Given the general relationships for quantum time evolution in Part 1 of these posts, it is natural to ask about how to express these relationships in a basis that is more suited for computation and physical understanding. That can be done by taking the general relationship for time development
\[ \left| \psi (t_2) \right> = U(t_2, t_1) \left| \psi (t_1) \right> \]
and the projecting this relation into the position basis $$\left| \vec r \right>$$ with the definition that the traditional Schrodinger wave function is given by
\[ \left< \vec r | \psi (t) \right> = \psi(\vec r, t) \; .\]
The rest of the computation proceeds by a strategic placement of the closure relation for the identity operator, $$Id$$,
\[ Id = \int d^3 r_1 \left| \vec r_1 \right>\left< \vec r_1 \right| \]
in the position basis, between $$U(t_2,t_1)$$ and $$\left| \psi(t_1) \right>$$ when $$U(t_2,t_1) \left| \psi(t_1) \right>$$ is substituted for $$\left| \psi(t_2) \right>$$
\[ \left< \vec r_2 | \psi(t_2) \right> = \left< \vec r_2 \right| U(t_2,t_1) \left| \psi(t_1) \right> = \\ \int d^3r_1 \left<\vec r_2 \right| U(t_2,t_1) \left| \vec r_1 \right> \left< \vec r_1 \right| \left. \psi(t_1) \right> \; .\]
Recognizing the form of the Schrodinger wave function about in both the left- and right-hand sides, the equation becomes
\[ \psi(\vec r_2, t_2) = \int d^3r_1 \left<\vec r_2 \right| U(t_2,t_1) \left| \vec r_1 \right> \psi(\vec r_1, t_1) \; .\]
If the matrix element of the evolution operator between $$\vec r_2$$ and $$\vec r_1$$ is defined as
\[ \left<\vec r_2 \right| U(t_2,t_1) \left| \vec r_1 \right> \equiv K(\vec r_2, t_2; \vec r_1, t_1) \; , \]
then the structure of the equation is now
\[ \psi(\vec r_2, t_2) = \int d^3r_1 K(\vec r_2, t_2; \vec r_1, t_1) \psi(\vec r_1, t_1) \; .\]
What meaning can be attached to this equation, which, for convenience, will be referred to as the boxed equation? Well it turns out that the usual textbooks on Quantum Mechanics are not particularly illuminating on this front. For example, Cohen-Tannoudji et al, usually very good in their pedagogy, have a presentation in Complement $$J_{III}$$ that jumps immediately from the boxed equation to the idea that $$K(\vec r_2, t_2; \vec r_1, t_1)$$ is a Greens function. While this idea is extremely important, it would be worthwhile to slow down the development and discuss the interpretation of the boxed equation both mathematically and physically.
Let’s start with the mathematical aspects. The easiest way to understand the meaning of the boxed equation is to start with a familiar example from classical mechanics – the simple harmonic oscillator.
The differential equation for the position, $$x(t)$$, of the simple harmonic oscillator is given by
\[ \frac{d^2}{dt^2} x(t) + \omega^2_0 x(t) = 0 \; ,\]
where $$\omega^2_0 = k/m$$ and where $$k$$ and $$m$$ are the spring constant and mass of the oscillator. The general solution of this equation is the well-known form
\[ x(t) = x_0 \cos(\omega_0 (t-t_0)) + \frac{v_0}{\omega_0} \sin(\omega_0 (t-t_0)) \, \]
with $$x_0$$ and $$v_0$$ being the initial position and velocity at $$t_0$$, respectively. To translate this system into a more ‘quantum’ form, the second-order differential equation needs to be translated into state-space form, where the state, $$\bar S$$, captures the dynamical variables (here the position and velocity)
\[ \bar S = \left[ \begin{array}{c} x \\ v \end{array} \right] \; ,\]
(the time dependence is understood) and the corresponding differential equation is written in the form
\[ \frac{d}{dt} {\bar S} = {\bar f}\left( \bar S,t\right) \; .\]
For the simple harmonic oscillator, the state-space form is explicitly
\[ \frac{d}{dt} \left[ \begin{array}{c} x \\ v\end{array} \right] = \left[ \begin{array}{cc} 0 & 1 \\ -\omega^2_0 & 0 \end{array} \right] \left[ \begin{array}{c} x \\ v\end{array} \right] \; , \]
with solutions of the form
\[ \left[ \begin{array}{c} x \\ v\end{array} \right] = \left[ \begin{array}{cc} \cos(\omega_0 (t-t_0)) & \frac{1}{\omega_0} \sin(\omega_0 (t-t_0)) \\ -\omega_0 \sin(\omega_0 (t-t_0)) & \cos(\omega_0 (t-t_0)) \end{array} \right] \left[ \begin{array}{c} x \\ v \end{array} \right] \\ \equiv M(t-t_0)\left[ \begin{array}{c} x \\ v \end{array} \right] \; .\]
The matrix $$M(t-t_0)$$ plays the role of the evolution operator (also known as the state transition matrix by engineers and the fundamental matrix by mathematicians), moving solutions forward or backward in time as needed because the theory is deterministic.
If the dynamical variables are denoted collectively by $$q_i(t)$$ where the index $$i=1, 2$$ labels the variable in place of the explicit names $$x(t)$$ and $$v(t)$$, then the state-space evolution equation can be written compactly as
\[ q_i(t) = \sum_{j} M_{ij}(t-t_0) q^{0}_j \;, \]
where $$q^{0}$$ is the collection of initial conditions for each variable (i.e. $$q^{0}_1 = x0$$, $$q^{0}_2 = v0$$). As written, this compact form can be generalized to an arbitrary number of dynamic variables by allowing the indices $$i$$ and $$j$$ to increase their range appropriately.
The final step is then to imagine that the number of dynamic variables goes to infinity in such a way that there is a degree-of-freedom associated with each point in space. This is the typical model used in generalizing a discrete dynamical system such as a long chain of coupled oscillators to a continuum system that describes waves on a string. In this case, the indices $$i$$ and $$j$$ is now replaced by a label indicating the position ($$x$$ and $$x’$$), the sum is replaced by an integral, and we have
\[ q(x,t) = \int dx’ M(t-t_0;x,x’) q(t_0;x’) \; ,\]
which except for the obvious minor differences in notation is the same form as the boxed equation.
Thus we arrive at the mathematical meaning of the boxed equation. The kernel $$K(\vec r_2, t_2; \vec r_1; t_1)$$ takes all of the dynamical values of the system at a given time $$t_1$$ and evolves them up to time $$t_2$$. The time $$t_1$$ is arbitrary since the evolution is deterministic, so that any particular configuration can be regarded as the initial conditions for the ones that follow. Each point in space is considered a dynamical degree-of-freedom and all points at earlier times contribute to its motion through the matrix multiplication involved in doing the integral. That is why the boxed equation involves to integration over time.
The final step is to physically interpret what the kernel means. From its definition as the matrix element between $$\vec r_2$$ and $$\vec r_1$$ of the evolution operator, the kernel is the probability amplitude that a particle moves from $$\vec r_1$$ to $$\vec r_2$$ during its evolution during the time span $$[t_1,t_2]$$. Or in other words, the conditional probability density that a particle can be found at $$\vec r_2$$ at time $$t_2$$ given that it started at position $$\vec r_1$$ at time $$t_1$$ is
\[ Prob(\vec r_2,t_2 | \vec r_1, t_1 ) = \left| K(\vec r_2,t_2; \vec r_1, t_1) \right|^2 \]
Next week, I’ll interpret how a slight modification of the kernel can be interpreted as a Greens function.