Based on work by Planck and Einstein that showed how light waves could exhibit particle-like properties, de Broglie hypothesized that particles could also have wavelike properties.
Examples of observable wavelike behavior are interference and diffraction. For example, when light is shined through a barrier with two slits, as in Young's double-slit experiment, the light waves will diffract through the slits. The destructive and constructive interference between the light waves produces a pattern of dark and light areas on the detector.
De Broglie derived the following equation for the wavelength of a particle of mass m (in kilograms k, g), traveling at velocity v (in start fraction, m, divided by, s, end fraction), where lambda is the de Broglie wavelength of the particle in meters and h is Planck's constant, 6, point, 626, times, 10, start superscript, minus, 34, end superscript, space, start fraction, k, g, dot, m, start superscript, 2, end superscript, divided by, s, end fraction:
lambda, equals, start fraction, h, divided by, m, v, end fraction
Note that the de Broglie wavelength and particle mass are inversely proportional. The inverse relationship is why we don't notice any wavelike behavior for the macroscopic objects we encounter in everyday life. It turns out that the wavelike behavior of matter is most significant when a wave encounters an obstacle or slit that is a similar size to its de Broglie wavelength. However, when a particle has a mass on the order of 10, start superscript, minus, 31, end superscript kg, as an electron does, the wavelike behavior becomes significant enough to lead to some very interesting phenomena.
Concept check: The fastest baseball pitch ever recorded was approximately 46.7 start fraction, m, divided by, s, end fraction. If a baseball has a mass of 0.145 kg, what is its de Broglie wavelength?
Plugging in the appropriate values for mass and velocity into de Broglie's equation, we get:
This wavelength is 20 orders of magnitude smaller than the diameter of a proton! Because this wavelength is so small, we would not expect to observe baseballs behaving like a wave, for example, exhibiting diffraction patterns.

Example 1: Calculating the de Broglie wavelength of an electron

The velocity of an electron in the ground-state energy level of hydrogen is 2, point, 2, times, 10, start superscript, 6, end superscript, space, start fraction, m, divided by, s, end fraction. If the electron's mass is 9, point, 1, times, 10, start superscript, minus, 31, end superscript kg, what is the de Broglie wavelength of this electron?
We can substitute Planck's constant and the mass and velocity of the electron into de Broglie's equation:
The wavelength of our electron, 3, point, 3, times, 10, start superscript, minus, 10, end superscript, space meters, is on the same order of magnitude as the diameter of a hydrogen atom, ~1, times, 10, start superscript, minus, 10, end superscript, space meters. That means the de Broglie wavelength of our electron is such that it will often be encountering things with a similar size as its wavelength—for instance, a neutron or atom. When that happens, the electron will be likely to demonstrate wavelike behavior!

The quantum mechanical model of the atom

Standing waves

A major problem with Bohr's model was that it treated electrons as particles that existed in precisely-defined orbits. Based on de Broglie's idea that particles could exhibit wavelike behavior, Austrian physicist Erwin Schrödinger theorized that the behavior of electrons within atoms could be explained by treating them mathematically as matter waves. This model, which is the basis of the modern understanding of the atom, is known as the quantum mechanical or wave mechanical model.
The fact that there are only certain allowable states or energies that an electron in an atom can have is similar to a standing wave. We will briefly discuss some properties of standing waves to get a better intuition for electron matter waves.
You are probably already familiar with standing waves from stringed musical instruments. For example, when a string is plucked on a guitar, the string vibrates in the shape of a standing wave such as the one shown below.
Standing wave animation showing two wavelengths of a wave. The nodes, which have the same amplitude at all times, are marked with red dots. There are five nodes.
A standing wave. Image credit: from Wikimedia Commons, public domain
Notice that there are points of zero displacement, or nodes, that occur along the standing wave. The nodes are marked with red dots. Since the string in the animation is fixed at both ends, this leads to the limitation that only certain wavelengths are allowed for any standing wave. As such, the vibrations are quantized.

Schrödinger's equation

How are standing waves related to electrons in an atom, you may ask?
On a very simple level, we can think of electrons as standing matter waves that have certain allowed energies. Schrödinger formulated a model of the atom that assumed the electrons could be treated at matter waves. While we won't be going through the math in this article, the basic form of Schrödinger's wave equation is as follows:
$\hat{H}\psi=E\psi$
$\psi$ is called a wave function; $\hat{H}$ is known as the Hamiltonian operator; and E is the binding energy of the electron. Solving Schrödinger's equation yields multiple wave functions as solutions, each with an allowed value for E.
A standing wave that forms a circle, with the wavelength labeled as the distance between two adjacent maximum amplitude points. Below, an example of where the wavelength does not fit the radius of the circle such that the waves overlap on one side of the circle as an example of destructive interference.
In the standing wave, top, exactly five full wavelengths fit within the circle. When the circumference of the circle does not allow an integer number of wavelengths, bottom, the resulting destructive interference results in cancellation of the wave.
Interpreting exactly what the wave functions tell us is a bit tricky. Due to the Heisenberg uncertainty principle, it is impossible to know for a given electron both its position and its energy. Since knowing the energy of an electron is necessary for predicting the chemical reactivity of an atom, chemists generally accept that we can only approximate the location of the electron.
How do chemists approximate the location of the electron? The wave functions that are derived from Schrödinger's equation for a specific atom are also called atomic orbitals. Chemists define an atomic orbital as the region within an atom that encloses where the electron is likely to be 90% of the time. In the next section, we will discuss how electron probabilities are determined.
The Heisenberg uncertainty principle, developed by physicist Werner Heisenberg, states that there is an inherent limitation to how precisely we can know both the position and the momentum—or energy—of a particle at a given time. That is to say, the more precisely we know the position of an electron, the less we know about its momentum, and vice versa. This can be stated mathematically as follows:
delta, x, dot, delta, p, is greater than or equal to, start fraction, h, divided by, 4, pi, end fraction
Here, delta, x represents the uncertainty in the electron's position; delta, p represents the uncertainty in the electron's momentum; and h is Planck's constant, 6, point, 626, times, 10, start superscript, minus, 34, end superscript, space, J, dot, s. From the inequality, we can see that delta, x and delta, p are inversely proportional. The inverse proportionality means that as the uncertainty in position decreases, the uncertainty in momentum increases, and vice versa.
Thus, we can never know both where an electron is and its energy all at the same time.

Orbitals and probability density

The value of the wave function $\psi$ at a given point in space—x, comma, y, comma, z—is proportional to the amplitude of the electron matter wave at that point. However, many wave functions are complex functions containing i, equals, square root of, minus, 1, end square root, and the amplitude of the matter wave has no real physical significance.
Luckily, the square of the wave function, $\psi^2$, is a little more useful. This is because the square of a wave function is proportional to the probability of finding an electron in a particular volume of space within an atom. The function $\psi^2$ is often called the probability density.
The probability density for an electron can be visualized in a number of different ways. For example, $\psi^2$ can be represented by a graph in which varying intensity of color is used to show the relative probabilities of finding an electron in a given region in space. The greater the probability of finding an electron in a particular volume, the higher the density of the color in that region. The image below shows the probability distributions for the spherical 1s, 2s, and 3s orbitals.
Probability distributions for 1s, 2s, and 3s orbitals. Greater color intensity indicates regions where electrons are more likely to exist. Nodes indicate regions where an electron has zero probability of being found. Image credit: UCDavis Chemwiki, CC BY-NC-SA 3.0 US
Notice that the 2s and 3s orbitals contain nodes—regions in which an electron has a 0% probability of being found. The existence of nodes is analogous to the standing waves we discussed in the previous section. The alternating colors in the 2s and 3s orbitals represent regions of the orbital with different phases, which is an important consideration in chemical bonding.
Another way of picturing probabilities for electrons in orbitals is by plotting the surface density as a function of the distance from the nucleus, r.
A radial probability graph showing surface probability $\psi^2r^2$ vs. r. Electrons occupying higher-energy orbitals have greater probabilities of being found farther from the nucleus. Image credit: UC Davis Chemwiki, CC BY-NC-SA 3.0 US
The surface density is the probability of finding the electron in a thin shell with radius r. This is called a radial probability graph. On the left is a radial probability graph for the 1s, 2s, and 3s orbitals. Notice that as the energy level of the orbital increases from 1s to 2s to 3s, the probability of finding an electron farther from the nucleus increases as well.
When Schrödinger's equation is solved, the wave function, $\psi$, that is obtained is associated with a particular orbital. Each orbital has a set of four quantum numbers that come out of Schrödinger's equation. Together, the four quantum numbers act like the zip code for an electron, defining its orbital inside the atom. The four quantum numbers are as follows:
• n, the principal quantum number, is the major factor in determining the energy of an orbital. Orbitals with the same n value are said to share the same electron shell.
• l, the angular quantum number, defines the shape of the orbital. Orbitals with the same n value but different values of l are called subshells.
• m, start subscript, l, end subscript, the magnetic quantum number, is related to the orbital's orientation in space.
• m, start subscript, s, end subscript, the spin quantum number, indicates the spin of an electron. Electrons can be spin-up left parenthesis, m, start subscript, s, end subscript, equals, plus, start fraction, 1, divided by, 2, end fraction, right parenthesis or spin-down left parenthesis, m, start subscript, s, end subscript, equals, minus, start fraction, 1, divided by, 2, end fraction, right parenthesis.
For more details on how quantum numbers can be assigned, see the following video on quantum numbers and video on writing quantum numbers for the first four shells.

Shapes of atomic orbitals

So far we have been examining s orbitals, which are spherical. As such, the distance from the nucleus, r, is the main factor affecting an electron's probability distribution. However, for other types of orbitals such as p, d, and f orbitals, the electron's angular position relative to the nucleus also becomes a factor in the probability density. This leads to more interesting orbital shapes, such as the ones in the following image.
Schematics showing the general shapes of s, p, d, and f orbitals. Image credit: UCDavis Chemwiki, CC BY-NC-SA 3.0 US
The p orbitals are shaped like dumbbells that are oriented along one of the axes—x, comma, y, comma, z. The d orbitals can be described as having a clover shape with four possible orientations—with the exception of the d orbital that almost looks like a p orbital with a donut going around the middle. It's not even worth attempting to describe the f orbitals!

Electron spin: the Stern-Gerlach experiment

The last quantum phenomenon we will discuss is that of electron spin. In 1922, German physicists Otto Stern and Walther Gerlach hypothesized that electrons behaved as tiny bar magnets, each with a north and south pole. To test this theory, they fired a beam of silver atoms between the poles of a permanent magnet with a stronger north pole than south pole.
According to classical physics, the orientation of a dipole in an external magnetic field should determine the direction in which the beam gets deflected. Since a bar magnet can have a range of orientations relative to the external magnetic field, they expected to see atoms being deflected by different amounts to give a spread-out distribution. Instead, Stern and Gerlach observed the atoms were split cleanly between the north and south poles. Watch the following awesome video to see the hypothesis and experiment in action!
These experimental results revealed that unlike regular bar magnets, electrons could only exhibit two possible orientations: either with the magnetic field or against it. This phenomenon, in which electrons can exist in only one of two possible magnetic states, could not be explained using classical physics! Scientists refer to this property of electrons as electron spin: any given electron is either spin-up or spin-down. We sometimes represent electron spin by drawing electrons as arrows pointing up, $\uparrow$, or down, $\downarrow$.
One consequence of electron spin is that a maximum of two electrons can occupy any given orbital, and the two electrons occupying the same orbital must have opposite spin. This is also called the Pauli exclusion principle.

Summary

• Louis de Broglie proposed that all particles could be treated as matter waves with a wavelength lambda given by the following equation:
lambda, equals, start fraction, h, divided by, m, v, end fraction
• Erwin Schrödinger proposed the quantum mechanical model of the atom, which treats electrons as matter waves.
• Schrödinger's equation, $\hat{H}\psi=E\psi$, can be solved to yield a series of wave function $\psi$, each of which is associated with an electron binding energy, E.
• The square of the wave function, $\psi^2$, represents the probability of finding an electron in a given region within the atom.
• An atomic orbital is defined as the region within an atom that encloses where the electron is likely to be 90% of the time.
• The Heisenberg uncertainty principle states that we can't know both the energy and position of an electron. Therefore, as we learn more about the electron's position, we know less about its energy, and vice versa.
• Electrons have an intrinsic property called spin, and an electron can have one of two possible spin values: spin-up or spin-down.
• Any two electrons occupying the same orbital must have opposite spins.