## The Quantum–Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, as Bohr had argued, Erwin Schrödinger extended de Broglie’s work by incorporating the de Broglie relation into a wave equation, deriving what is today known as the Schrödinger equation.

When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra, and he did so without having to invoke Bohr’s assumptions of stationary states and quantized orbits, angular momenta, and energies; quantization in Schrödinger’s theory was a natural consequence of the underlying mathematics of the wave equation.

Like de Broglie, Schrödinger initially viewed the electron in hydrogen as being a physical wave instead of a particle, but where de Broglie thought of the electron in terms of circular stationary waves, Schrödinger properly thought in terms of three-dimensional stationary waves, or **wavefunctions**, represented by the Greek letter psi, *ψ*. A few years later, Max **Born** proposed an interpretation of the wavefunction *ψ* that is still accepted today: Electrons are still particles, and so the waves represented by *ψ* are not physical waves but, instead, are complex probability amplitudes.

The square of the magnitude of a wave-function \({\text{∣}\psi \text{∣}}^{2}\) describes the probability of the quantum particle being present near a certain location in space. This means that wavefunctions can be used to determine the distribution of the electron’s density with respect to the nucleus in an atom. In the most general form, the Schrödinger equation can be written as:

\(\hat{H}\psi =E\psi \)

\(\hat{H}\) is the Hamiltonian operator, a set of mathematical operations representing the total energy of the quantum particle (such as an electron in an atom), *ψ* is the wavefunction of this particle that can be used to find the special distribution of the probability of finding the particle, and \(E\) is the actual value of the total energy of the particle.

Schrödinger’s work, as well as that of Heisenberg and many other scientists following in their footsteps, is generally referred to as **quantum mechanics**.

## Optional Video: Schrödinger’s Cat

Austrian physicist Erwin Schrödinger, one of the founders of quantum mechanics, posed this famous question: If you put a cat in a sealed box with a device that has a 50% chance of killing the cat in the next hour, what will be the state of the cat when that time is up? Chad Orzel investigates this thought experiment.