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2.1: Time-Evolution with a Time-Independent Hamiltonian

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    107216
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    The time evolution of the state of a quantum system is described by the time-dependent Schrödinger equation (TDSE):

    \[i \hbar \frac {\partial} {\partial t} \psi ( \overline {r} , t ) = \hat {H} ( \overline {r} , t ) \psi ( \overline {r} , t ) \label{1.1}\]

    \(\hat{H}\) is the Hamiltonian operator which describes all interactions between particles and fields, and determines the state of the system in time and space. \(\hat{H}\) is the sum of the kinetic and potential energy. For one particle under the influence of a potential

    \[\hat {H} = - \frac {\hbar^{2}} {2 m} \hat {\nabla}^{2} + \hat {V} ( \overline {r} , t ) \label{1.2}\]

    The state of the system is expressed through the wavefunction \(\psi ( \overline {r} , t )\). The wavefunction is complex and cannot be observed itself, but through it we obtain the probability density

    \[P = | \psi ( \overline {r} , t ) |^{2},\]

    which characterizes the spatial probability distribution for the particles described by \(\hat{H}\) at time \(t\). Also, it is used to calculate the expectation value of an operator \(\hat{A}\)

    \[ \begin{align} \langle \hat {A} (t) \rangle &= \int \psi^{*} ( \overline {r} , t ) \hat {A} \psi ( \overline {r} , t ) d \overline {r} \\[4pt] &= \langle \psi (t) | \hat {A} | \psi (t) \rangle \label{1.3} \end{align}\]

    Physical observables must be real, and therefore will correspond to the expectation values of Hermitian operators (\(\hat {A} = \hat {A}^{\dagger}\)).

    Our first exposure to time-dependence in quantum mechanics is often for the specific case in which the Hamiltonian \(\hat{H}\) is assumed to be independent of time: \(\hat {H} = \hat {H} ( \overline {r} )\). We then assume a solution with a form in which the spatial and temporal variables in the wavefunction are separable:

    \[\psi ( \overline {r} , t ) = \varphi ( \overline {r} ) T (t) \label{1.4}\]

    \[i \hbar \frac {1} {T (t)} \frac {\partial} {\partial t} T (t) = \frac {\hat {H} ( \overline {r} ) \varphi ( \overline {r} )} {\varphi ( \overline {r} )} \label{1.5}\]

    Here the left-hand side is a function only of time, and the right-hand side is a function of space only (\(\overline {r}\), or rather position and momentum). Equation \ref{1.5} can only be satisfied if both sides are equal to the same constant, \(E\). Taking the right-hand side we have

    \[\frac {\hat {H} ( \overline {r} ) \varphi ( \overline {r} )} {\varphi ( \overline {r} )} = E \quad \Rightarrow \quad \hat {H} ( \overline {r} ) \varphi ( \overline {r} ) = E \varphi ( \overline {r} ) \label{1.6}\]

    This is the Time-Independent Schrödinger Equation (TISE), an eigenvalue equation, for which \(\varphi ( \overline {r} )\) are the eigenstates and \(E\) are the eigenvalues. Here we note that

    \[\langle \hat {H} \rangle = \langle \psi | \hat {H} | \psi \rangle = E,\]

    so \(\hat{H}\) is the operator corresponding to \(E\) and drawing on classical mechanics we associate \(\hat{H}\) with the expectation value of the energy of the system. Now taking the left-hand side of Equation \ref{1.5} and integrating:

    \[\begin{align} i \hbar \frac {1} {T (t)} \frac {\partial T} {\partial t} &= E \\[4pt] \left( \frac {\partial} {\partial t} + \frac {i E} {\hbar} \right) T (t) &= 0 \label{1.7} \end{align}\]

    which has solutions like this:

    \[T (t) = \exp ( - i E t / \hbar ) \label{1.8}\]

    So, in the case of a bound potential we will have a discrete set of eigenfunctions \(\varphi _ {n} ( \overline {r} )\) with corresponding energy eigenvalues \(E_n\) from the TISE, and there are a set of corresponding solutions to the TDSE.

    \[\psi _ {n} ( \overline {r} , t ) = \varphi _ {n} ( \overline {r} ) \underbrace{\exp \left( - i E _ {n} t / \hbar \right)}_{\text{phase factor}} \label{1.9}\]

    Phase Factor

    For any complex number written in polar form (such as \(re^{iθ}\)), the phase factor is the complex exponential factor (\(e^{iθ}\)). The phase factor does not have any physical meaning, since the introduction of a phase factor does not change the expectation values of a Hermitian operator. That is

    \[ \langle \phi |A|\phi \rangle = \langle \phi |e^{-i\theta}Ae^{i\theta}|\phi \rangle \]

    Since the only time-dependence in \(\psi _ {n} \) is a phase factor, the probability density for an eigenstate is independent of time:

    \[P = \left| \psi _ {n} (t) \right|^{2} = \text {constant}.\]

    Therefore, the eigenstates \(\varphi ( \overline {r} )\) do not change with time and are called stationary states.

    However, more generally, a system may exist as a linear combination of eigenstates:

    \[\begin{align} \psi ( \overline {r} , t ) &= \sum _ {n} c _ {n} \psi _ {n} ( \overline {r} , t ) \\[4pt] &= \sum _ {n} c _ {n} e^{- i E _ {n} t h} \varphi _ {n} ( \overline {r} ) \label{1.10} \end{align}\]

    where \(c_n\) are complex amplitudes, with

    \[\sum _ {n} \left| c _ {n} \right|^{2} = 1. \nonumber\]

    For such a case, the probability density will oscillate with time. As an example, consider two eigenstates

    \[ \begin{align} \psi ( \overline {r} , t ) &= \psi _ {1} + \psi _ {2} \nonumber \\[4pt] &= c _ {1} \varphi _ {1} e^{- i E _ {1} t / h} + c _ {2} \varphi _ {2} e^{- i E _ {2} t / h} \label{1.11} \end{align}\]

    For this state the probability density oscillates in time as

    \[\begin{align} P (t) & = | \psi |^{2} \nonumber \\[4pt] &= \left| \psi _ {1} + \psi _ {2} \right|^{2} \nonumber \\[4pt] & = \left| c _ {1} \varphi _ {1} \right|^{2} + \left| c _ {2} \varphi _ {2} \right|^{2} + c _ {1}^{*} c _ {2} \varphi _ {1}^{*} \varphi _ {2} e^{- i \left( \alpha _ {2} - \omega _ {1} \right) t} + c _ {2}^{*} c _ {1} \varphi _ {2}^{*} \varphi _ {1} e^{+ i \left( a _ {2} - \omega _ {1} \right) t} \nonumber \\[4pt] & = \left| \psi _ {1} \right|^{2} + \left| \psi _ {2} \right|^{2} + 2 \left| \psi _ {1} \psi _ {2} \right| \cos \left( \omega _ {2} - \omega _ {1} \right) t \label{1.12} \end{align}\]

    where \(\omega _ {n} = E _ {n} / \hbar\). We refer to this 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 Equation \ref{1.12} may also include a time-independent phase factor \(\phi\) that arises from the complex expansion coefficients.

    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

    Figure 1.png

    If we create a superposition of these states with Equation \ref{1.11}, the time-dependent probability density oscillates, with \(\langle x (t) \rangle\) bearing similarity to the classical motion. (Here \(c_0 = 0.5\) and \(c_1 = 0.87\).)

    Figure 2.png

    Readings

    1. Cohen-Tannoudji, C.; Diu, B.; Lalöe, F., Quantum Mechanics. Wiley-Interscience: Paris, 1977; p. 405.
    2. Nitzan, A., Chemical Dynamics in Condensed Phases. Oxford University Press: New York, 2006; Ch. 1.
    3. Schatz, G. C.; Ratner, M. A., Quantum Mechanics in Chemistry. Dover Publications: Mineola, NY, 2002; Ch. 2.

    2.1: Time-Evolution with a Time-Independent Hamiltonian is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Andrei Tokmakoff via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.