The time-dependent variational Monte Carlo (t-VMC) method is a

quantum Monte Carlo Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the ...

approach to study the dynamics of closed, non-relativistic

quantum system Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

s in the context of the quantum

many-body problem The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...

. It is an extension of the

variational Monte Carlo 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. The basic building block is a generic wave function , \Psi(a) \rangle ...

method, in which a time-dependent pure quantum state is encoded by some variational

wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...

, generally parametrized as :

\Psi(X,t) = \exp \left ( \sum_k a_k(t) O_k(X) \right )

where the complex-valued

a_k(t)

are time-dependent variational parameters,

X

denotes a many-body configuration and

O_k(X)

are time-independent operators that define the specific

ansatz In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...

. The time evolution of the parameters

a_k(t)

can be found upon imposing a

variational principle In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those func ...

to the

. In particular one can show that the optimal parameters for the evolution satisfy at each time the equation of motion :

i \sum_\langle O_k O_\rangle_t^c \dot_=\langle O_k \mathcal\rangle_t^c,

where

\mathcal

is the

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

of the system,

\langle AB \rangle_t^c=\langle AB\rangle_t-\langle A\rangle_t\langle B\rangle_t

are connected averages, and the quantum expectation values are taken over the time-dependent variational

, i.e.,

\langle\cdots\rangle_t \equiv\langle\Psi(t), \cdots, \Psi(t)\rangle

. In analogy with the

Variational Monte Carlo 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. The basic building block is a generic wave function , \Psi(a) \rangle ...

approach and following the

Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...

for evaluating integrals, we can interpret

\frac

as a

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

function over the multi-dimensional space spanned by the many-body configurations

X

. The

Metropolis–Hastings algorithm In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. This seque ...

is then used to sample exactly from this probability distribution and, at each time

t

, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories

a(t)

of the variational parameters are then found upon numerical integration of the associated

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

References

* * * {{cite thesis , author = G. Carleo , title = Spectral and dynamical properties of strongly correlated systems , url = http://www.sissa.it/cm/phdsection/past_phd_thesis/2011/Carleo.pdf , type= PhD Thesis , pages = 107–128 , year = 2011 Quantum mechanics Quantum Monte Carlo