Wave function | EPFL Graph Search

In quantum physics, a wave function (or wavefunction), represented by the Greek letter Ψ, is a mathematical description of the quantum state of an isolated quantum system. In the Copenhagen interpretation of quantum mechanics, the wave function is a complex-valued probability amplitude; the probabilities for the possible results of the measurements made on a measured system can be derived from the wave function. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively). The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of

A Framework for Numerical Simulations of Structure Formation

Claude Becker

In the absence of a full analytical treatment of nonlinear structure formation in the universe, numerical simulations provide the critical link between the properties of the underlying model and the features of the observed structures. Currently N-body simulations are the main tool to study structure growth. We explore an alternative framework for numerical simulations of structure formation. The underlying idea is to replace the long-range gravitational force in the Vlasov-Poisson system by a purely local interaction. To this end we trade the classical phase space distribution for its quantum mechanical counterpart, the Wigner distribution function. Its dynamical equation is equivalent to the Schrödinger equation and reduces to the Vlasov equation in the formal semi-classical limit. The proposed framework allows in principle to simulate systems with arbitrary phase space distributions and could for instance be beneficial for simulations of warm dark matter, where the velocity dispersion is important. We discuss several methods to obtain a set of wavefunctions whose Wigner distribution is close to a given initial phase space distribution function. An auxiliary gauge field is introduced to mediate the gravitational interaction, thereby obtaining a local Schrödinger-Maxwell system. We also use the ideas of lattice gauge theories to obtain a fully gauge-invariant discretization of the equations of motion. Their iterative solution was implemented in a three-dimensional simulation code. We discuss its computational complexity and memory requirements. Several testbed simulations were performed with this method. We compared the gravitational collapse of a Gaussian wavefunction with an independent numerical solution of the spherically symmetric Schrödinger-Newton system. The results were found to be in good agreement. Finally, a simple example of the growth of cosmic perturbations is investigated within our framework. We conclude by outlining various possible directions to optimize and develop our method.

EPFL2011

From hidden order to antiferromagnetism: Electronic structure changes in Fe-doped URu2Si2

Ji Dai, Emmanouil Frantzeskakis, Kevin Huang

In matter, any spontaneous symmetry breaking induces a phase transition characterized by an order parameter, such as the magnetization vector in ferromagnets, or a macroscopic many electron wave function in superconductors. Phase transitions with unknown order parameter are rare but extremely appealing, as they may lead to novel physics. An emblematic and still unsolved example is the transition of the heavy fermion compound URu2Si2 (URS) into the so-called hidden-order (HO) phase when the temperature drops below T-0 = 17.5 K. Here, we show that the interaction between the heavy fermion and the conduction band states near the Fermi level has a key role in the emergence of the HO phase. Using angle-resolved photoemission spectroscopy, we find that while the Fermi surfaces of the HO and of a neighboring antiferromagnetic (AFM) phase of well-defined order parameter have the same topography, they differ in the size of some, but not all, of their electron pockets. Such a nonrigid change of the electronic structure indicates that a change in the interaction strength between states near the Fermi level is a crucial ingredient for the HO to AFM phase transition.

NATL ACAD SCIENCES2021

A new framework for numerical simulations of structure formation

Claude Becker, Alexey Boyarsky, Oleg Ruchayskiy, Matthieu Schaller

The diversity of structures in the Universe (from the smallest galaxies to the largest superclusters) has formed under the pull of gravity from the tiny primordial perturbations that we see imprinted in the cosmic microwave background. A quantitative description of this process would require description of motion of zillions of dark matter particles. This impossible task is usually circumvented by coarse graining the problem: one either considers a Newtonian dynamics of 'particles' with macroscopically large masses or approximates the dark matter distribution with a continuous density field. There is no closed system of equations for the evolution of the matter density field alone and instead it should still be discretized at each time step. In this work, we describe a method of solving the full six-dimensional Vlasov-Poisson equation via a system of auxiliary Schrodinger-like equations. The complexity of the problem gets shifted into the choice of the number and shape of the initial wavefunctions that should only be specified at the beginning of the computation (we stress that these wavefunctions have nothing to do with quantum nature of the actual dark matter particles). We discuss different prescriptions to generate the initial wavefunctions from the initial conditions and demonstrate the validity of the technique on two simple test cases. This new simulation algorithm can in principle be used on an arbitrary distribution function, enabling the simulation of warm and hot dark matter structure formation scenarios.

Oxford Univ Press2014

A Framework for Numerical Simulations of Structure Formation

Claude Becker

EPFL2011