Numerical Quantum Mechanics

[Ogier Dudley]

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We introduce a method to solve nonlinear open quantum dynamics of a particle in situations where its state undergoes significant expansion in phase space while generating small quantum features at the phase-space Planck scale. Our approach involves simulating two steps. First, we transform the Wigner function into a time-dependent frame that leverages information from the classical trajectory to efficiently represent the quantum state in phase space. Next, we simulate the dynamics in this frame using a numerical method that implements this time-dependent nonlinear change of variables. To demonstrate the capabilities of our method, we examine the open quantum dynamics of a particle evolving in a one-dimensional weak quartic potential after initially being ground-state cooled in a tight harmonic potential. This approach is particularly relevant to ongoing efforts to design, optimize, and understand experiments targeting the preparation of macroscopic quantum superposition states of massive particles through nonlinear quantum dynamics.

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While quantum mechanics has been originally motivated by fundamental questions in theoretical physics, it now plays central role in applications: from control theory to quantum computing, physical chemistry, material theory and increasingly biology and drug design. Yet, equations of quantum mechanics present a formidable computational challenge, which will form a core of the workshop. The last few years have witnessed the emergence of a raft of modern computational methodologies that allow for very precise computational approximation of the equations of quantum mechanics while respecting features of physical significance. The new computational technologies bring together ideas from numerical mathematics, but also from differential geometry, harmonic analysis and other branches of pure mathematics, as well as ideas that have emerged in the applied community.

This simple example shows that the internal game universe can become indeterministic, even though the external one might be utterly deterministic. However, this example does not fully capture the weirdness of quantum mechanics, because in each one of the two alternate states Lara could find herself in (surviving the puzzle or being killed by it), she does not experience any effects from the other state at all, and could reasonably assume that she lives in a classical, deterministic universe.

p.s. It appears a real-time strategy game Achron incorporates time travel by using the fact that such real-time strategy games are mostly just simulations using preprogramed unit behaviors. So players may edit the orders they gave at an earlier point in time, which prompts the game to rerun the simulation from that point on. Of course, the opposing player may also adjust this past commands too.

The analogy between a classical duplication event for a conscious observer and the quantum mechanical many-worlds interpretation is a good one, but it has philosophical fine points regarding the probability measure, which you glossed over.

We are currently working on a cooperative game concept on quantum mechanics in a way that will assist us to build something that not only the external and internal universe has to cooperate but the 2 players that will co-ordinate and collaborate to perform various actions in to the internal environment will feel 100% brain stimulated. Not only they will need to behave as two players in 1 role but they will also have to perform hard advance estimations on possible scenarios the game will evolve.

"This is an interesting a valuable book for someone who wants to obtain a solid understanding of the fundamental aspects of quantum mechanics, an appreciation of some aspects of the numerical solution of practical applications, and an introduction or better to a very broad spectrum of topics."-Chemistry in Asutralia Magazine of the Royal Astralian Chemical Institute

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.AbstractNon-Archimedean mathematics is an approach based on fields which contain infinitesimal and infinite elements. Within this approach, we construct a space of a particular class of generalized functions, ultrafunctions. The space of ultrafunctions can be used as a richer framework for a description of a physical system in quantum mechanics. In this paper, we provide a discussion of the space of ultrafunctions and its advantages in the applications of quantum mechanics, particularly for the Schrdinger equation for a Hamiltonian with the delta function potential.

This workshop focuses on numerical and theoretical aspects arising from the modelling and simulation of quantum mechanical problems, coming from several applications (in electronics for ex.) or more fundamental problems (Bose-Einstein condensates). The aim of this workshop is to bring together researchers working in related (theoretical as well as numerical) fields, and to consolidate as well as to foster new interactions and collaborations within this community.

It will take place within the Institut de Mathmatiques de Toulouse, in the Amphitheater Laurent Schwartz. A detailed map of campus is available here : Access to campus. The amphitheater is located in building no. 23 on the campus map (rue Sbastienne Guyot).

In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. Quantum numbers are closely related to eigenvalues of observables. When the corresponding observable commutes with the Hamiltonian, the quantum number is said to be "good", and acts as a constant of motion in the quantum dynamics.

To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantum numbers includes the principal, azimuthal, magnetic, and spin quantum numbers. To describe other systems, different quantum numbers are required. For subatomic particles, one needs to introduce new quantum numbers, such as the flavour of quarks, which have no classical correspondence.

A consequence of space quantization was that the electron's orbital interaction with an external magnetic field would be quantized. This seemed to be confirmed when the results of the Stern-Gerlach experiment reported quantized results for silver atoms in an inhomogeneous magnetic field. The confirmation would turn out to be premature: more quantum numbers would be needed.[7]

Bohr, with his Aufbau or "building up" principle, and Pauli with his exclusion principle connected the atom's electronic quantum numbers in to a framework for predicting the properties of atoms.[9] When Schrdinger published his wave equation and calculated the energy levels of hydrogen, these two principles carried over to become the basis of atomic physics.

Good quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian, quantities that can be known with precision at the same time as the system's energy. Specifically, observables that commute with the Hamiltonian are simultaneously diagonalizable with it and so the eigenvalues a \displaystyle a and the energy (eigenvalues of the Hamiltonian) are not limited by an uncertainty relation arising from non-commutativity. Together, a specification of all of the quantum numbers of a quantum system fully characterize a basis state of the system, and can in principle be measured together. Many observables have discrete spectra (sets of eigenvalues) in quantum mechanics, so the quantities can only be measured in discrete values. In particular, this leads to quantum numbers that take values in discrete sets of integers or half-integers; although they could approach infinity in some cases.

The tally of quantum numbers varies from system to system and has no universal answer. Hence these parameters must be found for each system to be analyzed. A quantized system requires at least one quantum number. The dynamics (i.e. time evolution) of any quantum system are described by a quantum operator in the form of a Hamiltonian, H. There is one quantum number of the system corresponding to the system's energy; i.e., one of the eigenvalues of the Hamiltonian. There is also one quantum number for each linearly independent operator O that commutes with the Hamiltonian. A complete set of commuting observables (CSCO) that commute with the Hamiltonian characterizes the system with all its quantum numbers. There is a one-to-one relationship between the quantum numbers and the operators of the CSCO, with each quantum number taking one of the eigenvalues of its corresponding operator. As a result of the different basis that may be arbitrarily chosen to form a complete set of commuting operators, different sets of quantum numbers may be used for the description of the same system in different situations.

These quantum numbers are also used in the classical description of nuclear particle states (e.g. protons and neutrons).[citation needed] A quantum description of molecular orbitals requires other quantum numbers, because the symmetries of the molecular system are different.

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