In [quantum field theory](Quantum%20Field%20Theory%20(Index).md) and [quantum dynamics](Quantum%20Dynamics%20(index).md) the _propagator_ refers to one of two closely related quantities $D(q_F,t_F;q_I,t_I)$ or $K(q_F,t_F;q_I,t_I).$ In each case we are dealing with a [[Wavefunction]] that solves the [Schrödinger Equation](Quantum%20Mechanics/Quantum%20Dynamics/Schrödinger%20Equation.md). # Relationship between $D(q_F,t_F;q_I,t_I)$ and $K(q_F,t_F;q_I,t_I)$ $D(q_F,t_F;q_I,t_I)$ is a [Green's function](Green's%20function) for a the [Schrödinger Equation](Quantum%20Mechanics/Quantum%20Dynamics/Schrödinger%20Equation.md) and is thus expressed in terms of $K(q_F,t_F;q_I,t_I)$ as $D(q_F,t_F;q_I,t_I)=\frac{1}{i\hbar}\Theta(t_F-t_I)K(q_F,t_F;q_I,t_I)$ where $\Theta(t_F-t_I)$ is the [Heaviside function](Heaviside%20function.md). Here the Heaviside function can be thought of as enforcing causality since it is 0 for $t_F-t_I<0.$ # As a solution to the Schrödinger equation Given the form of the [Schrödinger equation](Quantum%20Mechanics/Quantum%20Dynamics/Schrödinger%20Equation.md), $i\hbar\frac{\partial \psi}{\partial t} = \hat{H}\psi$ we may solve the differential equation by rewriting it as $\bigg(\hat{H}-i\hbar\frac{\partial}{\partial t}\bigg)\hat{D}(t_F,t_I)=-i\hbar \hat{I} \delta(t_F-t_I).$ where in terms of coordinator space the equation is $\bigg(H_q-i\hbar\frac{\partial}{\partial t}\bigg)D(q_F,t_F;q_I,t_I)=-i\hbar \delta (q_F-q_I) \delta(t_F-t_I)$ where $D(q_F,t_F;q_I,t_I)$ is the _propagator_. Since it is a type of [[Green's function]] for the Schrödinger equation we often refer to it as the _Green's function_ in contrast with the propagator given as $K(q_F,t_F;q_I,t_I).$ # Constructing the propagator ## Path integral formulation [Path integral formulation](Path%20integral%20formulation.md) #QuantumMechanics/QuantumDynamics #QuantumMechanics/QuantumDynamics/PathIntegrals