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.
\[ I(t) = \int_{a(t)}^{b(t)} dx f(t,x) \]
then its derivative with respect to $$t$$ is
\[ \frac{d}{dt} I(t) = \int_{a(t)}^{b(t)} dx \frac{\partial}{\partial t} f(t,x) + f(b(t),x) \frac{\partial}{\partial t}b(t) – f(a(t),x) \frac{\partial}{\partial t}a(t) \]
Applying this to the formal solution for the evolution operator gives
\[ \frac{d}{dt} U(t,t_0) = \int_{t_0}^{t} dt’ \frac{\partial}{\partial t} \left( H(t’) U(t’,t_0) \right) + H(t) U(t,t_0) \frac{\partial}{\partial t} t \\ \\ = H(t) U(t,t_0) \; ,\]
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.