The laws of physics, as we understand them, can be divided into two realms: classical physics, which describes objects at human size scales or larger, and quantum physics, which describes objects at the atomic scale. These two sets of laws are very different. Classical physics states that an object like a ball has a position and momentum that can both be determined to arbitrary precision. But for a microscopic particle governed by quantum physics, such as an electron, position and momentum cannot be simultaneously known with absolute certainty. In quantum theory, a particle is described using a mathematical construct called a wavefunction, which determines the probabilities for the outcomes of any measurement performed on it. Quantum wavefunctions tend to spread out when there are no forces acting on the particle, meaning that the particle’s position becomes increasingly uncertain with the passage of time.
How do we define something to be microscopic or macroscopic? One interesting factor to consider is the role of gravity. Gravitation is the weakest of the four fundamental forces, so it is often ignored when dealing with quantum systems. However, the Hungarian physicist Lajos Diósi suggested in 1984 that the gravitational force of an object on itself, or “self-gravitation”, might play a key role in linking the microscopic and macroscopic worlds. As the gravitational force is always attractive, self-gravitation may help explain why macroscopic objects can be “localized” (i.e., possessing definite positions) rather than being spread out in space.
When studying gravitation, physicists normally just consider how one body is affected by a second body’s gravity, or vice versa. However, it is possible to model self-gravitation by combining Schrödinger’s equation, which describes the behaviour of quantum particles, with Newton’s law of gravitation. (It is noteworthy that Newton’s law of gravitation treated classically, which makes the theory a combination of quantum mechanical and classical – appropriately enough, since it is attempting to bridge the microscopic and macroscopic worlds.) The result is an equation called the Schrödinger-Newton equation, describing the dynamical behavior of quantum particles subject to self-gravitation.
The Schrödinger-Newton equation is a nonlinear partial integro-differential equation that is much more difficult to solve than a conventional Schrödinger equation. In fact, no analytical solutions are known to this day! As an undergraduate researcher working in the group of Tomasz Paterek in Nanyang Technological University (NTU), Singapore, I implemented a scheme for solving the Schrödinger-Newton equation numerically, by discretizing time into small time steps and converting the equation into an approximation known as Cayley’s form. I then used the numerical solver to analyze the physics described by the Schrödinger-Newton equation.
As a simple case study, I took a particle initially described by a spherically symmetric “Gaussian” wavefunction, representing a particle whose probable positions are clustered around a specific point. For this case, the solution to the conventional Schrödinger equation (i.e., in the absence of self-gravitation) can be derived analytically, allowing the effects of self-gravitation to be easily identified.
The results are shown in the figure. Starting from the same initial conditions, the particle governed by the Schrödinger-Newton equation collapses toward the origin as time progresses, whereas the particle governed by the conventional Schrödinger equation becomes increasingly spread out. This confirms Diósi’s prediction that self-gravitation can cause quantum wavefunctions to become localized.
Figure 1 Time-evolution of Gaussian particle’s wavefunction under Schrödinger-Newton equation with free particle’s wavefunction evolution for comparison.
This is not the whole story, however. Gravitational effects may exert another type of influence on a quantum wavefunction, tied to the cosmological expansion of the universe. Physicists and astronomers have determined that the universe is undergoing an accelerating expansion – meaning that space itself is expanding as time goes by, similar to the surface of a balloon that is being inflated. This process of accelerating expansion is associated with a type of energy, called “dark energy”, which permeates the entire universe but whose precise origin and nature remains unknown. Of the known energy-matter content of the universe, dark energy occupies about 68% (another 27% consists of dark matter, and the remaining 5% is the ordinary matter that we are able to account for). Dark energy has an “anti-gravity” effect, meaning that it tends to move objects away from each other.
Just as the Schrödinger-Newton describes quantum particles subject to self-gravitation, it is possible to derive a similar equation incorporating all three of the effects we have discussed: (i) quantum mechanics, (ii) self-gravitation, and (iii) cosmological expansion tied to dark energy. We call this the Schrödinger-Newton-Lambda equation, “Lambda” being shorthand for the effects of dark energy (the so-called “cosmological constant”) in Einstein’s theory of gravitation.
At present, researchers in the Paterek group are investigating the numerical solutions to the Schrödinger-Newton-Lambda equation. This has turned out to be a difficult and intricate task, for a great deal of care is needed in separating the effects of self-gravitation from the effects of cosmological expansion. If successful, this study may shed light on some of the longest-standing and most profound issues in fundamental physics.
About the author
Kelvin Onggadinata is a third year student pursuing BSc Physics & Applied Physics programme from the School of Physical and Mathematical Sciences, NTU Singapore. His article won the Merit Prize in the SPMS Science Writing Competition 2019.
