The **time-dependent variational Monte Carlo (t-VMC)** method is a quantum Monte Carlo approach to study the dynamics of closed, non-relativistic quantum systems in the context of the quantum many-body problem. It is an extension of the variational Monte Carlo method, in which a time-dependent pure quantum state is encoded by some variational wave function, generally parametrized as

where the complex-valued are time-dependent variational parameters, denotes a many-body configuration and are time-independent operators that define the specific ansatz. The time evolution of the parameters can be found upon imposing a variational principle to the wave function. In particular one can show that the optimal parameters for the evolution satisfy at each time the equation of motion

where is the Hamiltonian of the system, are connected averages, and the quantum expectation values are taken over the time-dependent variational wave function, i.e., .

In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret as a probability distribution function over the multi-dimensional space spanned by the many-body configurations . The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time , the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories of the variational parameters are then found upon numerical integration of the associated differential equation.

The **mathematical formulations of quantum mechanics** are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces, and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.

In quantum mechanics, the **uncertainty principle** is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, *x*, and momentum, *p*, can be predicted from initial conditions.

The **Schrödinger equation** is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In quantum mechanics, a **density matrix** is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent *mixed states*. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state.

In quantum mechanics, **perturbation theory** is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to the system. If the disturbance is not too large, the various physical quantities associated with the perturbed system can be expressed as "corrections" to those of the simple system. These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one. In effect, it is describing a complicated unsolved system using a simple, solvable system.

In physics, the **Schrödinger picture** is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are mostly constant with respect to time. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures.

The **path integral formulation** is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

In physics, the ** S-matrix** or

In quantum mechanics, the **canonical commutation relation** is the fundamental relation between canonical conjugate quantities. For example,

The **Schwinger–Dyson equations** (**SDEs**) or **Dyson–Schwinger equations**, named after Julian Schwinger and Freeman Dyson, are general relations between correlation functions in quantum field theories (QFTs). They are also referred to as the Euler–Lagrange equations of quantum field theories, since they are the equations of motion corresponding to the Green's function. They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite tower of SDEs.

In quantum mechanics, the **Hellmann–Feynman theorem** relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

**Photon polarization** is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The **theoretical and experimental justification for the Schrödinger equation** motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In computational physics, **variational Monte Carlo (VMC)** is a quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system.

In quantum mechanics, the **expectation value** is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the *most* probable value of a measurement; indeed the expectation value may have zero probability of occurring. It is a fundamental concept in all areas of quantum physics.

In non-equilibrium physics, the **Keldysh formalism** is a general framework for describing the quantum mechanical evolution of a system in a non-equilibrium state or systems subject to time varying external fields. Historically, it was foreshadowed by the work of Julian Schwinger and proposed almost simultaneously by Leonid Keldysh and, separately, Leo Kadanoff and Gordon Baym. It was further developed by later contributors such as O. V. Konstantinov and V. I. Perel.

The **kicked rotator**, also spelled as **kicked rotor**, is a paradigmatic model for both Hamiltonian chaos and quantum chaos. It describes a free rotating stick in an inhomogeneous "gravitation like" field that is periodically switched on in short pulses. The model is described by the Hamiltonian

The theory of **causal fermion systems** is an approach to describe fundamental physics. It provides a unification of the weak, the strong and the electromagnetic forces with gravity at the level of classical field theory. Moreover, it gives quantum mechanics as a limiting case and has revealed close connections to quantum field theory. Therefore, it is a candidate for a unified physical theory. Instead of introducing physical objects on a preexisting spacetime manifold, the general concept is to derive spacetime as well as all the objects therein as secondary objects from the structures of an underlying causal fermion system. This concept also makes it possible to generalize notions of differential geometry to the non-smooth setting. In particular, one can describe situations when spacetime no longer has a manifold structure on the microscopic scale. As a result, the theory of causal fermion systems is a proposal for quantum geometry and an approach to quantum gravity.

In quantum mechanics, **weak measurements** are a type of quantum measurement that results in an observer obtaining very little information about the system on average, but also disturbs the state very little. From Busch's theorem the system is necessarily disturbed by the measurement. In the literature weak measurements are also known as unsharp, fuzzy, dull, noisy, approximate, and gentle measurements. Additionally weak measurements are often confused with the distinct but related concept of the weak value.

**Neural Network Quantum States** is a general class of variational quantum states parameterized in terms of an artificial neural network. It was first introduced in 2017 by the physicists Giuseppe Carleo and Matthias Troyer to approximate wave functions of many-body quantum systems.

- G. Carleo; F. Becca; M. Schiró & M. Fabrizio (2012). "Localization and glassy dynamics of many-body quantum systems".
*Sci. Rep*.**2**: 243. arXiv: 1109.2516 . Bibcode:2012NatSR...2E.243C. doi:10.1038/srep00243. PMC 3272662 . PMID 22355756.

- G. Carleo; F. Becca; L. Sanchez-Palencia; S. Sorella & M. Fabrizio (2014). "Light-cone effect and supersonic correlations in one- and two-dimensional bosonic superfluids".
*Phys. Rev. A*.**89**(3): 031602(R). arXiv: 1310.2246 . Bibcode:2014PhRvA..89c1602C. doi:10.1103/PhysRevA.89.031602. S2CID 45660254.

- G. Carleo (2011).
*Spectral and dynamical properties of strongly correlated systems*(PDF) (PhD Thesis). pp. 107–128.

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