---
title: Time-Dependent Hartree Equations
url: https://www.emergentmind.com/topics/time-dependent-hartree-equations
type: topic
---

# Time-Dependent Hartree Equations

Searching arXiv for recent and foundational papers on time-dependent Hartree equations to support the encyclopedia article.
Time-dependent Hartree equations are coupled nonlinear Schrödinger equations that govern the evolution of product-state approximations in quantum dynamics. In the nonrelativistic setting they arise as the time-dependent Dirac–Frenkel projection of the many-body Schrödinger flow onto the manifold of uncorrelated product states, and they provide an effective mean-field description in which the nonlinear potential is generated self-consistently by the instantaneous density [2207.13928]. In closely related contexts they also appear as limiting equations for large fermionic systems, as the single-configuration limit of multiconfigurational propagation schemes, and as the direct-field reduction of Hartree–Fock-type theories [2507.12390], [1802.01058], [2406.06845].

## 1. Variational origin and basic equations

The basic variational setting is the manifold of product states
$$
\mathcal M=\{\varphi_1\otimes\cdots\otimes\varphi_N\mid\varphi_j\in L^2(\Bbb R^d)\},
$$
together with the Dirac–Frenkel condition that the residual $i\partial_t u-Hu$ be orthogonal to the tangent space of $\mathcal M$ [2207.13928]. In this formulation, the Hartree equations are the projected dynamics obtained by restricting the full many-body Schrödinger equation to uncorrelated product states.

For the simplest two-factor case, the resulting system is
$$
\begin{cases}
i\partial_t\phi^x = H_x\,\phi^x \;+\;\displaystyle\int w(x,y)\,|\phi^y(t,y)|^2\,dy\;\phi^x,\\[1ex]
i\partial_t\phi^y = H_y\,\phi^y \;+\;\displaystyle\int w(x,y)\,|\phi^x(t,x)|^2\,dx\;\phi^y,
\end{cases}
$$
with
$$
H_x=-\tfrac12\Delta_x+V_1(x),\qquad H_y=-\tfrac12\Delta_y+V_2(y).
$$
This form makes explicit that each factor evolves under a one-body Hamiltonian plus a mean field generated by the density of the other factor [2207.13928].

In the $N$-orbital convolution form, the system reads
$$
i\partial_t\psi_j(t,x) = -\tfrac12\Delta_x\psi_j(t,x)
+\bigl(V*\rho(t,\cdot)\bigr)(x)\,\psi_j(t,x),\qquad j=1,\dots,N,
$$
with
$$
\rho(t,x)=\sum_{k=1}^N|\psi_k(t,x)|^2,\qquad
(V*\rho)(x)=\int_{\Bbb R^d}V(x-y)\,\rho(y)\,dy.
$$
Two cases singled out in the literature are the convolution/Coulomb case $V(x)=\pm|x|^{-1}$ in $d=3$, and a more general coupling $w(x,y)$ with polynomial growth and no convolution structure [2207.13928].

A recurring point in the literature is that the nonlinear Hartree coupling need not be perturbative. The general Cauchy theory in [2207.13928] is formulated precisely for a setting in which “the nonlinear coupling cannot be considered as a perturbation.” This distinguishes the time-dependent Hartree system from analyses that rely on small-coupling arguments.

## 2. Functional framework, conservation laws, and the Cauchy problem

The well-posedness theory developed for the Hartree approximation uses one-body Hamiltonians
$$
H_i=-\tfrac12\Delta_\xi+V_i(\xi)\ge0
$$
with $V_i\in C^\infty$ and graph norms
$$
\mathcal H_x^k=\{\,\phi\in L^2(\Bbb R^{d_1})\;:\;H_x^{k/2}\phi\in L^2\},\qquad
\|\phi\|_{\mathcal H_x^k}^2=\|\phi\|_{L^2}^2+\|H_x^{k/2}\phi\|_{L^2}^2,
$$
and similarly in the $y$ variable [2207.13928]. The working space is
$$
X_T^{\alpha,\beta}
= L^\infty\bigl([0,T];\mathcal H_x^\alpha\times\mathcal H_y^\beta\bigr).
$$

For general couplings, the analysis assumes a relative bound
$$
|w(x,y)|\;\le\;c_0\bigl(V_1(x)+V_2(y)+C\bigr)\qquad (0<c_0<1)
$$
and commutator bounds of the form
$$
\bigl|\langle [H_x^k,w(\cdot,y)]f_1,f_2\rangle_{L^2_x}\bigr|
\le C\,\bigl(1+V_2(y)\bigr)\|f_1\|_{\mathcal H_x^k}\|f_2\|_{\mathcal H_x^k},
$$
with analogous estimates in $y$ [2207.13928]. In the Coulomb/convolution setting, the theory covers $d_1=d_2=3$ with $w(x,y)=W(x-y)$ and $W\in L^p+L^\infty$ for some $p>3/2$, together with at-most-quadratic smooth parts of $V_1,V_2$ to ensure local Strichartz estimates [2207.13928].

The main results include a global existence and uniqueness theorem for the Coulomb/convolution case:
$$
(\phi^x,\phi^y)\in C(\Bbb R;L^2)^2\;\cap\;L^q_{\rm loc}(\Bbb R;L^r)^2,
$$
with
$$
1=2/r+1/p,\qquad 2/q=3(1/2-1/r),
$$
and conservation of the $L^2$ norms [2207.13928]. For general polynomial coupling, if
$$
(\phi^x_0,\phi^y_0)\in\mathcal H_x^\alpha\times\mathcal H_y^\beta,\qquad
\|\phi^x_0\|_{L^2}=\|\phi^y_0\|_{L^2}=1,
$$
then there exists a unique global solution
$$
\Phi=(\phi^x,\phi^y)\in C(\Bbb R_+;L^2\times L^2)\;\cap\;\bigcap_{T>0}X_T^{\alpha,\beta}.
$$
The total energy
$$
E(t)=\langle H_x\phi^x,\phi^x\rangle+\langle H_y\phi^y,\phi^y\rangle
+\iint w(x,y)\,|\phi^x(x)|^2\,|\phi^y(y)|^2\,dx\,dy
$$
is time-independent [2207.13928].

The proof strategy differs between the Coulomb and polynomial settings. For Coulomb/convolution couplings, the Duhamel formulation is combined with local Strichartz estimates and a fixed-point argument in a Banach space controlled by admissible $L_t^qL_x^r$ norms [2207.13928]. For general polynomial couplings, the argument uses a recursive linearization inspired by the Cauchy problem for symmetric quasilinear hyperbolic equations. A linear step is solved with Kato–Rellich, high-order graph norms are estimated through commutator bounds, convergence is obtained in $L^2$ by contraction for small $T$, and globalization follows from energy coercivity [2207.13928].

## 3. Semiclassical propagation and coherent-state structure

A complementary line of analysis studies the time-dependent Hartree equation in $\mathbb R^d$ with external potential $V(x)$ and interaction kernel $W(x)$:
$$
\begin{cases}
i\hbar\,\partial_t \psi^\hbar(t,x)
=
-\frac{\hbar^2}{2m}\,\Delta_x\psi^\hbar(t,x)
+
V(x)\,\psi^\hbar(t,x)
+
\bigl(W*|\psi^\hbar(t,\cdot)|^2\bigr)(x)\,\psi^\hbar(t,x),\\[6pt]
\psi^\hbar(0,x)=\psi_0^\hbar(x),\quad
\|\psi_0^\hbar\|_{L^2(\R^d)}=1.
\end{cases}
$$
The assumptions are $V\in C^3_b(\R^d)$ and $W\in C^3(\R^d)$, with $W$ real-valued, even, and all derivatives up to order three bounded in $L^\infty$ [1010.4889].

For coherent-state initial data centered at $(q_0,p_0)$,
$$
\psi_0^\hbar(x)
=
\frac{1}{(\pi\hbar)^{d/4}}\,
\exp\!\Bigl[
-\frac{|x-q_0|^2}{2\hbar}
+\frac{i}{\hbar}\,p_0\cdot(x-q_0)
\Bigr],
$$
the solution admits a semiclassical approximation by a time-dependent coherent state whose center follows a classical Hamiltonian flow [1010.4889]. The effective classical Hamiltonian is
$$
H(q,p)\;=\;\frac{p^2}{2m} \;+\;V(q)
\;+\;\int_{\R^d}W\bigl(q-y\bigr)\,\bigl|A(0,y)\bigr|^2dy,
$$
and the phase contains both the classical action and the Hartree contribution [1010.4889].

A central quantitative result is the $L^2$ error estimate
$$
\bigl\|\psi^\hbar(t,\cdot)\;-\;\psi^\hbar_{\rm sc}(t,\cdot)\bigr\|_{L^2(\R^d)}
\;\le\;
C\,e^{C\,t}\,\sqrt\hbar.
$$
The derivation uses subtraction of the semiclassical ansatz from the exact solution, a source estimate of order $O(\sqrt\hbar)$ arising from Taylor remainders of $V$ and $W$, unitarity of the linear propagator, and Grönwall’s inequality [1010.4889].

The same analysis yields a full formal asymptotic expansion
$$
A\!\bigl(t,\xi;\hbar\bigr)
=\sum_{k=0}^N\hbar^{k/2}\,A^{(k)}(t,\xi)\;+\;O(\hbar^{(N+1)/2}),
$$
with recursively determined coefficients. The leading profile solves a “harmonic-Hartree” transport equation involving $V''(q(t))$ and $W''(0)$, while the next term solves a linear inhomogeneous Schrödinger equation driven by $V'''(q(t))$, $W'''(0)$, and the lower-order profile [1010.4889]. This places the time-dependent Hartree equation within a semiclassical framework in which coherent states remain accurate up to an $O(\sqrt\hbar)$ error.

## 4. Mean-field derivation for dense fermions

In a distinct asymptotic regime, the time-dependent Hartree equations emerge as the effective large-$N$ dynamics of dense interacting fermions. The microscopic system considered in [2507.12390] consists of spinless fermions in $\mathbb R^3$ with antisymmetric wave function
$$
\Phi_\tau\in L^2_a(\mathbb R^{3N}),
$$
evolving under
$$
i\,\partial_\tau\Phi_\tau =\;H\,\Phi_\tau,\qquad
H=\sum_{i=1}^N(-\Delta_i)\;+\;\sum_{1\le i<j\le N}v(x_i-x_j),
$$
where $v\in C^2(\mathbb R^3)$ is real-valued, radial, and $N$-independent [2507.12390].

Because the fermions are confined to a volume of order one, typical momenta are $O(N^{1/3})$, the kinetic energy is $\sim N^{5/3}$, and the interaction energy is $\sim N^2$ [2507.12390]. The nontrivial mean-field time scale is obtained by
$$
\tau=\;t\,\varepsilon,\qquad \varepsilon=N^{-2/3}.
$$
On this rescaled time scale, the limiting dynamics is the $N$-orbital Hartree system
$$
i\,\partial_t\varphi_k^t \;=\;h(t)\,\varphi_k^t,\qquad
h(t)=-\Delta +\bigl(v*\rho_t\bigr),\qquad
\rho_t(x)=\sum_{k=1}^N\bigl|\varphi_k^t(x)\bigr|^2,
$$
with orthonormal initial orbitals $\varphi_k^0$ [2507.12390].

A central technical step is a time-dependent gauge transformation
$$
G(t,x_1,\dots,x_N) =\exp\!\Bigl(i\,t\sum_{1\le i<j\le N}v(x_i-x_j)\Bigr),
\qquad
\Psi_t =G(t)\,\Phi_t.
$$
The gauged state satisfies
$$
i\,\partial_t\Psi_t =H^g(t)\,\Psi_t,
$$
with
$$
H^g(t) =\sum_{i=1}^N\Bigl(i\nabla_i + t\sum_{j\ne i}f_{ij}\Bigr)^2,\qquad
f_{ij}=-\nabla v(x_i-x_j).
$$
The corresponding gauged Hartree orbitals
$$
\psi_k^t =\exp\!\bigl(i\,t\,(v*\rho_t)\bigr)\,\varphi_k^t
$$
solve a one-body gauged equation with Hamiltonian $h^g(t)$ expressed in terms of $\bar f=f*\rho_t$ and related partial traces [2507.12390]. The paper’s interpretation is explicit: the gauge transformation eliminates the dominant interaction contribution and replaces it by more regular “magnetic-type” terms.

The proof introduces a counting functional for excitations outside the Slater space, an auxiliary dynamics generated by a quadratic truncation of the gauged Hamiltonian, propagation-of-regularity bounds for the gauged Hartree orbitals, and a weighted Duhamel estimate comparing the full and auxiliary evolutions [2507.12390]. The final estimate is a trace-norm approximation of the one-particle reduced density matrix:
$$
\sup_{\|M\|=1} \Bigl|\Tr\bigl(M\,\gamma^{\Phi_t}\bigr)-\tfrac1N\Tr\bigl(M\,p^{\varphi^t}\bigr)\Bigr|
\;\le\; \exp\bigl(e^{C(1+t)^2}\bigr)
\max\{N^{5/24-\delta_1/4},\,N^{1/12-\delta_2/4},\,N^{1/12-\delta_1/8},\,N^{-1/24}\}.
$$
This gives a rigorous derivation of the time-dependent Hartree equations in a strongly interacting dense fermion regime [2507.12390].

A common misconception is that Hartree mean-field limits are available only in weak-coupling or semiclassical scalings. The dense-fermion result does not fit that template: the interaction is not scaled by $N^{-1}$, and the proof instead relies on the gauge-transformed formulation [2507.12390]. This suggests that the effective Hartree description can persist in regimes where the raw two-body interaction is parametrically larger than in standard mean-field analyses.

## 5. Relation to Hartree–Fock, relativistic Hartree, and pairing theories

The time-dependent Hartree equation is closely related to, but not identical with, time-dependent Hartree–Fock. In the electronic TDHF formulation, the many-electron state is approximated by a single Slater determinant of time-dependent spin-orbitals,
$$
|ψ(t)\rangle=|φ_1(t)φ_2(t)\dots φ_N(t)\rangle,
$$
and each orbital satisfies
$$
i\,\partial_t\,|φ_i(t)\rangle = \hat h(t)\,|φ_i(t)\rangle,
$$
with mean-field Hamiltonian $\hat h(t)=\hat H_F[\rho(t)] + \hat H_D\cdot E(t)$ [2309.00737]. In a finite basis,
$$
\rho_{pq}(t)=\sum_{i=1}^N C_{pi}^*(t)\,C_{qi}(t),
$$
and the Fock matrix is
$$
F_{pq}(t)=h_{pq} + \sum_{r,s}\rho_{rs}(t)\big[(pq|rs)-(pr|sq)\big].
$$
The exchange contribution, encoded by the second term in brackets, is absent in the Hartree model [2309.00737].

This difference appears especially clearly in the relativistic framework. In the time-dependent relativistic Hartree–Fock model with spherical symmetry, the single-particle Dirac equation is
$$
i\,\partial_t\,\psi_i(t,x) = \int d^3x' \, h(x,x';t)\,\psi_i(t,x'),
$$
with
$$
h=h_{\rm kin}+h_D+h_E,
$$
where $h_D$ is the local direct Hartree mean field and $h_E$ is the nonlocal exchange term [2406.06845]. If one omits all Fock terms, $h_E\to0$, then
$$
i\,\partial_t\,\psi_i
=
\bigl[-\,i\,\alpha\!\cdot\!\nabla+\beta M+\beta\Sigma_S(t,x)+\Sigma_0(t,x)\bigr]\psi_i
$$
becomes “exactly the time-dependent relativistic Hartree (TD-RMF) equation” [2406.06845].

Pairing theories provide another reduction. In canonical-basis TDHFB, the generalized density matrix obeys
$$
i\,\frac{d}{dt}R=[{\cal H},R],
$$
with normal density $\rho$ and pairing tensor $\kappa$ [1007.0785]. Under the diagonal approximation for the pair potential in the canonical basis, the equations simplify to coupled evolution equations for canonical orbitals, occupations, and pair amplitudes. If $\Delta_k(t)\equiv0$ for all $k$, then $\rho_k$ freezes at $0$ or $1$, $\kappa_k=0$, and the single-particle states satisfy
$$
i\,\partial_t\,\phi_k(t)=h(t)\,\phi_k(t),
$$
which is “precisely the usual TDHF (or in a local-density approximation, the time-dependent Hartree) equation”
$$
i\partial_t\psi_i(\vec r,t)=\bigl[-\tfrac{\nabla^2}{2m}+U[\rho]\bigr]\psi_i(\vec r,t)
$$
[1007.0785].

The algorithmic literature reflects the same distinction. A hybrid quantum–classical TDHF implementation can propagate the one-body mean-field Hamiltonian
$$
\hat h(t)=\sum_{i,j}G_{ij}(t)\,a_i^\dagger a_j
$$
with Trotterization, matchgate rotations, and Yang–Baxter-equation-based circuit compression to constant depth per time step [2309.00737]. The paper explicitly notes that TDHF is free-fermionic and classically simulatable in polynomial time, so the quantum formulation is positioned as a benchmark for real-time mean-field dynamics rather than as a reinterpretation of Hartree itself [2309.00737].

## 6. Multiconfigurational generalizations and the Hartree limit

The time-dependent Hartree equations occupy a limiting position within multiconfigurational theories. In the standard MCTDH ansatz for an $f$-mode wavefunction,
$$
\Psi(\{r_\kappa\},t) = \sum_{j_1=1}^{n_1}\cdots\sum_{j_f=1}^{n_f}
A_{j_1\cdots j_f}(t)\,\prod_{\kappa=1}^f\varphi^{(\kappa)}_{j_\kappa}(r_\kappa,t),
$$
the Dirac–Frenkel principle yields coupled equations for the coefficient tensor $A$ and the single-particle functions (SPFs) [1802.01058]. The SPF equations involve the reduced single-particle density matrices and their inverses,
$$
\dot{\bm\varphi}^{(\kappa)}(t)
=\frac{1}{i\hbar}\,\bigl[1-P^{(\kappa)}(t)\bigr]\,
\bigl[\rho^{(\kappa)}(t)\bigr]^{-1}\,
\langle H\rangle^{(\kappa)}(t)\,\bm\varphi^{(\kappa)}(t),
$$
which constitute a standard numerical bottleneck [1802.01058].

If one chooses $n_\kappa=1$ for every mode, then there is only one SPF per mode and $A\equiv1$. The ansatz collapses to
$$
\Psi(r_1,\dots,r_f,t)=\prod_{\kappa=1}^f\varphi^{(\kappa)}(r_\kappa,t),
$$
and the MCTDH equations reduce to the familiar time-dependent Hartree equations
$$
i\hbar\,\dot\varphi^{(\kappa)}
=
\Bigl[-\tfrac{\hbar^2}{2m_\kappa}\partial_{r_\kappa}^2
+V^{(\kappa)}_{\rm eff}(r_\kappa,t)\Bigr]\varphi^{(\kappa)}
$$
[1802.01058]. In this sense, MCTDH generalizes TDH by replacing a product-state ansatz with a time-dependent multiconfiguration expansion.

For indistinguishable particles, the multiconfigurational extension is MCTDH-X, comprising MCTDH-B for bosons and MCTDH-F for fermions [1908.03578]. The wavefunction is expanded in time-dependent permanents or Slater determinants built from optimized orbitals:
$$
|\Psi(t)\rangle=\sum_{\vec n} C_{\vec n}(t)\,|\vec n;t\rangle.
$$
The coefficient equation is linear in $C$,
$$
i\,\dot C_{\vec n}(t)
=
\sum_{\vec n'} \langle \vec n;t|\hat H|\vec n';t\rangle C_{\vec n'}(t),
$$
while the orbital equation is nonlinear because it depends on the one- and two-body reduced density matrices,
$$
\rho_{kq}(t)=\langle\Psi(t)|b_k^\dagger b_q|\Psi(t)\rangle,\qquad
\rho_{kslq}(t)=\langle\Psi(t)|b_k^\dagger b_s^\dagger b_l b_q|\Psi(t)\rangle
$$
[1908.03578]. Restricted-active-space variants further truncate the CI space and introduce orbital-rotation terms between active subspaces [1908.03578].

Recent numerical analysis revisits MCTDH from the perspective of tangent-space projectors and projector splitting. The projector-splitting algorithm replaces explicit reduced-density inversion by an auxiliary set of non-orthogonal SPFs and QR or Cholesky factorizations of matricized coefficient tensors [1802.01058]. A closely related tensor-network extension, MPS-MCTDH, rewrites the coefficient tensor in matrix-product-state form and propagates local site tensors with effective Hamiltonians constructed by DMRG-like sweeps [1808.01110]. The mean-field operators entering the SPF equations are then evaluated by contractions over the MPS representation [1808.01110].

The multiconfigurational literature therefore clarifies the position of TDH. It is not a rival to MCTDH-type methods but their lowest-rank product-state limit. A plausible implication is that many numerical innovations developed for MCTDH—projector splitting, tangent-space formulations, and tensor-network parametrizations—can be interpreted as systematic strategies for moving beyond the Hartree manifold while retaining variational time propagation [1802.01058], [1808.01110].

## 7. Applications and scope across physical settings

The application domain of time-dependent Hartree-type equations is broad, but the precise physical role depends on the variant under consideration. In the nonrelativistic product-state setting, the Hartree system is a model of dimension reduction for quantum dynamics and a leading-order mean-field approximation whose mathematical well-posedness supports subsequent approximation and error analyses [2207.13928]. In the semiclassical regime, it describes the propagation of coherent states under self-consistent nonlinear potentials and admits explicit control of the departure from a classical-flow-based wave-packet ansatz [1010.4889].

In relativistic nuclear dynamics, the Hartree reduction appears when exchange is omitted from the time-dependent relativistic Hartree–Fock equation. The resulting TD-RMF system retains only the local Hartree self-energies $\Sigma_S$ and $\Sigma_0$ generated by the instantaneous scalar density and vector current [2406.06845]. The spherical TD-RHF framework was then used to study the isoscalar giant monopole resonance of $^{208}$Pb, with numerical checks of conservation of total binding energy, particle number, and time-reversal invariance [2406.06845]. These are properties of the RHF formulation, but the Hartree reduction is the direct-field limit of that framework.

In multiconfigurational many-body physics, the Hartree limit serves as a reference point against which correlation effects are measured. MCTDH-X benchmarks against the Harmonic Interaction Model and its time-dependent version show convergence to exact solutions as the number of orbitals $M$ increases, and bosonic as well as fermionic applications include Bose-condensate dynamics, photoionization cross sections, time delays, high-harmonic generation, and transient absorption [1908.03578]. The specific claim here is not that the Hartree equations alone capture those correlated phenomena, but that the time-dependent Hartree picture is embedded within a hierarchy of increasingly correlated variational theories [1908.03578].

Across these settings, the central structural feature remains unchanged: the equation is nonlinear because the potential depends on the evolving density, yet it remains one-body at the level of each orbital or factor. The literature reviewed here presents that structure in several mathematically distinct forms—product-state Dirac–Frenkel projection [2207.13928], semiclassical coherent-state propagation [1010.4889], dense-fermion mean-field derivation [2507.12390], relativistic direct-field reduction [2406.06845], and multiconfigurational product-state limit [1802.01058].

Source: https://www.emergentmind.com/topics/time-dependent-hartree-equations