5.1: Time–Dependent Hamiltonians
- Page ID
- 28767
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(|n\rangle\).