2.1: Dynamics with a Time-Independent Hamiltonian

​ This book concerns itself primarily with describing time-dependent properties of a molecular quantum system, with an emphasis on dynamics. Dynamics refers to a unified description of the particles and fields of a system as a function of time and position, which we can express in terms of the time-dependent state of the system \(|\Psi(\mathbf{r}, t)\rangle\). The time evolution of \(|\Psi(\mathbf{r}, t)\rangle\) is described by the time-dependent Schrödinger equation (TDSE): 

\[\mathrm{i} \hbar \dfrac{\partial}{\partial t} \psi(\mathbf{r}, t)=\hat{H}(\mathbf{r}, t) \psi(\mathbf{r}, t)\]

In this differential equation, the Hamiltonian \(\hat{H}\), which describes all interactions between particles and fields, is the quantity that determines how the state of the system evolves in time and space. For many interacting nuclei and electrons and possibly time-varying external potentials, this is a daunting equation to solve, and we will approach this with different strategies.

We start by summarizing the case in which the Hamiltonian \(\hat{H}\) is assumed to be independent of time: \(\hat{H}=\hat{H}(\mathbf{r})\). We then assume a solution that is a simple product of two functions, one of which is a function of time, and the other of space:

\[\Psi(\mathbf{r}, t)=\varphi(\mathbf{r}) T(t)\]

Inserting this into eq. (2.1.1) allows us to rewrite the TDSE as

\[\mathrm{i} \hbar \dfrac{1}{T(t)} \dfrac{\partial}{\partial t} T(t)=\dfrac{\hat{H}(\mathbf{r}) \varphi(\mathbf{r})}{\varphi(\mathbf{r})}\]

Here we have separated variables leaving the left-hand side as a function only of time, and the right-hand side as a function only of space. Equation (2.1.3) can only be satisfied if both sides are equal to the same constant, \(E\). Taking the right-hand side, we obtain,

\[\hat{H}(\mathbf{r}) \varphi(\mathbf{r})=E \varphi(\mathbf{r})\]

This is the time-independent Schrödinger equation (TISE), an eigenvalue equation, for which \(\varphi(\mathbf{r})\) are the eigenstates and \(E\) are the eigenvalues. Here we note that \(\langle\hat{H}\rangle= \left\langle\psi \right| \! \hat{H} \! \left| \psi \right\rangle=E\), so \(\hat{H}\) is the operator corresponding to \(E\) and drawing on classical mechanics we associate \(\langle\hat{H}\rangle\) with the expectation value of the energy of the system.

Now taking the left-hand side of eq. (2.1.3) we find,

\[\mathrm{i} \hbar \dfrac{1}{T(t)} \dfrac{\partial T}{\partial t}=E\]

Integrating over time \(t\) we have,

\[T(t)=\exp (-\mathrm{i} E t / \hbar)\]

Considering the set of eigenfunctions \(\varphi_{n}(\mathbf{r})\) with corresponding energy eigenvalues \(E_{n}\) obtained from solving the TISE, we obtain a set of corresponding solutions to the TDSE.

\[\psi_{n}(\mathbf{r}, t)=\varphi_{n}(\mathbf{r}) \exp \left(-\mathrm{i} E_{n} t / \hbar\right)\]

The time-dependence of the system is complex and oscillatory with a frequency dictated by the energy of the system. Since the probability density for the eigenstate \(\left|\psi_{n}(\mathbf{r}, t)\right|^{2} d \mathbf{r}\) is independent of time, the eigenstates are called stationary states. More generally, a system may exist as a linear superposition of eigenstates:

\[\Psi(\mathbf{r}, t)=\sum_{n} c_{n} \psi_{n}(\mathbf{r}, t)=\sum_{n} c_{n} e^{-\mathrm{i} E_{n} t / \hbar} \varphi_{n}(\mathbf{r})\]

where the coefficients \(c_{n}\) are complex numbers satisfying the normalization condition \( \sum_{n}\left|c_{n}\right|^{2}=1\). In this case, the probability density will oscillate with time. As an example, consider a linear combination of two eigenstates:

\[\Psi(\mathbf{r}, t)=\psi_{1}+\psi_{2}=c_{1} \varphi_{1}(\mathbf{r}) e^{-\mathrm{i} E_{1} t / \hbar}+c_{2} \varphi_{2}(\mathbf{r}) e^{-\mathrm{i} E_{2} t / \hbar}\]

For this state the probability density oscillates in time at a frequency dictated by the energy difference between states:

\[\begin{align}\begin{aligned}
P(t) & =|\Psi|^{2}=\left|\psi_{_{1}}+\psi_{_{2}}\right|^{2} \\
 &=\left|c_{_{1}} \varphi_{_{1}}\right|^{2}+\left|c_{_{2}} \varphi_{_{2}}\right|^{2}+c_{_{1}}^{*} c_{_{2}} \varphi_{_{1}}^{*} \varphi_{_{2}} e^{-i\left(E_{2}-E_{1}\right) t / \hbar}+c_{_{2}}^{*} c_{_{1}} \varphi_{_{2}}^{*} \varphi_{_{1}} e^{+i\left(E_{2}-E_{1}\right) t / \hbar} \\
 &=\left|\psi_{_{1}}\right|^{2}+\left|\psi_{_{2}}\right|^{2}+2\left|\psi_{_{1}} \psi_{_{2}}\right| \cos \left(\omega_{21} t\right)
\end{aligned}\end{align}\]

where the last line uses the following definition:

\[\omega_{mn}=\frac{E_{m}-E_{n}}{\hbar}\]

We refer to the state of the system that gives rise to this time-dependent oscillation in probability density as a coherent superposition state, or coherence. More generally, the oscillation term in eq. (2.1.10) may also include a time-independent phase factor \(\phi\) that arises from the complex nature of the expansion coefficients \(c_{i}\).

As an example, consider the superposition of the ground and first excited states of the quantum harmonic oscillator. The basis wavefunctions, \(\psi_{0}(x)\) and \(\psi_{1}(x)\), and their stationary probability densities \(P_{i}=\left\langle\psi_{i}(x) | \psi_{i}(x)\right\rangle\) are shown in Figures 2.1.2 a and b. If we create a superposition of these states with eq. (2.1.9), here with \(c_{0}=c_{1}=\sqrt{2}\), the time-dependent probability density oscillates, with \(\langle x (t)\rangle\) tracking the classical motion, as shown in Figure 2.1.2c and the accompanying video. 

 

Figure \(\PageIndex{2}\): (a) Real part of wavefunction for the ground (red) and first excited state (blue) of the harmonic oscillator. (b) Probability densities corresponding to (a). (c) Time-dependent superposition of the ground and first excited states of the harmonic oscillator for \(c_{0}=c_{1}=0.707\). Curves are offset and separated in time in increments of \(2 \pi / 7 \omega\). Here \(\alpha^{2}=2 \hbar / m \omega\) for the harmonic oscillator. 

Figure \(\PageIndex{1}\)

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