Solving the Schrödinger equation with imagination
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Lire en français [2]Solving the Schrödinger equation with imagination
It was in 1925 that Erwin Schrödinger postulated the equation that bears his name, the governing equation in quantum mechanics, based on which it is possible – in principle – to calculate everything that can be known about a physical system: the properties of atoms, reactivity of molecules, response of a material, etc. A century on, the Schrödinger equation remains essential in both physics and chemistry.
However, in practice it cannot be solved exactly. Right from the start, theorists have kept exploring mathematical [7] shortcuts that could draw them closer to solutions.
In this century-long quest for ever more effective approximations, three French researchers have just made a potentially major breakthrough. Their radical approach involves replacing, on paper, the real system to be studied with an equivalent that has no physical reality, but that is, on the face of it, much more simple. Their initial results suggest potential advances in modelling a number of complex quantum systems that have remained beyond reach so far.
The origins of quantum mechanics
The Austrian physicist was originally trying to mathematically describe the matter waves imagined by Louis de Broglie in 1924. In a series of six articles released two years later, the future Nobel Prize winner published his equation, along with a number of his solutions for it.
In particular, he exactly solved the case of the hydrogen atom, the simplest atom, consisting of a proton and an electron. This was an undeniable success, for not only did Schrödinger use a solid basis to determine the energy levels for the electron introduced by Niels Bohr 10 years earlier, but for the first time he gave a mathematical description of its atomic orbitals, also known as “wave functions”.
A few months later, Max Born showed how these wave functions could be used to determine the probability of the electron’s presence around the nucleus in various possible quantum states. Quantum mechanics was beginning to take shape.
While the laws of physics had already been established, in 1930 Paul Dirac pointed out that the chief difficulty resided in calculating them. It quickly became evident that the physical systems for which it is possible to solve the Schrödinger equation are an exception. In other words, to describe more than an electron, approximations are needed to enable calculations, all while capturing the essential physical phenomena being studied.
Powerful approximations
Density functional theory (DFT) has made it possible to develop the most powerful approximations. Llewellyn Thomas and Enrico Fermi laid the foundations for this theory. In 1927, they proposed modelling an assembly of electrons, or more generally quantum particles, such as a quantum fluid – with major success in describing multi-electron atoms, white dwarves in astrophysics [12], and electrons in solids.
More specifically, such an approach involves foresaking an exhaustive determination of the relevant system’s wave function (which would amount to knowing the position of all of the particles that make up the system), instead focusing exclusively on their density, in other words how these particles are distributed, which depends solely on the three coordinates of space.
Two crucial theorems, demonstrated in 1964 by Pierre Hohenberg and Walter Kohn (winner of the 1998 Nobel Prize in chemistry), justify the relevance and validity of this approximation. Firstly, the density of a particle system contains all of the information needed to calculate the energy of its ground state, which is to say its stationary state of lowest energy. Secondly, there is a mathematical object known as a “functional”, from which the density of the ground state and its energy can be determined.
“This result is fairly counterintuitive,” comments Toulouse. “It involves a thorough reformulation of Schrödinger’s equation for a system’s ground state.”
DFT consecrated
The form of this functional remains unknown. In other words, there is no operational method for concretely extracting a system’s physical properties.
However, in 1965 Walter Kohn and Lu Jeu Sham proposed a decisive reformulation: to consider a system of interacting electrons (within an atom, molecule, or material) as a fictitious system of non-interacting particles complemented by what is known as an exchange-correlation term. Calculations can then be performed, all while preserving the complexity of the initial problem and ensuring that the virtual system’s density remains faithful to that of the real system. The art of physics and chemistry thus consists in proposing relevant approximations for this exchange-correlation term.
In the ensuing decades, with enhanced computing power and continually improving approximations, the method became essential for quantum chemistry and materials physics. “We went from an abstract theory to a tool that can make increasingly accurate predictions,” underscores Lewin. This is reflected in the fact that in 2025 alone, approximately 89,000 publications were based on application of DFT.
Systems resistant to calculation
Numerous physical and chemical systems continue to resist attempts by theorists to put them into equations. Two years ago, in connection with the MaQui Project [14] – which models quantum systems with new mathematical approaches1 – the CNRS researchers Éric Cancès, Mathieu Lewin, and Julien Toulouse identified three of these intractable systems: atoms exposed to attosecond pulses, whose highly-excited electrons dynamically explore a great number of quantum states; heavy atoms, of interest for nuclear research, for which relativistic effects are important; and some non-periodic materials made of graphene.
“These three situations are a priori very different,” points out Cancès. “There are nevertheless subtle links between them, especially with respect to the very strong correlations between particles, and the quite difficult theoretical and numerical issues they raise, whose resolution would open up substantial prospects.”
The three researchers and their respective students tackled the attosecond physics [16] problem head-on. “It was the most open problem for the three of us, the one for which, at first glance, we had no avenues,” Cancès says.
Lewin adds: “In concrete terms, this involved making progress on the time-dependent version of DFT, which was developed beginning in the 1980s to move beyond the description of a system’s ground state, in an effort to address unbalanced situations involving excited states, but whose theoretical foundations remain incomplete and whose performance deteriorates as one moves away from balance.”
One of the difficulties – with no connection to balance – arises from the fact that the exchange-correlation term adopts forms that are particularly difficult to handle. “In an unbalanced system, particles move a lot,” Lewin explains. “To model them, we must, on paper, immerse the fictitious system of electrons in an extremely complex environment – physicists speak of potential – and do so within a formulation that often does not faithfully reproduce the dynamics of the real system.”
An imaginary potential to simplify reality
The researchers chose to pursue a radical proposition: since they were to manipulate the equations describing a virtual system, they decided to take this logic a step further. To reproduce the real dynamic, instead of making virtual electrons move in a convoluted potential, the scientists introduced a new term – “imaginary potential” – that enables particles to appear and disappear.
“This way we can change electronic density without having to move particles too much,” comments Toulouse. “It has little to do with physics, but it is very effective.”
To test this method, the team applied it to two modelled systems (referred to as “toy models”) whose solution is precisely known: a laser-excited helium atom model, and a dual-atom model where a laser creates an electron transfer from one atom to the other, as observed with some molecules.
“In both cases, we showed that the imaginary potential has a much simpler expression than the real potential of usual time-dependent DFT,” Lewin enthuses. “That is why we formulated the hypothesis that it would be simpler to propose interesting approximations for more realistic models,” adds Toulouse. This is precisely what the three researchers will now focus on.
Their work featured in two articles recently published in the journals Physical Review Letters [18] and Physical Review A [19]. Will this help make advances in solving new problems in physics and chemistry? It is too early to tell, but one thing is certain: by proposing a description of a real system’s quantum dynamics using a model whose physical realism is deliberately set aside, the researchers maintain the tradition of describing the material world using highly counterintuitive mathematical objects. Just like Erwin Schrödinger did a hundred years ago.
See also
Rebel with a cause [20]
Enter the matrices! [21]
1905: Einstein initiates the quantum age [22]
- 1. This work received government funding in connection with the research with risk accelerator programme, managed by the French National Research Agency (ANR) as part of France 2030 under the reference number ANR-24-RRII-0001.
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