Schrödinger’s Atomic Model: Wavefunctions and Orbitals Explained

An electron sits inside a hydrogen atom. In the old picture, you might imagine it moving around the nucleus like a tiny planet around the Sun. That model seems reasonable. After all, planets orbit things.

But this picture creates a problem. If an electron is a charged particle moving in a circle, classical physics predicts it should lose energy and spiral into the nucleus. Atoms should collapse almost instantly.

They do not.

So what picture replaces the neat little orbit? Schrödinger’s contribution was to stop asking, “Where is the electron travelling?” and start asking, “What patterns are possible for an electron, and where are we likely to find it?”

Before the mathematics, make a prediction.

Imagine an electron inside a hydrogen atom. If it is not following a fixed circular path, does that mean we have no information about it?

The answer is no. We lose the idea of an exact path, but we gain a different kind of information: a map of where the electron is likely to be found.

This change is the foundation of the modern atomic model.

01From electron paths to electron waves

To understand Schrödinger’s idea, we first need the problem that came before it.

The Niels Bohr model of hydrogen solved an important mystery. It explained why hydrogen only produced certain wavelengths of light rather than a continuous range. Electrons could only exist in certain energy levels.

You can read more about that earlier model here:

The Bohr Model of Hydrogen: Quantised Orbits Explained

However, Bohr’s model still treated the electron as a tiny object moving around the nucleus. It gave useful predictions, but it did not explain why only those orbits were allowed or what an electron actually was between measurements.

A clue came from Louis de Broglie, who proposed that matter could have wave properties. Electrons were not simply particles; they could behave like waves.

The de Broglie Matter Waves: HSC Physics Explained Clearly

A wave is not located at one point in the same way a small ball is. A water wave spreads across a region. A sound wave fills a space.

Schrödinger used this idea and developed an equation that describes possible electron waves inside atoms.

The key idea:

An electron in an atom is described by a wavefunction.

The wavefunction does not tell us the exact location of the electron. Instead, it contains information about the probability of finding the electron in different places.

02The wavefunction: a map of possibilities

The symbol used for the wavefunction is the Greek letter psi:

\[ \psi \]

The wavefunction is a mathematical description of the electron’s quantum state. It can have positive and negative values, but these values are not directly the probability.

The probability comes from the square of the wavefunction’s magnitude:

\[ \text{Probability density} \propto |\psi|^2 \]

This means regions where \( |\psi|^2 \) is large are regions where an electron is more likely to be detected.

A useful analogy is a weather map.

A weather map does not say, “It is raining at exactly this one dot.” Instead, it shows regions where rain is more or less likely. A wavefunction works similarly: it describes a pattern of possibilities.

The analogy breaks down because weather is a classical prediction about many complicated factors, while quantum probability is built into the nature of the system. An electron is not hiding somewhere with a known location waiting for us to find it.

Before measurement, the wavefunction describes the possible outcomes.

03Orbitals are not orbits

This is the most common mistake students make.

An orbit is a path. It answers:

“Where does the object travel?”

An orbital is a region of high probability. It answers:

“Where are we likely to detect the electron?”

These are completely different ideas.

An orbital is not a fuzzy track that the electron follows. The electron does not draw the shape of the orbital as it moves.

Instead, the shape comes from solving Schrödinger’s equation for different allowed states.

For example:

The first mental model is useful, but incomplete:

• Useful model: the electron exists in a cloud-like region.
• More precise model: the cloud represents \( |\psi|^2 \), calculated from a quantum wavefunction.
• Full model: the wavefunction is found by solving Schrödinger’s equation.

04Schrödinger’s equation: finding allowed electron states

The full Schrödinger equation is beyond the calculations required in HSC Physics, but the idea behind it is essential.

A simplified way to think about the equation is:

\[ \text{Energy information} + \text{wave behaviour} \rightarrow \text{allowed electron states} \]

For an electron in an atom, the equation only allows certain wave patterns. Each allowed pattern has a specific energy.

This is similar to a guitar string.

A guitar string cannot vibrate in any random shape. It can only form standing wave patterns that fit the fixed ends of the string.

An atom works differently, but the same idea appears: only certain wave patterns are allowed.

The important difference is that an electron wave is not a physical string vibrating through space. The wavefunction is a mathematical description of quantum behaviour.

05Quantum numbers: naming the allowed states

The solutions to Schrödinger’s equation are described using quantum numbers.

The main quantum numbers used to describe electron states are:

The principal quantum number \(n\) is similar to Bohr’s energy levels. Higher values generally mean higher energy and a larger average distance from the nucleus.

The other quantum numbers describe the possible wave patterns within that energy level.

This explains something Bohr’s model could not: electrons are not just sitting on numbered tracks. They exist in allowed quantum states.

06How Schrödinger changed the atomic model

The development can be summarised like this:

Schrödinger did not make electrons “waves instead of particles” in a simple everyday sense.

The better statement is:

Electrons are quantum objects that are described using wavefunctions, and measurements reveal probabilities associated with those wavefunctions.

09The misconception: “The electron must be somewhere, so it must have a hidden path”

This prediction feels reasonable because everything we see in everyday life behaves this way.

A tennis ball is always somewhere. A car is always somewhere on a road.

But electrons are not tiny tennis balls.

In quantum mechanics, the wavefunction gives the probabilities for possible measurement outcomes. The theory does not assume there is a classical path waiting to be discovered.

A student might think:

“If the electron cloud is spread out, the electron must be physically smeared across space.”

That is also too simple.

The cloud is not a photograph of a tiny cloud of electron material. It is a representation of probability density calculated from the wavefunction.

The model is powerful because it predicts measurements, not because it gives a familiar picture of a miniature solar system.

10Questions and solutions

Question 1

A student says: “An electron in a p orbital follows a figure-eight path around the nucleus.” Is this statement correct? Explain.

Solution 1

The statement is incorrect because an orbital is not a path. A p orbital represents a probability distribution showing where an electron is likely to be detected.

The mistake comes from treating the electron like a planet. In Schrödinger’s model, the electron is described by a wavefunction, and the p shape comes from the regions where \( |\psi|^2 \) is larger.

The electron does not trace the outline of the p orbital as it moves.

Question 2

Explain why Schrödinger’s model was a better atomic model than Bohr’s model for describing electrons in atoms.

Solution 2

Schrödinger’s model is better because it explains electrons as wavefunctions with probability distributions rather than particles moving in fixed circular paths.

Bohr’s model successfully explained hydrogen energy levels and spectral lines, but it did not explain the three-dimensional shapes of electron regions or why only certain states are allowed.

Schrödinger’s equation produces allowed quantum states, which explain orbitals and electron probabilities.

Question 3

A hydrogen atom contains an electron in an \(n=2\) state. The electron moves to an \(n=1\) state.

Describe what happens to the electron’s energy and wavefunction.

Solution 3

The electron loses energy and changes to a different wavefunction.

The \(n=1\) state has lower energy than the \(n=2\) state, so the energy difference is released as a photon.

The electron does not spiral inward like a classical particle. Instead, the atom changes from one allowed quantum state to another, meaning the probability distribution describing the electron changes.

Question 4

Two students discuss orbitals.

Student A says: “The electron is spread through the entire orbital.”

Student B says: “The electron has one exact position, but we do not know it.”

Which explanation is closer to Schrödinger’s model?

Solution 4

Neither explanation is completely correct, but Student A is closer if interpreted carefully.

The orbital is a region where the electron has a higher probability of being detected. It does not mean the electron is a physical cloud filling that space.

Student B uses a classical idea that assumes the electron has a definite hidden position. Schrödinger’s model does not describe electrons with exact paths and positions before measurement.

The key idea is that the wavefunction describes possible outcomes and probabilities, not an unknown classical location.

Question 5

A scientist proposes a new atomic model where electrons still move in perfect circular paths, but the paths are so small that current experiments cannot detect them.

The scientist argues this model is acceptable because it gives the same predictions as Schrödinger’s model for hydrogen energy levels.

Assess this argument.

Solution 5

The argument is not acceptable because matching one prediction is not enough; the model must also explain the wider evidence.

The tempting route is to accept the circular paths because Bohr’s model also predicted hydrogen energy levels successfully. However, the circular-path model fails to explain why electrons have different orbital shapes and probability distributions.

Schrödinger’s model explains more observations because it connects allowed energies to wavefunctions and orbitals. A model that adds hidden circular paths without improving predictions adds an assumption but does not explain the evidence.

The important constraint is that a scientific model must explain the pattern of observations, not just reproduce one result.

11Where this idea leads next

Once electrons are understood as wavefunctions rather than tiny planets, several other atomic ideas become clearer.

The next step is understanding how changes between these allowed states produce specific photons, which explains atomic spectra such as the Balmer series. The quantum model also provides the foundation for understanding electron configurations and why different elements have different chemical behaviour.