Time evolution operator in QM - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-09-23T09:37:16Z https://physics.stackexchange.com/feeds/question/434783 https://creativecommons.org/licenses/by-sa/4.0/rdf https://physics.stackexchange.com/q/434783 4 Time evolution operator in QM N.Beck https://physics.stackexchange.com/users/209754 2018-10-15T19:41:26Z 2018-10-15T20:39:05Z <p>I am reading a introduction to quantum mechanics right now. There is a part about the time evolution operator:</p> <p><span class="math-container">\begin{align*} i\hbar \partial_t \,\psi(\vec r, t) = \hat H(t)\, \psi(\vec r,t) \end{align*}</span> is the time dependent Schrödinger-equation. If we assume that for each <span class="math-container">$\psi(\vec r, t_0)$</span> there is a unique solution <span class="math-container">$\psi(\vec r, t)$</span>, then we can define an operator </p> <p><span class="math-container">$$U(t,t_0): \mathcal H \to \mathcal H,\, \psi(\vec r, t_0) \mapsto \psi(\vec r, t)$$</span> </p> <p>This operator is linear, since the Schrödinger equation is linear and it is unitary, since <span class="math-container">$\partial_t \langle\psi(\vec r, t)| \psi(\vec r, t)\rangle = 0$</span>. I am totally happy with that. I can also accept, that <span class="math-container">$U(t,t_0) = e^{-i(t-t_0)\hat H/ \hbar}$</span>, if <span class="math-container">$\hat H$</span> is time independent, where <span class="math-container">$e^{-i(t-t_0)\hat H/ \hbar}$</span> is defined over how it acts on the eigenvectors of <span class="math-container">$\hat H$</span>. </p> <p>But I have no idea, what the next sentence in my book means, and there is no good explanation. Is says there:</p> <blockquote> <p>The differential equation, together with the initial condition (<span class="math-container">$U(t_0,t_0) = Id$</span>) is equivalent to the integral equation: <span class="math-container">\begin{align*} U(t,t_0) = 1 - \frac{i}{\hbar } \int_{t_0}^t ds\, \hat H(s) U(s,t_0) \end{align*}</span> </p> </blockquote> <p>So my problem is basically, I don't understand this at all :/. How can I integrate operators, what does that even mean? Are there any good examples, where this integral makes sense? This is probably a really stupid question, but I am happy if someone could spare two minutes to help me.</p> https://physics.stackexchange.com/questions/434783/-/434796#434796 6 Answer by DanielC for Time evolution operator in QM DanielC https://physics.stackexchange.com/users/40726 2018-10-15T20:39:05Z 2018-10-15T20:39:05Z <p>This is a very good question, but a mathematical one. The expression you quoted from the book is the part-"summation" of the Dyson expansion of the unitary evolution operator. To quote from Reed and Simon, theorem X.69 (Vol. II, p. 282) </p> <blockquote> <p>Let <span class="math-container">$t\mapsto H(t)$</span> a strongly continuous map of <span class="math-container">$\mathbb{R}$</span> into the <em>bounded</em> self-adjoint operators on a Hilbert space <span class="math-container">$\mathcal{H}$</span>. Then there is a unitary propagator on <span class="math-container">$\mathcal{H}$</span> so that, for all <span class="math-container">$\psi\in\mathcal{H}$</span>, <span class="math-container">$$\phi_s (t) = U(t,s) \psi$$</span> satisfies <span class="math-container">$$\frac{d}{dt} \phi_s (t) = -i H(t) \phi_s (t) \ ; \ \phi_s (s) = \psi$$</span></p> </blockquote> <p>The proof starts by explicitely exhibiting the unitary propagator as</p> <p><span class="math-container">$$U(t,s) \phi = 1 +\sum_{n=1}^{\infty} (-i)^n \int_{s}^{t} \int_{s}^{t_1} ... \int_{s}^{t_n} H(t_1)... H(t_n) \phi \ dt_n ... \ dt_1$$</span></p> <p>What the book did is just ~resum~ the infinite expression to the right of <span class="math-container">$H(t_1)$</span> into another U (the minus vs. plus sign after the unit vector comes from the different convention for evolution). Now we have no longer an integral of a product of operators, but of Hilbert space-valued functions. This is just an iteration of a Bochner-type integral. </p>