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5.1: Time–Dependent Hamiltonians

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    Recall that for a system described by a Hamiltonian, \(\hat{H}_0\), which is time–independent, the most general state of the system can be described by a wavefunction \(|\Psi , t \rangle\) which can be expanded in the energy eigenbasis \(\{|n \rangle \}\) as follows:

    \[|\Psi , t\rangle = \sum_n c_n \text{ exp}(−iE_nt/\hbar )|n\rangle \nonumber\]

    where the coefficients, \(c_n\), are time-independent, and \(E_n\) denotes the eigenvalue corresponding to the energy eigenstate \(|n \rangle\) of \(\hat{H}_0\).

    When we generalise to the case where the Hamiltonian is of the form

    \[\hat{H} = \hat{H}_0 + \hat{V} (t) \nonumber\]

    we can again expand in \(|n \rangle\), the time-independent eigenbasis of \(\hat{H}_0\)

    \[|\Psi , t\rangle = \sum_n c_n(t) \text{ exp}(−iE_nt/\hbar )|n\rangle \nonumber\]

    but the coefficients, \(c_n\), will now in general be time-dependent.

    The wavefunction satisfies the time-dependent Schrödinger equation;

    \[i\hbar \frac{\partial}{\partial t} |\Psi , t \rangle = \hat{H} |\Psi , t\rangle \nonumber\]

    so that we can substitute the expansion of \(|\Psi , t\rangle\) to determine the equations satisfied by the coefficients \(c_n(t)\). Writing \(E_n = \hbar \omega_n\) and denoting the time derivative of \(c_n\) by \(\dot{c}_n\) we obtain

    \[i\hbar \sum_n (\dot{c}_n − i\omega_n c_n) \text{ exp}(−i\omega_n t)|n \rangle = \sum_n (c_n \hbar \omega_n + c_n\hat{V} ) \text{ exp}(−i\omega_nt)|n \rangle \nonumber\]

    which simplifies immediately to give

    \[\sum_n (i\hbar \dot{c}_n − c_n\hat{V} ) \text{ exp}(−i\omega_n t)|n \rangle = 0 \nonumber\]

    We now premultiply this equation with another eigenstate of \(\hat{H} 0, \langle m|\), to give

    \[i\hbar \dot{c}_m \text{ exp}(−i\omega_m t) − \sum_n c_n V_{mn} \text{ exp}(−i\omega_nt) = 0 \nonumber\]

    giving the following set of coupled, first–order differential equations for the coefficients:

    \[\boxed{i\hbar \dot{c}_m = {\sum}_n c_nV_{mn} \text{ exp}(i\omega_{mn}t)} \nonumber\]

    where \(\omega_{mn} = \omega_m − \omega_n\) and \(V_{mn} = \langle m|\hat{V} |n \rangle\).

    This tells us how the coefficient \(c_m\) varies with time, i.e. the probability that a measurement will show the system to be in the \(m^{th}\) eigenstate. It is exact, but not terribly useful because we must, in general, solve an infinite set of coupled differential equations.

    It is worth dwelling on the importance of the quantity \(V_{mn}\). This ‘matrix element’ is an integral which tells us how much the potential \(\hat{V}\) mixes states \(|m \rangle\) and \(|n\rangle\). If it is zero (which it often is, by symmetry) then \(\hat{V}\) cannot induce a transition between states \(|m \rangle\) and \(|n\rangle\).

    5.1: Time–Dependent Hamiltonians is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland 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.

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