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

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.

This post will be the beginning of my extended effort to organize material on the time evolution operator, quantum propagators, and Greens functions.  The aim of this is to put into a self-consistent and self-contained set of posts the background necessary to gnaw away at a reoccurring confusion I have had over these items from their presentations in the literature as to the names, definitions, and uses of the objects. In particular, the use of the Schrodinger, Heisenberg, and Interaction pictures.

Once this organization is done, I hope to use these methods to serve as a springboard for research into methods of applying quantum mechanical techniques to classical dynamical systems. In particular, the use of the Picard iteration (aka the Dyson’s expansion) for time-varying Hamiltonians.

The references that I will be using are:

[1] Quantum Mechanics – Volume 1, Claude Cohen-Tannoudji, Bernard Diu, and Frank Laloe
[2] Quantum Mechanics, Leonard Isaac Schiff
[3] Principles of Quantum Mechanics, R. Shankar
[4] Modern Quantum Mechanics, J.J. Sakurai

Starting simply, in this post I will be reviewing the definition and properties of the evolution operator.

Adapting the material in [1] (p. 236, 308-311), the Schrodinger equation in a representation-free form is:

\[ i \hbar \frac{d}{dt} \left| \psi(t) \right> = H(t) \left| \psi(t)\right>\]

From the structure of the Schrodinger equation, the evolution of the state $$\left|\psi(t)\right>$$ is entirely deterministic being subject to the standard, well-known theorems about the existence and uniqueness of the solution.  For the skeptic that is concerned that $$\left|\psi(t)\right>$$ can be infinite-dimensional I don’t have much in the way of justification except to say three things. First that the Schrodinger equation in finite dimensions (e.g. two state systems) maps directly to the usual cases of coupled linear systems dealt with in introductory classes on differential equations. Second, it is common practice for infinite-dimensional systems (i.e., PDEs) to be discretized for numerical analysis, and so the resulting structure is again a finite-dimensional linear system, although with arbitrary sizes. That is to say, the practitioner can refine the mesh used arbitrarily until either his patience or his computer gives out. It isn’t clear that such a process necessarily converges but the fact that there isn’t a hue and cry of warnings in the community suggests that convergence isn’t a problem. Finally, for those cases where the system is truly infinite-dimensional, with no approximations allowed, there are theorems about the Cauchy problem and how to propagate forward in time from initial data and how the resulting solutions are deterministic. How to match up a evolution operator formalism to these types of problems (e.g., heat conduction) may be the subject of a future post. One last note, I am unaware of a single physical system that involves time evolution that can’t be manipulated (especially for numerical work) into the form $$\frac{d}{dt} \bar S = \bar f(\bar S; t)$$ where $$\bar S$$ is the abstract state and $$\bar f$$ is the vector field that is the function of the state and time. The Schrodinger equation is then an example where $$\bar f(\bar S;t)$$ is a linear operation.

From the theory of linear systems, the state at some initial time $$\left|\psi(t_0)\right>$$ is related to the state at the time $$t$$ by

\[ \left|\psi(t)\right> = U(t,t_0) \left|\psi(t_0)\right>\]

To determine what equation $$U(t,t_0)$$ obeys, simply substitute the above expression into the Schrodinger equation to yield
\[i \hbar \frac{d}{dt}\left[ U(t,t_0) \left|\psi(t_0)\right> \right] = H \; U(t,t_0) \left|\psi(t_0)\right> \; ,\]

and since $$\left|\psi(t_0)\right>$$ is arbitrary, the equation of motion or time development equation for the evolution operator is
\[i \hbar \frac{d}{dt} U(t,t_0) = H \; U(t,t_0) \; .\]

The required boundary condition is
\[ U(t_0,t_0) = Id \; ,\]
where $$Id$$ is the identity operator of the correct size to be consistent with the state dimension. That is to say that $$Id$$ finite dimensional matrix with rank equal to the state of $$\left| \psi(t) \right>$$ or it is infinite dimensional.

Some obvious properties can be deduced without an explicit expression for $$U(t,t_0)$$ by regarding as variable $$t_0$$. Assume that $$t_0$$ takes on a particular value $$t’$$ then the evolution operator can relate the state at that time to some later time $$t”$$ as
\[ \left| \psi(t”)\right> = U(t”,t’) \left| \psi(t’)\right> \; .\]
Now let $$t_0$$ take on the value $$t”$$ and connect the state at this time to some other time $$t$$ by
\[ \left| \psi(t)\right> = U(t,t”) \left| \psi(t”)\right> \; .\]
By composing these two expressions, the state at $$t’$$ can be related to the state at $$t$$ with a stop off at the intermediate time $$t”$$ resulting in the general composition relation
\[ U(t,t’) = U(t,t”) U(t”,t’) \; .\]
Using the same type of arguments the inverse of the evolution operator can be seen to be
\[U^{-1}(t,t_0) = U(t_0,t) \; \]
which can also be expressed as
\[ U(t,t_0) U(t_0,t) = U(t,t) = Id \; .\]

Formal solution for the equation of motion of the evolution operator is
\[ U(t,t_0) = Id – \frac{i}{\hbar} \int_{t_0}^{t} dt’ H(t’) U(t’,t_0) \]
which can be verified using the Liebniz rule for differentiation under the integral sign.

There are three cases to be examined (based on the material in [4] pages 72-3).

1. The Hamiltonian is not time dependent, $$H \neq H(t)$$. In this case, the evolution operator has an immediate closed form solution given by
\[ U(t,t_0) = e^{-\frac{i H (t-t_0)}{\hbar} } \; .\]

2. The Hamiltonian is time dependent but it commutes with itself at different times, $$H = H(t)$$ and $$\left[ H(t),H(t’) \right] = 0$$. This case also possesses an immediate closed solution but with a slight modification
\[ U(t,t_0) = e^{-\frac{i}{\hbar}\int_{t_0}^{t} dt’ H(t’)} \]

3. The Hamiltonian is time dependent and it does not commute with itself at different times, $$H = H(t)$$ and $$\left[H(t),H(t’)\right] \neq 0$$. In this case, the solution that exists is written in the self-iterated form
\[ U(t,t_0) = Id + \\ \sum_{n=1}^{\infty} \left(-\frac{i}{\hbar}\right)^n \int_{t_0}^{t} dt_1 \int_{t_0}^{t_1} dt_2…\int_{t_0}^{t_{n-1}} dt_n H(t_1) H(t_2)… H(t_n) \; .\]

The structure of the iterated integrals in case 3, is formally identical to the Picard iteration, a technique that is used in a variety of disciplines to construct solutions to initial value problems, at least in a limited time span. I am not aware of any formal proof that the convergence in Case 3 is guaranteed in the most general setting when $$H(t)$$ and $$U(t,t_0)$$ are infinite dimensional but the iterated solution is used in quantum scattering and so the method is worth studying.

Next week, I’ll be exploring the behavior of a related object called the quantum propagator.