2.4: Equation of Motion for the Time-Evolution Operator

Our primary interest is with dynamical processes, and for this we need to work with a time-dependent Hamiltonian. Let’s start at the beginning by obtaining the equation of motion that describes the wavefunction and its time evolution through the time propagator. We are seeking equations of motion for quantum systems equivalent to Newton’s—or more accurately Hamilton’s—equations for classical systems. The question is, if we know the wavefunction at time \(t_{0} \) is \( \left|\psi (\mathbf{r}, t_{0} )\right\rangle\), how does it change with time? How do we determine \(\left|\psi (\mathbf{r}, t_{0}) \right\rangle\) for some later time \(t>t_{0}\)? We will use our intuition here, relying heavily on correspondence to classical mechanics. To simplify notation, in the following discussion we will not explicitly show the spatial dependence \( (\mathbf{r}) \) of the wavefunction.

To derive a deterministic equation of motion, we start by asserting causality—that \(\left|\psi(t_{0})\right\rangle\) precedes and determines \(|\psi(t)\rangle\). As above, we define a "time-evolution operator" or "propagator" \(\hat{U}\) that acts on the wavefunction to the right to propagate the system forward in time from \(t_0\) to \(t\):

\[|\psi(t)\rangle=\hat{U}(t, t_{0})\left|\psi(t_{0})\right\rangle\]

Additionally, the wavefunction must be continuously differentiable in time, and therefore the state is unchanged when the initial and final time-points are the same

\[\hat{U}(t, t)=1\]

If we take the system to be deterministic, then it stands to reason that we should get the same wavefunction whether we evolve to a target time in one step \(\left(t_{0} \rightarrow t_{2}\right)\) or multiple steps \(\left(t_{0} \rightarrow t_{1} \rightarrow t_{2}\right)\). Therefore, we can write the composition property

\[\hat{U}(t_{2}, t_{0})=\hat{U}(t_{2}, t_{1}) \hat{U}(t_{1}, t_{0})\]

Note that since \(\hat{U}\) acts to the right, order matters:

\[\begin{align} \begin{aligned} \left|\psi (t_{2}) \right\rangle &=\hat{U}(t_{2}, t_{1}) \hat{U}(t_{1}, t_{0})\left|\psi(t_{0})\right\rangle
\\ &=\hat{U}(t_{2}, t_{0})\left|\psi (t_{1})\right\rangle \end{aligned} \end{align} \]

Equation (2.4.4) is already very suggestive of an exponential form for \(\hat{U}\). Furthermore, since time is continuous and the operator is linear, it also suggests that the time propagator is only dependent on a time interval rather than the absolute initial and final time points:

\[\hat{U}(t_{1}, t_{0})=\hat{U} (t_{1}-t_{0})\]

As a result of the linearity and the principle of superposition, \(\hat{U}\) is distributive. If we express the state of a system as the superposition

\[\left|\psi(t_{0})\right\rangle=c_{1} \! \left|\varphi_{1}(t_{0})\right\rangle+c_{2} \! \left|\varphi_{2}(t_{0})\right\rangle\]

then

\[\begin{align} \begin{aligned} |\psi(t)\rangle &=\hat{U}(t, t_{0})\left|\psi(t_{0})\right\rangle \\ &=\hat{U}(t, t_{0}) c_{1} \! \left|\varphi_{1}(t_{0})\right\rangle+\hat{U}(t, t_{0}) c_{2} \! \left|\varphi_{2}(t_{0})\right\rangle \\ &=c_{1}(t) \left|\varphi_{1}\right\rangle+c_{2}(t)\left|\varphi_{2}\right\rangle \end{aligned} \end{align}\]

While the coefficients \(c_{i}(t)\) evolve with time, the total population of the system must be conserved, and therefore

\[\sum_{n}\left|c_{n}(t)\right|^{2}=\sum_{n}\left|c_{n}(t_{0})\right|^{2}\]

Thus \(\hat{U}\) is not dependent on \(|\psi\rangle\), and the differential equation of motion for \(\hat{U}\) is linear in time. Note that eq. (2.4.8) implies that \(\hat{U}\) must be unitary.

\[P= \langle\psi(t) | \psi(t)\rangle=\left\langle\psi(t_{0})\right| \hat{U} \vphantom{A}^{\dagger} \hat{U} \left| \psi(t_{0})\right\rangle\]

which holds if \(U^{\dagger}=U^{-1}\). Also, the inverse of the time-propagator is the time reversal operator. From eq. (2.4.3):

\[\begin{align} \hat{U}(t, t_{0}) \hat{U}(t_{0}, t)=1 \\[3pt] \therefore \hat{U} \vphantom{A}^{-1}(t, t_{0})=\hat{U}(t_{0}, t) \end{align}\]

Now we are able to find an equation of motion that describes the time-evolution operator using the change of the system for an infinitesimal time-step, \(\delta t: \hat{U}(t+\delta t, t)\). Since

\[\lim _{\delta t \rightarrow 0} \hat{U}(t+\delta t, t)=1\]

we expect that for small enough \(\delta t, \hat{U}\) will change linearly with \(\delta t\). This is based on analogy to thinking of deterministic motion in classical systems. Setting \(t_{0}\) to \(0\), so that \(\hat{U}(t, t_{0})=\hat{U}(t)\), we can write

\[\hat{U}(t+\delta t)=\hat{U}(t)-\mathrm{i} \hat{\Omega}(t) \delta t\]

\(\hat{\Omega}\) is a time-dependent Hermitian operator, which is required for \(\hat{U}\) to be unitary. We can now write a differential equation for the time-development of \(\hat{U}\) starting with

\[\frac{d \hat{U}(t)}{d t}=\lim _{\delta t \rightarrow 0} \frac{\hat{U}(t+\delta t)-\hat{U}(t)}{\delta t}\]

So from eq. (2.4.13) we have

\[\frac{\partial \hat{U}(t, t_{0})}{\partial t}=-\mathrm{i} \hat{\Omega} \hat{U}(t, t_{0})\]

We can now see that the operator needed a complex argument. Otherwise, probability density would not be conserved; it would rise or decay. Rather it oscillates through different states of the system.

We note that \(\hat{\Omega}\) has units of frequency. Since quantum mechanics fundamentally associates frequency and energy as \(E=\hbar \kern.05em \omega\), and since the Hamiltonian is the operator corresponding to the energy, and responsible for time evolution in classical Hamiltonian mechanics, we write

\[\hat{\Omega}=\frac{\hat{H}}{\hbar}\]

With that substitution we have an equation of motion for \(\hat{U}\):

\[\mathrm{i} \hbar \frac{\partial}{\partial t} \hat{U}(t, t_{0})=\hat{H} \hat{U}(t, t_{0})\]

Multiplying from the right by \(\left|\psi(t_{0})\right\rangle\) gives the TDSE:

\[\mathrm{i} \hbar \frac{\partial}{\partial t} \left|\psi\right\rangle=\hat{H} \left|\psi\right\rangle\]

If we use the Hamiltonian for a free particle, \(-(\hbar^{2} / 2 m)(\partial^{2} / \partial x^{2})\), we will notice it looks similar to a classical wave equation except that it’s linear in time. Rather, eq. (2.4.18) looks like a diffusion equation with an imaginary diffusion constant.

We are also interested in the equation of motion for \(\hat{U} \vphantom{A}^{\dagger}\) which describes the time evolution of the conjugate wavefunctions. If we follow the same approach, recognizing that \(\hat{U} \vphantom{A}^{\dagger}\) acts to the left:

\[\langle\psi(t)|=\left\langle\psi(t_{0})\right| \hat{U} \vphantom{A}^{\dagger}(t, t_{0})\]

we get

\[-\mathrm{i} \hbar \frac{\partial}{\partial t} \hat{U} \vphantom{A}^{\dagger}(t, t_{0})=\hat{U} \vphantom{A}^{\dagger}(t, t_{0}) \hat{H}\]

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