Landmark Solutions

A contribution by Dr. Christopher Hilweg

Faculty of Physics, University of Vienna

A hundred years ago, in 1926, the Austrian physicist and alumnus of the University of Vienna, Erwin Schrödinger, derived a new dynamical law that changed how physicists think about the microscopic world and that would become one of the central pillars of modern physics: the Schrödinger equation. Today, physicists around the world make daily use of this equation as it continues to guide research, from understanding the smallest particles in nature to building new quantum technologies. 

In place of the trajectories of classical mechanics, Schrödinger’s framework uses a probabilistic description based on the so-called wavefunction, which encodes all physically accessible information about a quantum system and describes how it evolves in time.  Although quantum theory is broader than Schrödinger’s equation, Schrödinger’s insight remains the everyday language in which many scientists and technologists think and calculate. Within its domain, it is both a gateway to fundamental understanding and a practical tool that continues to guide experiments and design new quantum devices. For example, we can use the equation to describe an electron orbiting an atomic nucleus, a particle tunneling through a potential barrier, or even a vibrating nanomechanical oscillator cooled close to its quantum ground state. 

The best way to appreciate the Schrödinger equation's impact is to examine what it allows physicists to calculate. First, it determines the energy levels of atoms and molecules, providing the foundation of spectroscopy, chemistry and much of modern materials science. Second, it predicts quantum superposition, the ability of a system to exist simultaneously in multiple states, a feature that lies at the heart of quantum computing and quantum sensing. Third, it plays a crucial role in the description of quantum interference, the wave-like (self-)interaction of particles, and entanglement, the phenomenon Einstein famously dismissed as "spooky action at a distance."

These concepts are not abstract curiosities. They are the basis of technologies we use every day. Transistors, the building blocks of computers and smartphones, rely on quantum descriptions of electrons in semiconductors. Lasers, used in telecommunications and medicine, are explained by quantum transitions between atomic energy levels. MRI scanners harness quantum spin dynamics to image the human body. More recently, the same principles have enabled practical quantum cryptography, which promises communication that is fundamentally secure against eavesdropping. Experiments with entangled photons have demonstrated practical quantum key distribution systems, and quantum computers may eventually solve problems far beyond the reach of classical machines, from designing new materials to optimizing complex systems. All of these developments trace back, in one way or another, to the equation Schrödinger wrote down a century ago.

Yet much of contemporary research asks a deeper question: where does the Schrödinger equation break down? 

At the University of Vienna, a long tradition of foundational quantum experiments, stretching from the work of Anton Zeilinger to current research groups, continues to probe the equation's validity under increasingly extreme conditions.

One striking example is matter-wave interferometry. In these experiments, molecules containing hundreds or even thousands of atoms are sent through interferometers and allowed to interfere with themselves, exactly as the Schrödinger equation predicts. According to the equation, any object, no matter how large, should behave like a wave under the right conditions. Yet our everyday experience tells us that macroscopic objects behave classically. By testing quantum interference with ever-larger particles, researchers probe the boundary between the quantum and classical worlds, bringing Schrödinger's famous thought experiment of the cat closer to experimental reality.

Another frontier is quantum optomechanics. Here, the mechanical motion of macroscopic objects (tiny mirrors, vibrating membranes, levitated nanoparticles) is cooled so strongly that it becomes governed by quantum mechanics. Light is then used to measure and manipulate this motion with extraordinary precision. The Schrödinger equation predicts quantum superpositions even for these mechanical systems, and demonstrating this would push quantum mechanics into regimes that were unimaginable when the equation was first written.

Photons, the quanta of light, can also be used as carriers of quantum information, and here the relationship to the Schrödinger equation deserves closer attention. While the original Schrödinger equation describes non-relativistic massive particles, photons are massless and travel at the speed of light, thus their dynamics are not directly governed by this equation. But the conceptual framework it introduces carries directly over, with a lot of practical consequences. Photons are particularly attractive carriers of quantum information because they interact only weakly with their environment, preserving delicate quantum states over long distances. Experiments in Vienna have demonstrated multi-photon entanglement and advanced quantum information protocols that form building blocks for future quantum computers and secure communication networks.

Complementing these experimental efforts, theoretical physicists at Vienna are tackling fundamental questions about quantum mechanics from several angles. One line of research investigates how quantum coherence can be created and controlled in synthetic many-body systems, studying the interplay of topology, disorder and dissipation. Another uses tensor networks to classify the different types of entanglement that many-body systems can exhibit, probing the fundamental computational limits of our ability to understand them. A third pushes even deeper into the foundations: asking what the quantum framework implies for our concepts of causality, reference frames and the nature of reality itself, territory that becomes especially relevant as quantum mechanics confronts gravity.

Despite its tremendous success, quantum theory still leaves us with profound mysteries. The greatest may be how quantum mechanics relates to gravity.

General relativity describes gravity as the curvature of spacetime. Quantum theory describes matter through wavefunctions evolving according to the Schrödinger equation. No one has yet succeeded in reconciling these two frameworks, and even the right starting point remains contested. Experiments in quantum optomechanics, quantum optics and large-mass interferometry aim to test whether gravity influences quantum superpositions, or whether quantum states themselves can generate gravitational effects. If successful, such experiments could offer the first experimental clues toward a theory unifying quantum mechanics and gravity.

The Schrödinger equation is often presented in textbooks as a solved problem, a piece of mathematics whose implications we already understand. The reality is very different.

Every day, in laboratories around the world, physicists test its predictions in new regimes: with larger systems, more complex quantum states and unprecedented precision. Each experiment pushes the boundaries of what we know about the quantum world.

It is remarkable that a single equation written in 1926 continues to guide discoveries at the frontiers of physics, from molecules interfering with themselves to photons performing computations, from nanomechanical oscillators to the mathematical classification of quantum phases and to possible tests of quantum gravity. And perhaps, in the coming decades, experiments will reveal something even more profound: where this equation ultimately fails, and what deeper theory of nature lies beyond it. If that happens, the Schrödinger equation will have served its highest purpose, not as the final word, but as the compass that led us to the next horizon.

Find out more here about this exciting research!

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A contribution by Prof. Andrii Chumak

Faculty of Physics, University of Vienna

When Erwin Schrödinger published his wave equation in 1926, he had the electron in a Coulomb potential in mind. A century later, the formal structure of the Schrödinger eigenvalue problem underlies the quantum description of a very different kind of excitation — collective oscillations of magnetic order in a crystal. In the field of magnonics, the Schrödinger equation is one of the key theoretical pillars connecting static magnetic moments to the rich dynamics of spin waves.

A ferromagnet in its ground state has all spins aligned. Disturb a few, and the perturbation propagates through the lattice like a ripple on water. These propagating disturbances are spin waves; their quanta are magnons — bosonic quasiparticles that carry spin angular momentum without moving charge. In practice, much of magnonics is formulated through the Landau–Lifshitz (–Gilbert) equation or via Heisenberg equations of motion, yet at the microscopic level these descriptions ultimately rest on a quantum-mechanical foundation.

The Equation at Work

The primary application of the Schrödinger equation in magnonics is solving the eigenvalue problem Ĥ|ψ⟩ = E|ψ⟩ for the Heisenberg exchange Hamiltonian Ĥ = –J Σ Ŝᵢ·Ŝⱼ. Its solutions yield the magnon dispersion relation ω(k) — the energy–momentum relationship that tells us how fast spin waves travel and how they interact. Since a spin wave is a collective excitation, a direct solution of the Schrödinger equation for a system of ∼10²³ coupled spins is intractable. Instead, one applies the Holstein–Primakoff transformation, which recasts spin operators into bosonic creation and annihilation operators. After this step, the Hamiltonian becomes quadratic in bosonic operators and its diagonalisation yields independent harmonic modes; and spin waves emerge naturally as a gas of quasiparticles — magnons.

Beyond the energy spectrum, the time-dependent Schrödinger equation iℏ ∂|ψ⟩/∂t = Ĥ|ψ⟩ describes how a spin configuration evolves under external magnetic fields or microwave pulses — a cornerstone for spintronics and magnonic signal processing. It was also the quantum approach, treating excitations as discrete magnons rather than continuous waves, that explained why the magnetisation of a ferromagnet decreases at low temperatures following Bloch’s T³⁄₂ law — a result that classical physics could not reproduce.

This formal equivalence with textbook quantum mechanics has been extraordinarily productive. Concepts such as bound states, tunnelling, interference, and band gaps translate directly into magnetic phenomena engineered in the laboratory. Researchers have realised magnonic quantum wells, waveguides, and topological edge states simply by shaping the effective potential that spin waves experience.

From Quasiparticles to Quantum Information

The emerging field of quantum magnonics takes this programme to its ultimate conclusion: operating with single magnons in the quantum limit. Magnons cover gigahertz to terahertz frequencies, scale to nanometre footprints, interface naturally with microwave circuitry, and exhibit strongly nonlinear and nonreciprocal dynamics. These properties make them promising candidates for quantum information processing, coherent transport, and entangled-state preparation. Key experimental milestones already include coherent coupling of magnon modes to superconducting resonators, single-magnon detection and full Wigner-function tomography, as well as the generation of squeezed and entangled magnon states via parametric driving and magnonic nonlinearities.

The most critical requirement for all of these applications is the ability to support long magnon lifetimes. Recent experiments have reported that dipolar-exchange magnons in ultra-pure yttrium iron garnet can exhibit lifetimes reaching tens of microseconds at millikelvin temperatures — approaching the energy-relaxation times of superconducting transmon qubits — which considerably strengthens the case for the magnonic direction. With such coherence times, magnons can realistically serve as quantum interconnects, robust quantum memories, and carriers of nonclassical states, including Schrödinger cat states: superpositions of macroscopically distinct magnetic excitations that bear the name of the very physicist whose equation makes the whole enterprise possible.

A hundred years on, Schrödinger’s equation continues to animate new materials, new regimes, and new quantum technologies — one magnon at a time.

Further reading:

https://arxiv.org/abs/2601.12833

https://arxiv.org/abs/2505.22773

Find out more here about this exciting research!

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A contribution by Prof. Massimiliano Procura

Faculty of Physics, University of Vienna

The Schrödinger equation introduced a radically new way of thinking about physics: the state of a system is encoded in a complex amplitude, and its evolution is governed by a linear partial differential equation. Simple in form yet profound in consequence, this idea has shaped every subsequent development of quantum theory. Although originally formulated for non-relativistic systems, its conceptual framework underlies our understanding of quantum phenomena across all scales.

In quantum field theory - the language of the Standard Model of fundamental interactions - we describe fields rather than wave-functions, yet the essential structure persists, extending Schrödinger’s original framework to generalized multi-particle Hilbert spaces. Physical predictions are built from amplitudes whose phases interfere and whose magnitudes determine probabilities. This develops into the S-matrix, which describes particle collisions - including creation and annihilation processes - and underpins collider phenomenology.

Its legacy is also evident in the central role of evolution equations, which provide a controlled description of quantum dynamics. Moreover, in regimes where quantum field theory simplifies - such as near thresholds or for slowly moving particles - Schrödinger-type equations re-emerge explicitly, describing for example Hydrogen-like bound states with systematically improvable precision.

At the Large Hadron Collider, the need for accuracy brings all these elements together: interference, multi-scale dynamics, and the interplay of perturbative and non-perturbative physics all rely on the same principles of unitarity of time evolution and superposition.

A century after its formulation, the Schrödinger equation stands as a cornerstone of the conceptual and technical framework of quantum theory, informing theoretical constructions, computational methods, and the interpretation of experimental phenomena.
 

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A contribution by Prof. Caslav Brukner

Faculty of Physics, University of Vienna & IQOQI Vienna, ÖAW

Abstract

Schrödinger’s equation gave early quantum theory a visual language that looked like physics again: a wave evolving by a linear differential equation. This essay argues that the same success also seeded a recurring impulse to keep quantum theory “classical-looking” by treating the wave function as a physical wave. Schrödinger quickly realized that, for many-particle systems, the wave function is naturally defined on configuration space rather than ordinary physical space, blocking any straightforward reading of it as a literal classical wave. Read through Mach and Boltzmann, who shaped his intellectual outlook most deeply, his achievement appears double-edged: it provided an extraordinarily powerful picture for calculation and discovery, while also warning against taking that picture too literally. I argue that this tension never fully disappeared. It still reappears in modern physics whenever the wave function, or in quantum field theory the field itself, is treated as ontology rather than as part of a representation tied to measurement and observational context, a point sharpened by Bell-type no-go theorems. The centenary moral is: use pictures boldly, but demote them ontologically.

Find out more here about the exciting research in the group of Prof. Caslav Brukner!

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A contribution by Prof. Roberto Cerbino and Prof. Christoph Dellago

Daculty of Physics, University of Vienna

Abstract

In 1926, Erwin Schrödinger introduced the equation that now bears his name, iℏ ∂ψ / ∂t = Hψ, a linear law describing how a quantum state ψ evolves in time under the action of the Hamiltonian operator H. Schrödinger’s equation was quite the revolution: instead of tracking trajectories in space, dynamics became encoded in the evolution of a state in an abstract Hilbert space, opening a new window onto what physicists mean by the “state” of a system. A century later, this conceptual shift still permeates physics, including in unexpected places. Our own field of research, soft matter physics, deals with complex fluids, colloids, polymers, and biological materials. At first sight it appears far removed from quantum mechanics: the systems are large, warm, and dominated by thermal noise. The de Broglie wavelengths of the relevant mesoscopic objects of soft matter are tiny compared with their characteristic sizes, and quantum coherence plays essentially no direct role at the scales of interest. In fact, we describe them using an essentially classical toolkit: hydrodynamics for slow flows, elasticity for deformations, thermodynamics for phase behavior, statistical mechanics for fluctuations, coarsegraining and effective field theories for collective behavior, stochastic processes and kinetic theory for transport, liquid-state theory for correlations, and increasingly nonequilibrium and nonlinear dynamics for active and driven systems. 

Soft matter physics is therefore classical in both scale and formalism. Yet this classical character is only part of the story. The concepts and equations used in modern soft matter theory still carry the imprint of Schrödinger’s one-line reformulation of dynamics. The echoes of that equation can be recognized at three distinct levels: 

• at the microscopic level, where the stability and interactions of matter ultimately have quantum-mechanical origins;
• at the level of mathematical structure, where closely related differential operators govern phenomena as different as quantum waves and diffusive spreading;
• and at the conceptual level, in the transition from trajectory-based determinism to statebased and probabilistic descriptions in many-body physics.

Find out more here about the exciting research in the group of Prof. Roberto Cerbino and the group of Prof. Christoph Dellago.

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