Schrödinger's equation revolutionized our vision of matter

Representation of an atomic orbital forming a complex structure evoking a flower or a cloud of probability. — Visualization of an electronic orbital, a cloud of probability with complex and elegant shapes, resulting from the solutions of the Schrödinger equation.
Image source: astronoo.com (new window) - AI-generated image, public domain.

Scientific summary

The Astronoo article presents the Schrödinger equation and its founding role in the description of atomic structure. Published in 1926, this fundamental equation of quantum mechanics replaces the fixed orbits of the Bohr model with clouds of probability. The solutions of the equation, called wave functions, draw elegant three-dimensional shapes: the atomic orbitals s, p, d and f. The article explains how the square of the wave function, |Ψ|², gives the probability of finding the electron around the nucleus. The geometry of these orbitals determines the chemical properties of atoms and the structure of molecules.

How did Schrödinger's equation revolutionize our vision of the atom?

This article is devoted to the Schrödinger equation, a pillar of quantum mechanics, and to its ability to describe the structure of atoms. Before Schrödinger, the Bohr model imagined the electron as a planet orbiting the nucleus on a circular orbit. But this vision was incomplete. In 1926, Schrödinger proposed an equation that radically transforms this image: the electron is no longer a localized particle, but a probability wave. What does this equation reveal? It shows that the electron is not located at a precise place, but exists as a cloud of probability around the nucleus. The solutions of the equation, called wave functions, draw three-dimensional geometric shapes: atomic orbitals. These shapes are not mere mathematical curiosities: they determine how atoms bond to form molecules, thus explaining the structure of all matter. The article explores this probabilistic dance, from the equation to the elegant shapes of s, p, d and f orbitals, and shows how quantum mechanics replaced the certainty of orbits with the richness of probabilities.

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From certainty to probability: the atom according to Schrödinger

The electron, a wave rather than a planet

Imagine for a moment that in the atom, the electron is no longer a small planet wisely orbiting a sun-like nucleus, but a musical note, a vibration, a wave. That is the conceptual revolution triggered by Erwin Schrödinger (1887-1961) in 1926.

The wave function: a map of probabilities

The Schrödinger equation (fundamental equation of quantum mechanics describing the evolution of a particle through a wave function.) does not tell us where the electron is with certainty, but where it is likely to be. It transforms the possible atomic space into a dance floor where the dancer's figures draw the probable orbitals.

Understanding the Schrödinger equation without the headache

The Schrödinger equation is written: $\hat{H}\,\Psi(r,\theta,\phi)=E\,\Psi(r,\theta,\phi)$ where $(r, \theta, \phi)$ are spherical coordinates (r: the radial coordinate, θ: the polar coordinate, ϕ: the azimuthal coordinate). However, in its simplest form, it is written: $ \hat{H} \Psi = E \Psi $

To unlock the mysteries of the atom, a special key is needed. This key is what physicists call the "wave function". It has been given a chic little name: Ψ (pronounced "Psi", like the beginning of "psyche" or Neptune's trident).

How to read $ \hat{H} \Psi = E \Psi $ simply?

Imagine that Ψ is a kind of "navigation map" for the electron. This map does not say where the electron will go, but where it may go. More precisely, it describes the places where you have a chance of finding it, and the places where you will not find it.

On the left, $\hat{H}$ represents "everything that happens in the atom" (the attraction of the nucleus, energy, constraints). On the right, $E$ is the result: the energy the electron must have to be stable in this configuration.

Solving this equation is like looking for the right note on a piano so that it resonates perfectly in a room. You cannot play just anything: only certain notes (the energies $E$) and certain ways of vibrating (the functions Ψ) work harmoniously.

But what does Ψ look like?

This was the great dilemma of physicists until Max Born (1882-1970) had a brilliant intuition. He said: "Do not look at Ψ itself, look at its reflection, its square, $|\Psi|^2$". It is like smoke revealing a cloud of probability where the object was before the magician made it disappear. Thus, $|\Psi|^2$ indicates, point by point in the space around the nucleus, the probability of the electron's presence.

Clearly, where $|\Psi|^2$ is large is the atom's "garden", where the electron spends its time. Where $|\Psi|^2$ is zero is a "desert", the electron never comes there. The electron then becomes less a marble than a cloud, a trail of light, a blurred and elegant dance around the nucleus called an "orbital".

The patterns of the electronic dance: atomic orbitals

From Bohr's orbits to Schrödinger's orbitals

Applied to the hydrogen atom (a proton and an electron), the Schrödinger equation reveals a series of solutions, quantum states, each characterized by three numbers, often called quantum numbers. Each solution corresponds to an atomic orbital, a specific three-dimensional geometric shape that is the electron's privileged "dance zone".

Where Niels Bohr (1885-1962) saw electrons orbiting on well-ordered circular orbits, like planets around the Sun, the Schrödinger equation reveals a far richer and more poetic universe. The solutions draw geometric shapes of elegant complexity in space.

The shapes of atomic orbitals

The perfect sphere (s orbital): The simplest of them, called 1s, is a sphere. The electron has no preferred direction; it is present homogeneously all around the nucleus, like a perfectly symmetrical fog.
The dumbbells (p orbitals): As soon as we move a little away from the nucleus (second energy level), space becomes structured. Three identical "dumbbell" shapes then appear, each pointing in a direction of space (along the x, y and z axes). The electron then concentrates in two opposite lobes, on either side of a zone of silence (the nodal plane).
The flowers of matter (d and f orbitals): At even greater distances, the choreography becomes richer. The d orbitals adopt geometries evoking four-leaf clovers. The f orbitals, for their part, deploy multiple petals in even more sophisticated arrangements. True mathematical signatures of space, they are to chemistry what scales are to music: the fundamental structure that allows all combinations, all molecules, all the matter around us.

The geometry of orbitals: the key to chemistry

These shapes are not mere mathematical whims. They are the keystone of chemistry. The geometry of orbitals determines how atoms can bond with each other to form molecules.

The direction of the p orbitals, for example, explains the angular shape of the water molecule (H₂O). The richness of the d orbitals is responsible for the bright colors of the ions of transition metals.

Thus, the Schrödinger equation, through its solutions, is the hidden grammar that dictates the structure of all the matter around us.

N.B.:
The mental image of the "electron cloud" is more accurate than that of the "moving particle". An orbital is an instantaneous probability map. If you took an ultra-fast photograph of the atom, you would see the electron at one point; by superimposing millions of these photographs, you would see the shape of the orbital appear, like a trail of light left by a dancer on a dark stage.

References

FAQ: Everything about the Schrödinger equation and atomic orbitals

What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation of quantum mechanics, formulated by Erwin Schrödinger in 1926. It describes the time evolution of the wave function of a quantum system, such as an electron in an atom. In its stationary form, it is written $\hat{H}\Psi = E\Psi$, where $\hat{H}$ is the Hamiltonian operator (representing the total energy of the system), $\Psi$ the wave function and $E$ the energy of the stationary state. The solutions of this equation give the possible quantum states of the electron and their associated energy levels.

What is an atomic orbital?

An atomic orbital is a region of space around the nucleus where the probability of finding an electron is maximal. It is defined by the solutions of the Schrödinger equation for an atom. Each orbital is characterized by three quantum numbers (principal, secondary and magnetic) and has a specific geometric shape: spherical for s orbitals, dumbbell-shaped for p orbitals, clover-shaped for d orbitals, and more complex for f orbitals. These shapes determine the chemical properties of atoms and the geometry of molecules.

Why is the electron described by a wave function and not as a classical particle?

The electron cannot be described as a classical particle (a localized marble) because it exhibits wave properties, confirmed by experiments such as electron diffraction. The wave function $\Psi$ encodes all available information about the electron: its probable position, its momentum, its energy. The square of the wave function, $|\Psi|^2$, gives the probability of finding the electron at a given point. This probabilistic description is fundamental in quantum mechanics and replaces the notion of a precise trajectory.

What is the difference between Bohr's atomic model and Schrödinger's?

The Bohr model (1913) imagines the electron orbiting on well-defined circular orbits around the nucleus, like a planet around the Sun. This model, although it made it possible to explain the hydrogen spectrum, is incomplete. The Schrödinger model (1926) replaces these orbits with orbitals: three-dimensional clouds of probability. The electron no longer has a simultaneously well-defined position and velocity (Heisenberg's uncertainty principle). The Schrödinger model is more precise and makes it possible to understand the electronic structure of all atoms, not only hydrogen.

How do atomic orbitals determine the chemical properties of elements?

The geometry and energy of atomic orbitals determine how atoms interact and bond with each other. For example, the direction of p orbitals explains the angular shape of the water molecule (H₂O). The richness of d orbitals is responsible for the bright colors of transition metal ions. The filling of orbitals by electrons, according to the Pauli exclusion principle and Hund's rule, explains the periodic classification of elements and their chemical properties (reactivity, electronegativity, magnetism, etc.).

What is Heisenberg's uncertainty principle and what is its link with the Schrödinger equation?

Heisenberg's uncertainty principle (1927) states that it is impossible to know simultaneously with precision the position and momentum of a quantum particle: $\Delta x \cdot \Delta p \geq \hbar/2$. This principle follows directly from the wave nature described by the Schrödinger equation: a wave function that is very localized in space (precise position) corresponds to a broad distribution of momenta (velocities), and vice versa. Thus, the Schrödinger equation and the uncertainty principle are two complementary facets of the quantum description of matter, explaining why the electron cannot be represented by a well-defined classical trajectory.