dependent motion
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
Ψ
(
X
,
t
)
=
exp
(
∑
k
a
k
(
t
)
O
k
(
X
)
)
{\displaystyle \Psi (X,t)=\exp \left(\sum _{k}a_{k}(t)O_{k}(X)\right)}
where the complex-valued
a
k
(
t
)
{\displaystyle a_{k}(t)}
are time-dependent variational parameters,
X
{\displaystyle X}
denotes a many-body configuration and
O
k
(
X
)
{\displaystyle O_{k}(X)}
are time-independent operators that define the specific ansatz. The time evolution of the parameters
a
k
(
t
)
{\displaystyle a_{k}(t)}
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
i
∑
k
′
⟨
O
k
O
k
′
⟩
t
c
a
˙
k
′
=
⟨
O
k
H
⟩
t
c
,
{\displaystyle i\sum _{k^{\prime }}\langle O_{k}O_{k^{\prime }}\rangle _{t}^{c}{\dot {a}}_{k^{\prime }}=\langle O_{k}{\mathcal {H}}\rangle _{t}^{c},}
where
H
{\displaystyle {\mathcal {H}}}
is the Hamiltonian of the system,
⟨
A
B
⟩
t
c
=
⟨
A
B
⟩
t
−
⟨
A
⟩
t
⟨
B
⟩
t
{\displaystyle \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 wave function, i.e.,
⟨
⋯
⟩
t
≡
⟨
Ψ
(
t
)
|
⋯
|
Ψ
(
t
)
⟩
{\displaystyle \langle \cdots \rangle _{t}\equiv \langle \Psi (t)|\cdots |\Psi (t)\rangle }
.
In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret
|
Ψ
(
X
,
t
)
|
2
∫
|
Ψ
(
X
,
t
)
|
2
d
X
{\displaystyle {\frac {|\Psi (X,t)|^{2}}{\int |\Psi (X,t)|^{2}\,dX}}}
as a probability distribution function over the multi-dimensional space spanned by the many-body configurations
X
{\displaystyle X}
. The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time
t
{\displaystyle t}
, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories
a
(
t
)
{\displaystyle a(t)}
of the variational parameters are then found upon numerical integration of the associated differential equation.
You do not have permission to view the full content of this post. Log in or register now.
Ψ
(
X
,
t
)
=
exp
(
∑
k
a
k
(
t
)
O
k
(
X
)
)
{\displaystyle \Psi (X,t)=\exp \left(\sum _{k}a_{k}(t)O_{k}(X)\right)}
where the complex-valued
a
k
(
t
)
{\displaystyle a_{k}(t)}
are time-dependent variational parameters,
X
{\displaystyle X}
denotes a many-body configuration and
O
k
(
X
)
{\displaystyle O_{k}(X)}
are time-independent operators that define the specific ansatz. The time evolution of the parameters
a
k
(
t
)
{\displaystyle a_{k}(t)}
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
i
∑
k
′
⟨
O
k
O
k
′
⟩
t
c
a
˙
k
′
=
⟨
O
k
H
⟩
t
c
,
{\displaystyle i\sum _{k^{\prime }}\langle O_{k}O_{k^{\prime }}\rangle _{t}^{c}{\dot {a}}_{k^{\prime }}=\langle O_{k}{\mathcal {H}}\rangle _{t}^{c},}
where
H
{\displaystyle {\mathcal {H}}}
is the Hamiltonian of the system,
⟨
A
B
⟩
t
c
=
⟨
A
B
⟩
t
−
⟨
A
⟩
t
⟨
B
⟩
t
{\displaystyle \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 wave function, i.e.,
⟨
⋯
⟩
t
≡
⟨
Ψ
(
t
)
|
⋯
|
Ψ
(
t
)
⟩
{\displaystyle \langle \cdots \rangle _{t}\equiv \langle \Psi (t)|\cdots |\Psi (t)\rangle }
.
In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret
|
Ψ
(
X
,
t
)
|
2
∫
|
Ψ
(
X
,
t
)
|
2
d
X
{\displaystyle {\frac {|\Psi (X,t)|^{2}}{\int |\Psi (X,t)|^{2}\,dX}}}
as a probability distribution function over the multi-dimensional space spanned by the many-body configurations
X
{\displaystyle X}
. The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time
t
{\displaystyle t}
, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories
a
(
t
)
{\displaystyle a(t)}
of the variational parameters are then found upon numerical integration of the associated differential equation.
You do not have permission to view the full content of this post. Log in or register now.