This post takes a small detour from the main thread of the previous posts to make a quick exploration of the classical applications of the Greens function.
In the previous posts, the basic principles of quantum evolution have resulted in the development of the propagator and corresponding Greens function as a prelude to moving into the Feynman spacetime picture and its applications to quantum scattering and quantum field theory. Despite all of the bra-ket notation and the presence of $$\hbar$$, there has actually been very little presented that was peculiarly quantum mechanical, except for the interpretation of the quantum propagator as a probability transition amplitude. Most of the machinery developed is applicable to linear systems regardless of their origins.
Here we are going to use that machinery to explore how the knowledge of the propagator allows for the solution of an inhomogeneous linear differential equation. While the presence of an inhomogeneity doesn’t commonly show up in the quantum mechanics, performing this study will be helpful in several ways. First, it is always illuminating to compare applications of the same mathematical techniques in quantum and classical settings. Doing so helps to sharpen the distinctions between the two, but also helps to point out the commons areas where insight into one domain may be more easily obtained than in the other. Second, the term Greens function is used widely in many different but related contexts, so having some knowledge highlighting the basic applications is useful in being able to work through the existing literature.
Lets start with a generic linear, homogeneous, differential equation
\[ \frac{d}{dt} \left| \psi(t) \right> = H \left| \psi(t) \right> \; ,\]
where $$\left| \psi(t) \right>$$ is simply a state of some sort in either a finite- or infinite-dimensional system, and $$H$$ is some linear operator. Let the solutions of this equation, by the methods discussed in the last three posts, be denoted by $$\left| \phi(t) \right>_h$$ where the $$h$$ subscript means ‘homogeneous’.
Now suppose the actual differential equation that we want to solve involves an inhomogeneous term $$\left|u(t)\right>$$ that is not related to the state itself.
\[ \left( \frac{d}{dt} – H \right) \left| \psi(t) \right> = \left| u(t) \right> \; .\]
Such a term can be regarded as an outside driving force. How, then, do we solve this equation?
Recall that the homogeneous solution at some earlier time $$\left| \phi(t_0) \right>_h$$ evolves into a later time according to
\[ \left| \phi(t) \right>_h = \Phi(t,t_0) \left| \phi(t_0) \right>_h \; , \]
where the linear operator $$\Phi(t,t_0)$$ is called the propagator. Now the general solution of the inhomogeneous equation can be written in terms of these objects as
To demonstrate that this is true, apply the operator
\[ L \equiv \frac{d}{dt} – H(t) \]
to both sides. (Note that the any time dependence for the operator $$H(t)$$ has been explicitly restored for reasons that will become obvious below.) Since $$\left| \phi(t)\right>_h$$ is a homogeneous solution,
\[ L \left| \phi(t) \right>_h = 0 \]
and we are left with
\[ L \left| \psi(t) \right> = L \int_{t_0}^t dt’ \Phi(t,t’) \left| u(t’) \right> \; .\]
Now expand the operator on the right-hand side, bring the operator $$H(t)$$ into the integral over $$t’$$, and use the Liebniz rule to resolve the action of the time derivative on the limits of integration. Doing this gives
\[ L \left| \psi(t) \right> = \Phi(t,t) \left| u(t) \right> + \int_{t_0}^t dt’ \frac{d}{dt} \Phi(t,t’) \left| u(t’) \right> \\ – \int_{t_0}^t dt’ H(t) \Phi(t,t’) \left| u(t’) \right> \; .\]
Now recognize that $$\Phi(t,t) = Id$$ and that
\[ \frac{d}{dt} \Phi(t,t’) = H(t) \Phi(t,t’) \]
since $$\Phi(t,t’)$$ is propagator for the homogeneous equation. Substituting these relations back in simplifies the equation to
\[ L \left| \psi(t) \right> = \left| u(t) \right> + \int_{t_0}^t dt’ H(t) \Phi(t,t’) \left| u(t’) \right> \\ – \int_{t_0}^t dt’ H(t) \Phi(t,t’) \left| u(t’) \right> \; .\]
The last two terms cancel and, at last, we arrive at
\[ \left( \frac{d}{dt} – H \right) \left| \psi(t) \right> = \left| u(t) \right> \; , \]
which completes the proof.
It is instructive to actually carry this process out for a driven simple harmonic oscillator. In this case, the usual second-order form is given by
\[ \frac{d^2}{dt^2} x(t) + \omega_0^2 x(t) = F(t) \]
and the corresponding state-space form is
\[ \frac{d}{dt} \left[ \begin{array}{c} x \\ v \end{array} \right] = \left[ \begin{array}{cc} 0 & 1 \\ \omega_0^2 & 0 \end{array} \right] \left[ \begin{array}{c} x \\ v \end{array} \right] + \left[ \begin{array}{c} 0 \\ F(t) \end{array}\right] \; ,\]
from which we identify
\[ H = \left[ \begin{array}{cc} 0 & 1 \\ \omega_0^2 & 0 \end{array} \right] \; \]
and
\[ \left| u(t) \right> = \left[ \begin{array}{c} 0 \\ F(t) \end{array} \right] \; .\]
The propagator is given by
\[ \Phi(t,t’) = \left[ \begin{array}{cc} \cos(\omega_0 (t-t’)) & \frac{1}{\omega_0} \sin(\omega_0 (t-t’)) \\ -\omega_0 \sin(\omega_0 (t-t’)) & \cos(\omega_0 (t-t’)) \end{array} \right] \; , \]
and the driving integral becomes
\[ \int_0^t dt’ \left[ \begin{array}{c} \frac{1}{\omega_0} \sin\left( \omega_0 (t-t’) \right) F(t’) \\ \cos\left( \omega_0 (t-t’) \right) F(t’) \end{array} \right] \; ,\]
where $$t_0$$ has been set to zero for convenience.
The general solution for the position of the oscillator can then be read off from the first component as
\[ x(t) = x_h(t) + \int_0^t dt’ \frac{1}{\omega_0} \sin\left( \omega_0 (t-t’) \right) F(t’) \; . \]
This is essentially the form for the general solution, and is the same that results from the Greens function approach discussed in many classical mechanics texts (e.g., page 140 of Classical Dynamics of Particle and Systems, Second Edition, Marion). The only difference between the treatment here and a more careful treatment is the inclusion of a Heaviside function to enforce causality. Since this was discussed in detail in the last post and will also be covered in future posts, that detail was suppressed here for clarity.